Florian Dussap
email: florian.dussap@parisdescartes.fr

Anisotropic multivariate deconvolution using projection on the Laguerre basis

Keywords: anisotropic multivariate projection estimator, laguerre basis, model selection, nonparametric density estimation 2010 MSC: 62G07, 62H12

We investigate adaptive density estimation in the additive model Z = X + Y , where X and Y are independent d-dimensional random vectors with non-negative coordinates. Our goal is to recover the density of X from independent observations of Z, assuming the density of Y is known. In the d = 1 case, an estimation procedure using projection on the Laguerre basis have already been studied. We generalize this procedure in the multivariate case: we establish non-asymptotic upper bounds on the mean integrated squared error of the estimator and we derive convergence rates on anisotropic functional spaces. Moreover, we provide data-driven strategies for selecting the right projection space (for d = 1, we improve the previous projection procedure). We illustrate these procedures on simulated data, and in dimension d = 1 we compare our procedure with the previous adaptive projection procedure.

Statistical model and motivations

In this article, we study the problem of recovering the distribution of a random vector X when we only observe its sum with a noise vector Y with known distribution. This is a classical problem in nonparametric statistics (see references below), but we focus on the particular case where both X and Y have non-negative coordinates.

This assumption is quite unusual in deconvolution problems but is relevant for instance in reliability fields: we observe the failure times of several components in a system, each failure time being the sum of the lifetimes of two sub-components. In survival analysis, for d = 1, X would be the time of infection of a disease and Y the incubation time. The multivariate case can then be used to study the time of infection for multiple diseases.

More precisely, we consider the following statistical model:

Z i = X i + Y i , i = 1, . . . , n, (1)
where the X i 's, the Y i 's and Z i 's are random vectors in R d with non-negative coordinates. We assume that the X i 's are i.i.d. with unknown density f on R d + , and that the Y i 's are i.i.d. with known density g on R d

+ . Moreover, we assume that the X i 's and the Y i 's are independent. Our goal is to provide an adaptive procedure to estimate the density f from the observations Z 1 , . . . , Z n .

In the univariate case, there are a lot of papers about recovering the density of a random variable when it is observed with an additive known noise. Many authors use a kernel estimator introduced by [START_REF] Stefanski | Deconvolving kernel density estimators[END_REF]. [START_REF] Fan | On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems[END_REF] first introduced the notion of ordinary smooth and supersmooth noise (rate of decay of the characteristic function) to study optimal rates of convergence on Hölder spaces. When f is supersmooth and the noise is ordinary smooth, [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF] showed that the kernel estimator achieves a good rate of convergence and that this rate is optimal. When both f and the noise are supersmooth, the problem is more complicated and have been studied by Butucea and Tsybakov (2008a,b) from lower bound point of view.

To provide an adaptive estimator, different procedures were proposed. When f belongs to a Sobolev space, [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF] proposed a wavelet strategy that is adaptive and achieves optimal rates of convergence. For kernel estimators, [START_REF] Delaigle | Bootstrap bandwidth selection in kernel density estimation from a contaminated sample[END_REF] estimated the optimal bandwidth with a bootstrap procedure and showed its consistency. Hazelton andTurlach (2009, 2010) proposed a weighted kernel estimator with a data-driven way to choose the weights. Moreover, their weighted kernel estimator can be used in a multivariate setting. For projection with penalization strategies, [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] used a Shannon type basis to construct an adaptive estimator that is minimax in most cases. More recently, non-compact supported bases were used by [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF] (Laguerre basis), by [START_REF] Comte | Laguerre and Hermite bases for inverse problems[END_REF] (Laguerre basis and Hermite basis), and by Sacko (2019) (Hermite basis) to construct adaptive estimators on suitable functional spaces.

The Laguerre basis was also used in a regression setting to study the problem of Laplace deconvolution, see [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast-enhanced imaging[END_REF] and [START_REF] Vareschi | Noisy Laplace deconvolution with error in the operator[END_REF], or more recently [START_REF] Benhaddou | Anisotropic functional Laplace deconvolution[END_REF].

The multivariate deconvolution literature is more sparse. [START_REF] Masry | Multivariate probability density deconvolution for stationary random processes[END_REF] generalizes the kernel estimator for stationary random processes, with a dependence structure between the variables. The noise is assumed to have i.i.d. coordinates (isotropic noise) and no adaptive strategy is proposed, the author focuses on the problem of dependency between the variables. [START_REF] Youndjé | Optimal bandwidth selection for multivariate kernel deconvolution density estimation[END_REF] propose a cross-validation strategy to estimate the optimal bandwidth of the kernel estimator in the multivariate setting, and show it is asymptotically optimal under the assumption that the noise is isotropic and ordinary smooth. [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF] use a bandwidth selection procedure inspired by [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF]. Their procedure allows anisotropic noises, with both ordinary smooth and supersmooth components, and they derive rates of convergence for the pointwise risk and the L 2 risk, when f belongs to anisotropic Hölder, Nikol'skii or Sobolev classes. For ordinary smooth noise and when f belongs to anisotropic Nikol'skii classes, [START_REF] Rebelles | Structural adaptive deconvolution under Lp-losses[END_REF] provides an adaptive kernel estimator which is minimax for the L p -loss. Recently, [START_REF] Lepski | Oracle inequalities and adaptive estimation in the convolution structure density model[END_REF] studied a more general model (with direct and indirect observations of X) and provided an adaptive kernel estimator on anisotropic Nikol'skii classes, under the L p -loss.

Concerning our specific case of deconvolution with non-negative noise, the case d = 1 has already been studied by [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF] using a projection strategy on the Laguerre basis. We also use a projection strategy in the multivariate case. The main tool we use to construct our estimator is the theory of hypermatrices. Using the contraction product of hypermatrices, we show that it is possible to recover the coefficients of f from the observations. We recall the definitions of these objects in Section 2.

We provide rates of convergence for the MISE of our estimator on anisotropic functional spaces: Sobolev-Laguerre spaces and smooth Laguerre spaces. We propose a model selection procedure to produce an adaptive estimator, under mild assumptions on the noise density g. This procedure is inspired by the work of [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF] concerning bandwidth selection. It was introduced for model selection by [START_REF] Chagny | Warped bases for conditional density estimation[END_REF] for estimation of conditional density, in a two-dimensional setting. We show this procedure can be applied to our deconvolution problem, in a d-dimensional setting, and we establish an asymptotic oracle inequality for this procedure. Moreover, the proof is written to provide general steps that can be applied to other contexts.

Outline of the paper. In Section 2, we provide the notations and the key facts about hypermatrices we will use in the rest of the paper. In Section 3, we construct the estimator. In section 4, we provide non asymptotic MISE bounds and we derive convergence rates on Sobolev-Laguerre balls and smooth Laguerre balls. In Section 5, we give model selection procedures to construct an adaptive estimator and we establish oracle inequalities. In Section 6, we illustrate the procedures on simulated data. All the proofs are gathered in Section 7.

Preliminary on hypermatrices

Let α, β ∈ N d be multi-indices and x ∈ C d . We will use the following notations:

α + β := (α 1 + β 1 , . . . , α d + β d ), α × β := (α 1 β 1 , . . . , α d β d ), α β := α 1 β 1 , . . . , α d β d , x α := x α1 1 • • • x α d d , |α| := α 1 + • • • + α d , β α ⇐⇒ ∀q ∈ {1, . . . , d}, β q α q , m ∧ m := (m 1 ∧ m 1 , . . . , m d ∧ m d ) where m i ∧ m i := min(m i , m i ).
We denote 0 := (0, . . . , 0) and

1 := (1, . . . , 1). If k ∈ N d and ∈ N p are two multi-indices, we denote (k, ) ∈ N d+p their concatenation. Proposition 2.1 (Multibinomial theorem). If α ∈ N d is a multi-index and if x, y ∈ R d are vectors, then (x + y) α = β α α β x β y α-β where α β := α1 β1 × • • • × α d β d .
We recall some facts about hypermatrices; see [START_REF] Hogben | Handbook of Linear Algebra[END_REF] for more details. A hypermatrix is a multi-dimensional array of numbers. The order of a hypermatrix is its number of dimensions : an order-1 hypermatrix is a vector, an order-2 hypermatrix is matrix, and so on. m1,...,m d ) the set of order-d hypermatrices with m j components according to the j-th dimension. Moreover, if m ∈ (N * ) d and r ∈ (N * ) p are two multi-indices, since (m, r) denotes the concatenation of m and r, we denote R (m,r) := R (m1,...,m d ,r1,...,rp) . We choose to index the hypermatrix components starting from 0. So if T ∈ R m is a hypermatrix, its components are

If m = (m 1 , . . . , m d ) ∈ (N * ) d is multi-index, we denote by R m = R (
T k for k ∈ N d such that k m -1. Definition 2.2. Let m ∈ (N * ) d ,
, U R m := k m-1 T k U k and T 2 R m := k m-1 T 2 k .
For vectors (d = 1), we get the usual 2 norm and for matrices (d = 2), we get the Frobenius norm.

Definition 2.3 (Contraction product). Let m, p and r be multi-indices (with possibly different lengths) and denote d the length of p. Let T ∈ R (m,p) and U ∈ R (p,r) be two hypermatrices, their d-contraction product is the hypermatrice T × d U ∈ R (m,r) defined by:

[T × d U ] (k, ) := j p-1 T kj U j .
(2)

The contraction product generalizes the matrix product: if T , respectively U , are matrices m×p, respectiely p × r, their 1-contraction product is their matrix product. m) is a hypermatrix, then T induces a linear map (still denoted by T ):

Definition 2.4 (Frobenius norm and spectral norm

). Let m ∈ (N * ) d be a multi-index. If T ∈ R (m,
T : R m -→ R m x -→ T × d x .
We denote T * ∈ R (m,m) the hypermatrix defined by [T * ] k = T k . The linear map induced by T * is the adjoint linear map of the one induced by T :

∀x, y ∈ R m , y, T x R m = T * y, x R m .
We define the Frobenius norm and the spectral norm of T in the same way we define them for any endomorphism: T 2 F := Tr(T * T ) and ρ 2 (T ) := λ max (T * T ). It is easy to check that:

T 2 F = k, m-1 T 2 k , ρ(T ) = sup x∈R m \{0} T x R m x R m .
Lastly, we recall the classical inequalities between the Frobenius and the spectral norm:

1 m 1 • • • m d T 2 F ρ 2 (T ) T 2 F . (3)
Finally, we introduce the following definition that generalizes the concept of triangular matrices.

Definition 2.5. A hypermatrix T ∈ R (m,m) is said to be lower triangular if apart from multiindices , k m -1 such that k , we have T k = 0. An infinite hypermatrix T ∈ R N d ×N d is said to be lower triangular if apart from multi-indices , k ∈ N d such that k , we have T k = 0.

The estimation procedure

In the model (1), the Z i 's are i.i.d. random vectors on R d + , and they admit a density we denote by h. This density function is given by the convolution product of f and g:

(f * g)(x) := R d f (u) g(x -u) du = • • • [0,x1]ו••×[0,x d ] f (u) g(x -u) du.
We assume that f , g and h belong to L 2 (R d + ) and we expand them in the multivariate Laguerre basis. We recall that the one-dimensional Laguerre functions (ϕ k ) k∈N are defined by:

∀x ∈ R + , ϕ k (x) := √ 2 L k (2x) e -x , where L k (x) := k j=0 k j (-x) j j! ,
and that they form an orthonormal basis of L 2 (R + ). In the multivariate case, we tensorize the Laguerre basis. For k = (k 1 , . . . , k d ) ∈ N d a multi-index, we define the multivariate Laguerre function ϕ k on R d + as the tensor product of one-dimensional Laguerre functions:

ϕ k (x 1 , . . . , x d ) := ϕ k1 ⊗ • • • ⊗ ϕ k d (x 1 , . . . , x d ) = ϕ k1 (x 1 ) × • • • × ϕ k d (x d ). ( 4
)
The multivariate Laguerre functions (ϕ k ) k∈N d form a basis of L 2 (R d + ), and we expand the functions f , g and h in this basis:

f = k∈N d a k ϕ k , g = j∈N d b j ϕ j , h = ∈N d c ϕ . ( 5
)
The use of the Laguerre basis is relevant in this context, as the one-dimensional Laguerre functions verify the relation:

∀k, j ∈ N, ϕ k * ϕ j = 2 -1/2 (ϕ k+j -ϕ k+j+1 ), (6)
see (Abramowitz and Stegun, 1972, formula 22.13.14). Using this relation and h = f * g, by expanding the functions f , g and h on the Laguerre basis, we get a relation between their coefficients.

Proposition 3.1. If a, b and c are the coefficients defined in (5), then the following relation holds:

∀ ∈ N d , c = 2 -d/2 ε∈{0,1} d (-1) |ε| (a * b) -ε ,
where a * b is the discrete convolution product of a and b defined by

(a * b) k := j k a j b k-j if k ∈ N d and (a * b) k = 0 if k / ∈ N d .
This relation can be written as a discrete convolution product c = β * a with β ∈ N d defined by:

β k := 2 -d/2 ε∈{0,1} d (-1) |ε| b k-ε , (7)
where by convention b j = 0 if j / ∈ N d . Thus, we have a linear relation between the coefficients c and a:

∀ ∈ N d , c = k G k a k , where G k := β -k if k , 0 else. ( 8
)
If we consider G as an infinite hypermatrix [G k ] ,k∈N d ∈ R N d ×N d , then it is lower triangular according to Definition 2.5.

In the next proposition, we show that the linear relation between a and c is invertible.

a = k H k c k . ( 9
)
We denote

H k =: (G -1 ) k .
We can write the linear relations between a and c using hypermatrices and contraction products. For m ∈ (N * ) d , we denote by a m (resp. ,m) . Then, we have:

c m ) the hypermatrix [a k ] k m-1 ∈ R m (resp. [c ] m-1 ∈ R m ),

and we denote by

G m and G -1 m the hypermatrices [G k ] ,k m-1 and [(G -1 ) k ] ,k m-1 in R (m
c m = G m × d a m , a m = G -1 m × d c m , ( 10
)
where "× d " stands for the contraction product defined by (2).

Estimation procedure. For m ∈ (N * ) d , let S m be the vector space spanned by the functions ϕ k for k m -1, and let D m := m 1 • • • m d be its dimension. We estimate f by estimating f m the projection of f on S m . This projection is given by f m = k m-1 a k ϕ k , so the problem reduces to the estimation of a m . Because a m is related to c m by ( 10) and since c = E[ϕ (Z 1 )], we estimate f by: fm := ϕ (Z i ).

Non-asymptotic error bounds

We quantify the quality of the estimator fm by its MISE (Mean Integrated Squared Error):

E f -fm 2 L 2 .
In the next proposition, we decompose the MISE in a bias term and a variance term and give a bound on the MISE of fm .

Proposition 4.1. If f and g are L 2 (R d + ) functions, then we have the inequality:

∀m ∈ N d , E f -fm 2 L 2 f -f m 2 L 2 + 2 d D m ρ 2 (G -1 m ) n ∧ h ∞ G -1 m 2 F n . Remark 4.2. Using the Cauchy-Schwarz inequality, it holds h ∞ f L 2 g L 2 which is finite by assumption. Moreover, if g is bounded, we also have h ∞ g ∞ . Remark 4.3. The norm equivalence (3) implies that G -1 m 2 F D m ρ 2 (G -1 m )
, so the order of magnitude of the variance term is given by G -1 m 2

F . The minimum is important only for the small values of m (because of the constants).

Under an additional assumption, [START_REF] Comte | Laguerre and Hermite bases for inverse problems[END_REF] improved the variance bound in the one-dimensional case. We generalize their result to the multivariate case.

Proposition 4.4. We denote Y (j) 1 the j-th coordinate of Y 1 . We assume that for any nonempty subset J of {1, . . . , d}, we have:

M J (g) := E   j∈J 1 Y (j) 1   < +∞. ( 11
)
Then we have the following inequality:

∀m ∈ (N * ) d , E f -fm 2 L 2 f -f m 2 L 2 + c(g) √ D m ρ 2 (G -1 m ) n ∧ h ∞ G -1 m 2 F n ,
where c(g) is a positive constant depending on M J (g) for J ⊆ {1, . . . , d}.

To study the bias term, we assume that f belongs to a Sobolev-Laguerre space. In dimension d = 1, these functional spaces have been introduced by [START_REF] Bongioanni | What is a Sobolev space for the Laguerre function systems?[END_REF] to study the Laguerre operator. The connection with Laguerre coefficients was established later by [START_REF] Comte | Adaptive Laguerre density estimation for mixed Poisson models[END_REF]. Following the same idea, we define Sobolev-Laguerre balls on R d + . Definition 4.5 (Sobolev-Laguerre ball). Let L > 0 and s ∈ (0, +∞) d , we define the Sobolev-Laguerre ball of order s and radius L by:

W s (R d + , L) :=    f ∈ L 2 (R d + ) k∈N d a 2 k (f ) k s L    , with a k (f ) := f, ϕ k L 2 the Laguerre coefficients of f .
Assuming f to belong to W s (R d + , L), the bias term decreases to 0 with polynomial rate. Indeed, for m ∈ (N * ) d , we have:

f -f m 2 L 2 = k∈N d ∃q, kq mq a 2 k (f ) d q=1 k∈N d kq mq a 2 k (f ) k sq q k -sq q L d q=1 m -sq q . ( 12
)
The case where the Laguerre coefficients of f decrease with exponential rate is also interesting. We define new functional spaces, "smooth Laguerre spaces", in the following way.

Definition 4.6 (Smooth Laguerre ball). Let L > 0 and r ∈ (0, +∞) d , we define the smooth Laguerre ball of order r and radius L as:

S r (R d + , L) :=    f ∈ L 2 (R d + ) k∈N d a 2 k (f ) e r•k L    .
By the same argument as previously, if f belongs to S r (R d + , L), the bias term decreases to 0 with exponential rate:

∀m ∈ (N * ) d , f -f m 2 L 2 L d q=1 e -rqmq . ( 13
)
Now, we need to control the variance term in Proposition 4.1. In the one-dimensional case, this control can be provided under assumptions on the Laplace transform G of g and under assumptions on the derivatives of g, see lemma 3.6 in [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast-enhanced imaging[END_REF]. In the next proposition, we extend this result to the multivariate case. Moreover, we make assumptions only on the behavior of the Laplace transform of g, we do not need to study its differentials.

We recall that the Laplace transform of g is the function G defined on the domain P d + by:

G(s) := R d + e -s•x g(x) dx,
where P + stands for the set of complex numbers with non-negative real part. In addition, we extend the set C of complex numbers by adding a point at infinity denoted by ∞. The control of the Frobenius norm of G -1 m relies on the behavior of G when some of its arguments take the ∞ value.

Proposition 4.7. We assume that β ∈ 1 (N d ), with β defined in (7). We assume that G is non-zero on P d + and that there exists α ∈ (N * ) d such that the function:

K α (s) := (1 + s) α G(s), s ∈ P d + , can be extended as a non-zero function on (P + ∪ {∞}) d such that the restriction of K α on (iR ∪ {∞}) d is continuous. Then for m ∈ N d satisfying m 4, there exists a constant C > 0 depending on β such that G -1 m 2 F Cm 2α . Remark 4.8. If g ∈ W s (R d + , L) with L > 0 and s ∈ (1, +∞) d
, the Laguerre coefficients of g belong to 1 (N d ). Indeed, using Cauchy-Schwarz inequality,

k∈N d |b k | = k∈N d |b k | k s 2 k -s 2   k∈N d b 2 k k s   1 2   k∈N d k -s   1 2 < +∞,
because for every q ∈ {1, . . . , d}, s q > 1.

Remark 4.9. We notice that in dimension d = 1, these assumptions simply become:

1. The Laplace transform G does not vanish on P + . 2. The Fourier transform of g admits an asymptotic expansion g * (ω) = ω -α (K α + o( 1)) when |ω| goes to +∞, for some α ∈ N * and some non-zero constant K α .

It is easy to see that this second assumption is a consequence of the assumptions on the derivatives of g made in (Comte et al., 2017, subsection 2.5).

Remark 4.10. If the distribution of Y is a product of gamma distributions d q=1 Γ(α q , λ q ) with α ∈ (N * ) d , then G and K α are given by:

G(s) = d q=1 1 + s q λ q -αq and K α (s) = λ α d q=1 1 + s q λ q + s q αq
, so the assumptions of Proposition 4.7 are fulfilled.

Gathering ( 12) or ( 13) with Proposition 4.7, we obtain convergence rates for our estimator.

Theorem 4.11. Let s, r ∈ (0, +∞) d and L > 0. Assume that g satisfies the assumptions of Proposition 4.7 with α ∈ (N * ) d .

1. For m opt ∈ (N * ) d given by m opt,j ∝ n 1/ sj +sj d i=1 2α i s i
, there exists a constant C > 0 depending on s, L and g such that:

sup f ∈W s (R d + ,L) E f -fmopt 2 L 2 C n -1/ 1+ d i=1 2α i s i
.

2. For m opt ∈ (N * ) d given by m opt,j ∝ log n rj , there exists a constant C > 0 depending on r, L and g such that:

sup f ∈S r (R d + ,L) E f -fmopt 2 L 2 C (log n) d i=1 2αi
n .

Remark 4.12. Our convergence rates on Sobolev-Laguerre balls are similar to convergence rates found by [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF] on anisotropic Sobolev balls, in the context of deconvolution using a kernel estimator with an ordinary smooth noise.

Remark 4.13. We could have considered mixed regularities for f . For instance, if the coefficients of f satisfy:

k∈N d a 2 k k s1 1 • • • k sj 0 j0 exp(r j0+1 k j0+1 + • • • + r d k d ) L,
for some 1 j 0 < d, then we can give the following upper-bound on the bias term:

f -f m 2 L 2 L   j0 j=1 m -sj j + d j=j0+1 e -rj mj   .
By choosing m j ∝ log n rj for j > j 0 , the MISE of fm is then:

E f -fm 2 L 2 L j0 j=1 m -sj j + h ∞ N n j0 j=1 m 2αj j + L (d -j 0 ) n , with N n := n j>j 0
(log(n)/rj ) 2α j . We can now choose the j 0 -first components of m like in the case 1 of Theorem 4.11, but with a sample size N n , and we get:

E f -fmopt 2 L 2 C n j>j0 (log(n)/r j ) 2αj -1/ 1+ j 0 j=1 2α j s j
.

It is the rate we would find on a Sobolev-Laguerre ball in dimension j 0 , up to a logarithmic factor.

Adaptive estimation and oracle inequalities

In practice, we do not know the underlying regularity of f , so we can not compute the model m opt of Theorem 4.11. We want a data-driven procedure that automatically provides the biasvariance compromise, without assumptions on f . More precisely, we want to select m from the data such that the MISE of f m is close to the oracle MISE:

inf m E f -fm 2 L 2 . Let m * ∈ (N * ) d fixed such that D m * n.
We only look at sub-models of m * that are not "too big". More precisely, we consider the models that belong to the following model collection:

M n := m m * D m ρ 2 (G -1 m ) n log n . ( 14
)
We assume that the Y i 's distribution satisfies the following assumptions.

(A1)

The noise density g is bounded.

(A2) For every nonempty J ⊆ {1, . . . , d}, M J (g) defined by ( 11) is finite.

V (m) := c(g) √ D m ρ 2 (G -1 m ) n ∧ ( g ∞ ∨ 1) G -1 m 2 F log n n . ( 15
)
Moreover under assumption (A3), we can control the right deviation of fm -f m 2 L 2 when m belongs to M n . Lemma 5.1. Under assumptions (A1) to (A3), there exists a numerical constant a 0 (d) > 0 depending on the dimension d such that for every a > a 0 (d):

m∈Mn E fm -f m 2 L 2 -aV (m) + C n ,
with C a positive constant depending on g and a.

The one-dimensional case. When d = 1, we consider two procedures, indexed by i ∈ {1, 2}, the first procedure being the one studied by [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF]. In both procedures, the projection space is chosen by minimizing the penalized criterion:

mi := arg min m∈Mn -fm 2 L 2 + κ i pen i (m) , i ∈ {1, 2} (16)
where M n is defined by ( 14), κ i is a numerical constant to be adjusted and pen i is the penalty term. [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF] proposes the following penalty term:

pen 1 (m) := 2mρ 2 (G -1 m ) n ∧ ( g ∞ ∨ 1) G -1 m 2 F log n n , ( 17
)
and provides an oracle inequality for the estimator f m1 under assumptions (A1) and (A3), see Theorem 4.1 in Mabon's article. This choice is based on the bound of the variance term of Proposition 4.1. However if g is bounded, assumption (A2) holds automatically in dimension d = 1, so we can apply Proposition 4.4 to get a better bound on the variance term. Thus, we propose the penalty term:

pen 2 (m) := V (m) = c(g) √ m ρ 2 (G -1 m ) n ∧ ( g ∞ ∨ 1) G -1 m 2 F log n n . ( 18
)
We show that our estimator f m2 also satisfies an oracle inequality.

Theorem 5.2. We assume d = 1. Under assumtions (A1) to (A3), there exists a numerical constant κ 0 > 0 such that for every choice of κ > κ 0 , we have the following oracle inequality:

E f -f m2 2 L 2 4 inf m∈Mn f -f m 2 L 2 + κ pen 2 (m) + C n ,
with C a positive constant depending on κ and g.

The multivariate case. We use the procedure similar to [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF] for model selection introduced by Chagny (2013b) for the estimation of a conditional density function. We apply this procedure to our multivariate deconvolution problem, and we establish an oracle inequality. We choose m in the model collection M n defined in ( 14), minimizing:

m := arg min m∈Mn A(m) + κ 2 V (m) , ( 19
)
where V (m) is defined by (15) and A(m) is a term which has the order of the bias term (see the proof):

A(m) := max m ∈Mn fm -fm∧m 2 L 2 -κ 1 V (m ) + ,
and where κ 1 , κ 2 are two numerical constants to be adjusted.

Theorem 5.3 (Oracle inequality). Under assumptions (A1) to (A3), there exists a numerical constant κ 0 (d) > 0 depending on the dimension d such that for every choice of κ 1 , κ 2 satisfying κ 0 (d) < κ 1 κ 2 , we have the following oracle inequality:

E f -f m 2 L 2 C inf m∈Mn f -f m 2 L 2 + V (m) + C n ,
with C a positive constant depending on κ 1 and κ 2 , and C a positive constants depending on g, d and κ 1 .

Remark 5.4. It is important to notice that the proof of Theorem 5.3 only depends on the deconvolution structure through Lemma 5.1, to obtain a concentration bound on the variance term. The rest of the proof does not use the deconvolution structure again. Thus, the procedure can be applied to more general model selection problems.

Numerical illustrations

Estimators comparison in one-dimensional case

We want to compare the two estimators f m1 and f m2 defined in Section 5, with different distributions for X and Y , and with different sample size n. We compare their MISE:

R i = R i (f, g, n) := E f -f mi 2 L 2 , i ∈ {1, 2}.
We also compute the oracle risk:

R o = R o (f, g, n) := min 1 m m * E f -fm 2 L 2 ,
where m * is the maximal element of M n . We compute an approximation of • 2 L 2 with Simpson's rule on a bounded interval I of R + with 1000 points. We compute the expectations with an empirical mean over 500 samples (we use the same samples for R 1 , R 2 and R o ). Remark 6.1. Since f is non-negative, fm is replaced by ( fm ) + := max( fm , 0) in the following.

Distributions for X. We consider several distributions for X. In each case, we normalize the distribution for the variance to be 1. We use the same examples as [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF].

• Exponential E(1), I = [0, 5].

• Gamma Γ(20, √ 20), I = [0, 10]. • Rayleigh, f (x) = x σ 2 exp(-x 2 2σ 2 ) with σ 2 = 2 4-π , I = [0, 6]. • Weibull, f (x) = k λ x λ k-1 e -(x/λ) k , with k = 3 2 and λ = 1/ Γ(1 + 4 3 ) -Γ(1 + 2 3 ) 2 , I = [0, 5]. • Beta B(4, 5) with normalisation 9 √ 2 , I = [0, 8]. • Gamma mixture 0.4 Γ(2, 2) + 0.6 Γ(16, 4) with normalisation 1 √ 2.96 , I = [0, 5].
Distributions for Y . We choose gamma distributions for Y . We recall the density of the gamma distribution with parameters α 1 and λ > 0:

g(x) = λ α Γ(α) x α-1 e -λx , x 0.
with Γ the gamma function. These distributions satisfy assumptions (A1) to (A3), so we can use the procedure described in Section 5. We can compute the Laguerre coefficients of g exactly:

∀k ∈ N, b k = √ 2 λ λ + 1 α k j=0 k j α -1 + j j -2 1 + λ j , where α-1+j j := (α-1+j)×(α-2+j)ו••×α j! . Denoting b m the vectors [b k ] k m-1 , we can compute: b m = √ 2 λ λ + 1 α P m v m ,
where P m is the m × m matrix with components P ij := i j for 0 i, j m -1, and v m is the vector of size m:

v m := α -1 + j j -2 1 + λ j 0 j m-1 .
Therefore, we can compute easily and efficiently the matrix G m and its inverse. We choose two distributions for Y : the Γ(2, √ 20) distribution which has variance 1 10 and the Γ(2, √ 8) distribution which has variance 1 4 .

The constant c(g). This constant in the penalty pen 2 is not known, so we have to evaluate it numerically. Following the proof of the Proposition 4.4, we see this constant appears in the upper bound:

ρ 2 (G -1 m ) n m-1 k=0 E ϕ k (Z 1 ) 2 c(g) √ m ρ 2 (G -1 m ) n ,
which is true for every density of X 1 and for every m. Therefore,

c(g) = sup X1∼f sup m∈N * 1 √ m m-1 k=0 E ϕ k (X 1 + Y 1 ) 2 .
For every distribution of X we considered earlier and for every m from 1 to 50, we computed:

1 √ m m-1 k=0 E ϕ k (X 1 + Y 1 ) 2 ,
using an empirical mean over 1000 realizations of (X 1 , Y 1 ) to compute the expectation, then we took the maximum. For the two distributions we considered for Y 1 , the worst constant was about 0.4 and was reached for X 1 ∼ E(1). For the other distributions of X 1 , we found constants between 0.2 and 0.3. To be safe, we chose a bigger constant c(g) = 0.5 for both Γ(2, √ 20) and Γ(2, √ 8) distributions.

The constants κ. We use the same constant κ 1 = 0.03 as [START_REF] Mabon | Adaptive Deconvolution on the Non-negative Real Line[END_REF]. Concerning the constant κ 2 , we made several simulations to calibrate it and we chose κ 2 = 0.04. During the calibration of κ 2 , we realized that the collection M n did not have enough models. The selected dimensions were too small and lead to bad performances. Therefore, we considered the following collection:

M n := 1 m m * m ρ 2 (G -1 m ) 10 n log n .
Procedure. We take the maximal dimension to be m * = 20. We use two sample sizes: n = 200 and n = 2000. Given a distribution for X, a distribution for Y , a sample size n, and a subdivision Σ(I) of I, do:

1. Compute the matrix G 20 and its inverse.

Compute the collection M

n . 3. Compute f (x) for x ∈ Σ(I). Distribution of X MISE σ 2 Y = 1/10 σ 2 Y = 1/4 n = 200 n = 2000 n = 200 n = 2000 E(1)
m∈M n - m-1 k=0 â2 k + κ i pen i (m) , i ∈ {1, 2},
and compute f m1 (x) + and f m2 (x) + for x ∈ Σ(I).

(d) Compute J 1 = f -( f m1 ) + 2 L 2 and J 2 = f -( f m2 ) + 2 L 2 using Simpson's rule. (e) For m from 1 to 20, compute fm (x) + for x ∈ Σ(I); then compute J(m) = f - ( fm ) + 2 L 2 using Simpson's rule. 5. Compute R 1 (resp. R 2 )
as the mean of J 1 (resp. J 2 ) over the 500 samples. 6. For each m, compute the mean of J(m) over the 500 samples, then compute R o as the minimum of these quantities.

Results. We show our results in Table 1. We note that in the cases X ∼ E(1) and X ∼ Γ(20, √ 20), both estimators performed badly compared to the oracle. We see that our estimator f m2 is better when X has gamma, Rayleigh, beta and mixture gamma distribution. The estimator f m1 is better when X has Weibull and exponential distribution.

For illustration, Figure 1 shows several estimations when the distribution of X is a mixture gamma. It's a bimodal distribution, so it's interesting to see if the estimators are able to recover the two "peaks" and the "hollow" of the true density. For small samples (n = 200), the estimator f m1 seems to fail more often than the estimator f m2 . For large samples (n = 2000), both estimators locate well the two pikes (by overestimating the first one and underestimating the second one), but the estimator f m1 locates less well the hollow and underestimates more the second pike.

Model selection in two-dimensional case

In this subsection, our goal is to illustrate the procedure similar to [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF] on two examples, in the case d = 2.

Distributions for X. To generate a random vector X = (X (1) , X (2) ), we do the following: we generate a random variable W (1) with some distribution and we generate W (2) independently with some other distribution, and we compute:

X (1) X (2) = 1 0.1 0.2 1 W (1) W (2) ,
such that the coordinates of X are not independent. We consider the following distributions for W (1) and W (2) :

• Gamma Γ(3, 1) • Beta B(4, 5) with normalisation 9 √ 2 • Rayleigh, f (x) = x σ 2 exp(-x 2 2σ 2 ) with σ 2 = 2 4-π • Weibull, f (x) = k λ x λ k-1 e -(x/λ) k , with k = 3 2 and λ = 1/ Γ(1 + 4 3 ) -Γ(1 + 2 3 ) 2
Distribution for Y . We chose distributions of the form Γ(α 1 , λ 1 )⊗Γ(α 2 , λ 2 ). These distributions satisfy the assumptions (A1) to (A3) of the Section 5. Moreover, the Laguerre coefficients can be computed easily: if γ i is the density of the distribution Γ(α i , λ i ), then we have g = γ 1 ⊗ γ 2 , so:

∀k ∈ N 2 , b k = γ 1 , ϕ k1 L 2 × γ 2 , ϕ k2 L 2 . W distribution Gamma Beta Rayleigh Weibull
Gamma n = 500 7.78 12.99 11.73 20.30 (4.6, 4.8) (3.6, 7.0) (4.7, 5.9) (5.1, 6.4) n = 5000 1.36 3.40 1.96 6.30 (4.5, 5.6) (3.8, 8.9) (4.9, 6.5) (4.6, 8.5) Beta n = 500 11.75 22.04 18.73 28.15 (6.5, 3.9) (6.2, 6.3) (6.9, 5.0) (6.7, 5.4) 2: MISE computation over 100 samples of size n, for n = 500 and n = 5000. In rows: distribution of W (1) . In columns: distribution of W (2) . In each cell, we show the MISE (multiplied by 10 4 ) and the mean selected model.

Refer to Subsection 6.1 for the computation of γ

i , ϕ ki L 2 . We chose the distribution Γ(2, √ 20)⊗ Γ(2, √ 20) for Y .
The κ constants. After several simulations, we chose κ 1 = κ 2 = 10 -5 . However, this calibration is rough. Like in the one-dimensional case, the model collection M n defined by ( 14) was too small, so we considered the following collection:

M n := m m * D m ρ 2 (G -1 m ) 10 4 n log n .
Results. We took the maximal model to be m * = (12, 12). In each case, we simulated a sample of size n = 500 and n = 5000, then we computed the model m defined by ( 19). We computed

E f -f m 2
L 2 by approximating the double integral by a Riemann sum:

R 2 + f (x, y) -f m(x, y) 2 dx dy ≈ N1-1 i=0 N2-1 j=0 f (x i , y j ) -f m(x i , y j ) 2 ∆x i ∆y j
on a grid with step size 0.01, and by computing the expectation with an empirical mean over 100 samples. We show our results in Table 2. In each case, we compute the MISE and the mean selected model. We emphasize that in the majority of the samples, the selected model is anisotropic: the components of m are not equal. It is a property we expected from a procedure similar to [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF]. The estimator adapts to the regularity of f . For illustration, Figure 2 shows the result of the estimation when W (1) has a Gamma distribution and W (2) has a Weibull distribution, for a sample of size 5000. We show both the adaptive estimator f m and the estimator with no selection procedure f(12,12) , where we simply choose the maximum model. We see that the procedure selected the model m = (5, 8), which has a smaller dimension than the maximum model.

Proofs

Proofs of Sections 3 and 4

Proof of Proposition 3.1. Using the relation ( 6), we have:

f * g = k∈N d j∈N d a k b j (ϕ k * ϕ j ) = k∈N d j∈N d a k b j d q=1 2 -1/2 (ϕ kq+jq -ϕ kq+jq+1 ).
By setting q = k q + j q , we get:

f * g = 2 -d/2 ∈N d k∈N d k a k b -k d q=1 (ϕ q -ϕ q +1 ) = 2 -d/2 ∈N d (a * b) d q=1 (ϕ q -ϕ q +1 ).
Using the tensor product multi-linearity:

d q=1 (ϕ q -ϕ q +1 ) = d q=1 1 εq=0 (-1) εq ϕ q +εq = 1 ε1=0 • • • 1 ε d =0 d q=1 (-1) εq ϕ q +εq = ε∈{0,1} d (-1) |ε| ϕ +ε .
Thus, we get:

f * g = ∈N d (a * b)   2 -d/2 ε∈{0,1} d (-1) |ε| ϕ +ε   = ∈N d   2 -d/2 ε∈{0,1} d (-1) |ε| (a * b) -ε   ϕ .
Since h = f * g, by uniqueness of the Laguerre coefficients of h, we obtain the desired relation.

Proof of Proposition 3.2. First, we notice that

∀ , G = E[e -(Y (1) 1 +•••+Y (d) 1
) ] > 0. We proceed by induction on |k|.

• If |k| = 0, then k = 0 and a 0 = (G 0,0 ) -1 c 0 .

• Let r ∈ N, we suppose ( 9) is true for every

k ∈ N d such that |k| r. Let k ∈ N d with |k| = r + 1. From (8), c k = k =k G k a + G kk a k .

If

k with = k then | | < |k|, so we can use the induction assumption:

a k = (G kk ) -1     c k - k =k G k j H j c j     .
Thus, by setting H kk := (G kk ) -1 and H kj := (G kk ) -1 j k, =k G k H j for every j k, j = k, we've just proved ( 9) for all k such that |k| = r + 1.

Proof of Proposition

4.1. By Pythagoras theorem, f -fm 2 L 2 = f -f m 2 L 2 + f m -fm 2 L 2 .
We decompose the second term on the Laguerre basis:

f m -fm 2 L 2 = k m-1 (a k -âk ) 2 = a m -âm 2 R m .
We now give an upper bound on the last quantity in two different ways.

1. The first way is a bound using the spectral norm.

E f m -fm 2 L 2 = E G -1 m (c m -ĉm ) 2 R m ρ 2 (G -1 m ) E c m -ĉm 2 R m = ρ 2 (G -1 m ) E   k m-1 1 n n i=1 ϕ k (Z i ) -E ϕ k (Z 1 ) 2   = ρ 2 (G -1 m ) n k m-1 Var ϕ k (Z 1 ) ρ 2 (G -1 m ) n k m-1 E ϕ k (Z 1 ) 2 . Yet for every x ∈ R d + , ϕ k (x) 2 2 d and card{k ∈ N d | k m -1} = d q=1 m q , so: E f m -fm 2 L 2 ρ 2 (G -1 m ) n k m-1 E ϕ k (Z 1 ) 2 2 d d q=1 m q ρ 2 (G -1 m ) n . ( 20
)
2. For the second way, we develop the square.

E G -1 m (c m -ĉm ) 2 R m = E    m-1   k m-1 [G -1 m ] k (c k -ĉk )   2    = E   m-1 k m-1 k m-1 [G -1 m ] k (c k -ĉk ) [G -1 m ] k (c k -ĉk )   = m-1 k m-1 k m-1 [G -1 m ] k [G -1 m ] k Cov(ĉ k , ĉk ) = m-1 k m-1 k m-1 [G -1 m ] k [G -1 m ] k Cov ϕ k (Z 1 ), ϕ k (Z 1 ) n = 1 n m-1 Cov   k m-1 [G -1 m ] k ϕ k (Z 1 ), k m-1 [G -1 m ] k ϕ k (Z 1 )   = 1 n m-1 Var   k m-1 [G -1 m ] k ϕ k (Z 1 )   .
We control the variance by the expectation of the square:

Var   k m-1 [G -1 m ] k ϕ k (Z 1 )   E      k m-1 [G -1 m ] k ϕ k (Z 1 )   2    = R d +   k m-1 [G -1 m ] k ϕ k (z)   2 h(z) dz h ∞ k m-1 [G -1 m ] k ϕ k 2 L 2 = h ∞ k m-1 [G -1 m ] 2 k .
In the end, we obtain:

E f m -fm 2 L 2 h ∞ n m-1 k m-1 [G -1 m ] 2 k = h ∞ G -1 m 2 F n .
To prove Proposition 4.4, we will use the following lemma about the order of magnitude of the functions ϕ k for k ∈ N.

Lemma 7.1 [START_REF] Askey | Mean Convergence of Expansions in Laguerre and Hermite Series[END_REF]. There exists k 0 ∈ N and C > 0 such that for every k k 0 ,

ϕ k x 2 C ×                    1 if 0 x 1 k , k -1 4 x -1 4 if 1 k x δk, k -1 4 (ν -x) -1 4 if δk x ν -ν 1 3 , k -1 3 if ν -ν 1 3 x ν + ν 1 3 , k -1 4 (x -ν) -1 4 exp -η(x -ν) 3 2 ν -1 2 if ν + ν 1 3 x (1 + λ)ν, exp(-ξx) if (1 + λ)ν x,
where ν := 4k + 2, where δ et λ are small enough positive constants, and where η and ξ are fixed positive constants.

Proof of Proposition 4.4. Following the proof of Proposition 4.1, we see we need to improve the upper bound in (20). Using Lemma 7.1, we have for all k ∈ N,

|ϕ k (x)| 2 C      1 if 0 2x 1 k , 1/ √ kx if 1 k 2x δk, R k if 2x δk.
where

R k = o(1/ √ k) does not depend on x. Since "0 x 1 k " is equivalent to "1 1 √ kx ", we get |ϕ k (x)| 2 C( 1 √ kx + R k ). Thus for k = (k 1 , . . . , k d ) ∈ N d , we get: E ϕ k (Z) 2 = E   d j=1 ϕ kj (Z (j) ) 2   E   d j=1 C 1 k j Z (j) + R kj   E   d j=1 C 1 k j Y (j) + R kj   = C d √ k 1 • • • k d E 1 √ Y (1) • • • Y (d) + C o 1 √ k 1 • • • k d ,
where C depends on M J for J ⊂ {1, . . . , d}. Therefore:

k m-1 E ϕ k (Z) 2 c(g) D m ,
with c(g) a positive constant depending on M J for J ⊂ {1, . . . , d}.

Proof of Theorem 4.11. Using Proposition 4.1, the bound (12) on the bias term, and Proposition 4.7, if f belongs to W s (R d + , L), we have:

∀m ∈ (N * ) d , E f -fm 2 L 2 L d i=1 m -si i + h ∞ n d i=1 m 2αi i . By Remark 4.2, h ∞ is controlled by f L 2 g L 2 . Because f ∈ W s (R d + , L), we have f 2 L 2 L, so h ∞ √ L g L 2 . Thus, ∀m ∈ (N * ) d , E f -fm 2 L 2 L d i=1 m -si i + √ L g L 2 n d i=1 m 2αi i =: ψ n (m).
Let m opt minimizing ψ n , then the gradient of ψ n vanishes on m opt , so we have:

∀j ∈ {1, . . . , d}, L s j 2α j m -sj opt,j = √ L g L 2 n d i=1 m 2αi opt,i . (21)
Therefore, the coordinates of m opt must satisfy si 2αi m -si opt,i = sj 2αj m -sj opt,j for every i and j in {1, . . . , d}. Using these relations in (21), we obtain:

∀j ∈ {1, . . . , d}, m opt,j = C j (s, L, g) n 1/ sj +sj d i=1 2α i s i
, where C j (s, L, g) are constants depending on s, L and g. The minimum value of ψ n is then:

ψ n (m opt ) = C(s, L, g) n -1/ 1+ d i=1 2α i s i
, where C(s, L, g) is a constant depending in s, L and g. Now, if f belongs to S r (R d + , L), we use the bound (13) on the bias term:

∀m ∈ (N * ) d , E f -fm 2 L 2 L d i=1 e -rimi + √ L g ∞ n d i=1 m 2αi i =: φ n (m).
We minimize the function φ n (m) as we did above, and we find the following relations: ∀j ∈ {1, . . . , d}, r j m opt,j e -rj mopt,j = C j (r, L, g) (r j m opt,j )

d i=1 2αj n ,
where C j (r, L, g) are constants depending on r, L and g. Taking the log, we find:

r j m opt,j + d i=1 2α i -1 log(r j m opt,j ) = log n -log C j (r, L, g).
Thus, when n goes to +∞, we have r j m opt,j ∼ log n. Taking m opt ∈ (N * ) d such that m opt,j ∝ log n rj , we find:

φ n (m opt ) C(r, L, g) (log n) d i=1 2αi n ,
where C(r, L, g) is a constant depending on r, L and g.

Proposition 4.7 7.2.1. Preliminary results

To prove this proposition, we first need to extend the theory of Toeplitz matrices to hypermatrices. For more details about Toeplitz matrices, see [START_REF] Böttcher | Spectral properties of banded Toeplitz matrices[END_REF]. We say that T is an infinite lower triangular Toeplitz hypermatrix if it is lower triangular according to Definition 2.5 and if the value of T k depends only on the difference -k:

T = T k ,k∈N d , T k = a -k , with a = [a k ] k∈Z d and a k = 0 if k / ∈ N d .
So there is a bijection that takes a ∈ R N d and returns the corresponding Toeplitz hypermatrix T (a). We can see T (a) as a linear map on R N d :

x = x k k∈N d -→   k∈N d T k x k   ∈N d =   k a -k x k   ∈N d = a * x.
The associativity of the convolution product gives that for every a and b, we have T

Operator on

p (N d ). If a ∈ 1 (N d ) and if x is in p (N d ) with p ∈ [1, +∞], then a * x belongs to p (N d ) we have a * x p a 1 x p .
In other words, T (a) is an operator from p (N d ) to p (N d ) and its operator norm is bounded by a 1 . The 2 case. We will need to study T (a) as an operator on 2 (N d ) and give a bound on its norm with milder assumptions than a ∈ 1 (N d ). We define a subspace of L 2 (T d ) that plays an important role in this matter. Definition 7.2 (Hardy space). We define the Hardy space as the following subset of L 2 (T d ):

H 2 (T d ) := u ∈ L 2 (T d ) ∀k / ∈ N d , c k (u) = 0 ,
where c k (u) denotes the k-th Fourier coefficient of u T is the set of complex numbers with unitary module.

The map that takes a function and returns its Fourier coefficient is then an isometric bijection between H 2 (T d ) and 2 (N d ).

F : H 2 (T d ) -→ 2 (N d ) f -→ c k (f ) k∈N d .
We see that if u and v are H 2 (T d ) functions, the identity

F[u] * F[v] = F[u × v] translates for Toeplitz hypermatrices into T F[u] × T F[v] = T F[u × v] .
Under the additional assumption that F -1 [a] belongs to L ∞ (T d ), we show in the next proposition that T (a) defines an operator from 2 (N d ) to 2 (N d ).

Proposition 7.3. Let a ∈ 2 (N d ) such that u := F -1 [a] ∈ L ∞ (T d ).
Then T (a) defines an operator on 2 (N d ) and its operator norm is bounded by u ∞ .

Proof. Let x ∈ 2 (N d ) and let v := F -1 [x]. Then:

T (a)x 2 = F[u] * F[v] 2 = F[u × v] 2 = u × v L 2 u ∞ v L 2 = u ∞ x 2 .
Before we can prove Proposition 4.7, some facts need to be established. We denote by Ĉ := C ∪ {∞} the Riemann sphere. We will use the following functions:

• We make the assumption that β belongs to 1 (N d ) so that the power series B(z) := • We denote by G the Laplace transform of g. This function is defined on P d + , continuous on P d + , and holomorphic on {s ∈ C d | ∀q, Re(s q ) > 0}.

• If u is a L 1 (R d + ) or L 2 (R d +
) function, we denote by:

u * (ω) := R d + e iω•x u(x) dx, ω ∈ R d ,
its Fourier transform, where ω • x := i ω i x i . The Fourier transform of the Laguerre functions ϕ k , k ∈ N d can be computed from the case d = 1:

ϕ * k (ω) = d q=1 ϕ * kq (ω q ) = d q=1 (-1) kq √ 2 (1 + iω q ) kq (1 -iω q ) kq+1 .
We will need to understand the behaviour of the map z → 1+z 1-z .

Lemma 7.4. Let η : Ĉ → Ĉ be the homographic function η(s) := s-1 s+1 (with η(∞) = 1 and η(-1) = ∞).

1. The function η is invertible and its inverse is

η -1 (z) = 1+z 1-z . 2. The image of {s ∈ C | Re(s) > 0} by η is D. 3. The image of {s ∈ C | Re(s) = 0} by η is T \ {1}.
In the next proposition that generalizes lemma C.1 from [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast-enhanced imaging[END_REF], we show that the functions G and B are linked through the transformation η. 1-e iθ ) and we have:

∀θ ∈ R d , G 1 + e iθ 1 -e iθ = k∈N d β k e ik•θ = B(e iθ ),
with normal convergence of the series.

Remark

7.6. If g ∈ W s (R d + , L) with s ∈ (1, +∞) d , then β is 1 (N d ), see Remark 4.8.
Proof of Proposition 7.5. We start from the expression of β k and we use the Plancherel isometry:

β k = 2 -d/2 k 1 (-1) |ε| b k-ε = 2 -d/2 k 1 (-1) |ε| g, ϕ k-ε L 2 = 2 -d/2 k 1 (-1) |ε| 1 (2π) d g * , ϕ * k-ε L 2 = g * , 2 -d/2 (2π) d ε 1 (-1) |ε| ϕ * k-ε L 2 .
Let us compute the second factor in the scalar product. The reader is referred to Section 2 for the notations governing vectors and multi-indices computation.

2 -d/2 (2π) d ε 1 (-1) |ε| ϕ * k-ε (ω) = 2 -d/2 (2π) d ε 1 (-1) |ε| d q=1 (-1) kq-εq √ 2 (1 + iω q ) kq-εq (1 -iω q ) kq-εq+1 = (-1) |k| (2π) d ε 1 1 + iω 1 -iω k-ε 1 1 -iω 1 Using the multibinomial theorem (Proposition 2.1), = iω + 1 iω -1 k 1 (2π) d 1 -iω 1 + iω + 1 1 1 1 -iω 1 = iω + 1 iω -1 k 1 π d d q=1 1 1 + ω 2 q ,
We derive the following expression:

β k = 1 π d R d g * (ω) iω -1 iω + 1 k dω 1 • • • dω d (1 + ω 2 1 ) • • • (1 + ω 2 d )
.

The change of variable e -iθq = iωq-1 iωq+1 = η(iω q ) gives then:

β k = 1 (2π) d [0,2π] d g * e iθ + 1 i(e iθ -1) e -ik•θ dθ (22) = 1 (2π) d [0,2π] d G 1 + e iθ
1 -e iθ e -ik•θ dθ.

Therefore, if k ∈ N d , the k-th Fourier coefficient of the function θ → G( 1+e iθ 1-e iθ ) is β k . On the other hand, if k / ∈ N d , let us show that the Fourier coefficients vanish. Without loss of generality, we assume that k 1 < 0, and we compute the k-th Fourier coefficient:

1 (2π) d [0,2π] d G 1 + e iθ 1 -e iθ e -ik•θ dθ = 1 π d R d g * (ω) iω 1 -1 iω 1 + 1 k1 d q=2 iω q -1 iω q + 1 kq dω 1 • • • dω d (1 + ω 2 1 ) • • • (1 + ω 2 d ) = 1 π d R d g * (ω) i(-ω 1 ) -1 i(-ω 1 ) + 1 -k1 d q=2 iω q -1 iω q + 1 kq dω 1 • • • dω d (1 + ω 2 1 ) • • • (1 + ω 2 d ) = -1 π d R d g * (-ω 1 , ω 2 , . . . , ω d ) iω -1 iω + 1 k dω 1 • • • dω d (1 + ω 2 1 ) • • • (1 + ω 2 d )
,

where k := (-k 1 , k 2 , . . . , k d ). Let δ be the map on R d defined by δ(ω) := (-ω 1 , ω 2 , . . . , ω d ), then we have:

-1 π d R d g * δ(ω) iω -1 iω + 1 k dω 1 • • • dω d (1 + ω 2 1 ) • • • (1 + ω 2 d ) = -g * • δ, 2 -d/2 (2π) d ε 1 (-1) |ε| ϕ * k -ε L 2 = 2 -d/2 k 1 (-1) |ε| 1 (2π) d -g * • δ, ϕ * k -ε L 2 = 2 -d/2 k 1 (-1) |ε| g • δ, ϕ k -ε L 2 , because (g • δ) * = -g * • δ.
This last expression is always zero:

g • δ, ϕ k -ε L 2 = R d g(-x 1 , x 2 , . . . , x d ) ϕ k -ε (x) dx = 0, since the function g is zero of x 1 > 0 and ϕ k -ε is zero if x 1 < 0. Thus, the Fourier coefficients of θ → G( 1+e iθ 1-e iθ ) are [β k ] k∈N d . The function G is continuous on (iR ∪ {∞}) d , so the function G • (η -1
) ⊗d is continuous on T d by Lemma 7.4. Therefore, since the Fourier series θ → G( 1+e iθ 1-e iθ ) is normally convergent, this function is equal to its Fourier series at each point:

∀θ ∈ R d , G 1 + e iθ 1 -e iθ = k∈N d β k e ik•θ .
We need a last technical lemma before we start the proof of Proposition 4.7.

Lemma 7.7. Let α ∈ (N * ) d , then (1 -z) -α admits a power series expansion on D d given by:

∀z ∈ D d , (1 -z) -α = j∈N d α -1 + j j z j .
Moreover, denoting ζ j the j-th coefficient in the power series above, for m 4 we have (

T m (ζ) 2
1 -z) -α = d q=1 (1 -z q ) -αq = d q=1 ∞ jq=0 α q -1 + j q j q z jq q = j∈N d α -1 + j j z j .
Therefore,

T m (ζ) 2 F = j m-1 α -1 + j j 2 # (k, ) ∈ N d × N d k m -1, -k = j = d q=1 mq-1 jq=0 α -1 + j q j q 2 (m q -j q ),
and it reduces to the case d = 1 which was already solved in (Comte et al., 2017, appendix C). So if m q 4 for every q, then T

m (ζ) 2 F d q=1 m 2αq q = m 2α .

Proof of Proposition 4.7

From Proposition 7.5, we get G = B • η ⊗d . We define a function w on (D \ {1}) d by:

∀z ∈ (D \ {1}) d , w(z) := (1 -z) -α B(z).
This function is related to K α by the identity:

∀z ∈ (D \ {1}) d , w(z) = 2 -α K α (η -1 ) ⊗d (z) .
Thus, the function w can be extended as a function on D d , still denoted w, and according to our assumptions on K α and Lemma 7.4, w satisfies:

• w |T d is continuous; • w is continuous on (D \ {1}) d ;
• w is holomorphic on D d ;

• w doesn't vanish on D d .

Thus, the function w -1 = 1/w is well defined on D d and has the same properties. In particular, since it is holomorphic on D d , it admits a power series expansion:

∀z ∈ D d , w -1 (z) = k∈N d d k z k .
Let us have look on what is happening on T d . For r ∈ [0, 1[, we denote w -1 r the function on T d defined by: ∀t

∈ T d , w -1 r = w -1 (rt) = k∈N d d k r |k| t k .
On the one hand, the Fourier coefficients of w -1

r are d k r |k| (we set d k = 0 if k / ∈ N d ).
On the other hand, we can compute the Fourier coefficients and we get:

∀k ∈ Z d , d k r |k| = 1 (2π) d [0,2π] d w -1 (re iθ ) e -ik•θ dθ. Since w -1 is continuous on (D \ {1}) d , we have w -1 r → w -1 |T d a.
e. on T d . By dominated convergence, we obtain:

∀k ∈ Z d , d k = 1 (2π) d [0,2π] d w -1 (e iθ
[I N d ] k = 1 if = k, 0 else. Thus, for m ∈ N * , we get G -1 m = T m (ζ) × T m (d).
Taking the Frobenius norm, we obtain the following inequality:

G -1 m 2 F T m (ζ) 2 F × ρ 2 T m (d)
. From Proposition 7.3, we have:

ρ 2 T m (d) = sup x∈R m \{0} T m (d)x 2 R m x 2 R m sup x∈ 2 (N)\{0} T (d)x 2 2 x 2 2 = w -1 |T d 2 ∞ ,
and by Lemma 7.7, we have T

m (ζ) 2 F m 2α if m 4.

Proofs of Section 5

Proof of Theorem 5.2. The proof is identical to the one in Mabon ( 2017), but we use our Lemma 5.1 instead of Mabon's proposition 7.1.

Proof of Lemma 5.1. We will prove that fm -f m 2 L 2 can be written as the supremum of an empirical process, then we will use Talagrand's inequality (Theorem A.1). Let m ∈ (N * ) d , for every t ∈ S m we set:

G -1 m (t) := k m-1 G -1 m × d t m k ϕ k ,
where t m ∈ R m is the matrix of Laguerre coefficients of t. We define the empirical process ν n on the space S m * by:

ν n (t) := 1 n n i=1 G -1 m * (t)(Z i ) -E G -1 m * (t)(Z i ) .
Note that S m * contains every S m as a subspace for m ∈ M n and because of the triangular structure of G -1 , we have for every

t ∈ S m , G -1 m * (t) = G -1 m (t). For m ∈ M n , we have then: fm -f m 2 L 2 = k m-1 (â k -a k ) 2 = k m-1 ν 2 n (ϕ k ).
The map t → ν n (t) being linear on S m and (ϕ k ) k m-1 being a basis of S m , we have by Cauchy-Schwarz inequality:

k m-1 ν 2 n (ϕ k ) = sup t∈Bm ν 2 n (t), where B m := t ∈ S m t L 2 = 1 . Thus, we have: fm -f m 2 L 2 = sup t∈Bm ν 2 n (t).
We apply Talagrand's inequality 1 . We need the constants M, H and v to apply this inequality.

• Computation of M :

sup t∈Bm sup x∈R d + G -1 m (t)(x) = sup t∈Bm sup x∈R d + k m-1 G -1 m × d t m k ϕ k (x) 2 d/2 sup t∈Bm k m-1 G -1 m × d t m k 2 d/2 D m sup t∈Bm G -1 m × d t m R m = 2 d/2 D m ρ(G -1 m ). Thus, M 2 := 2 d D m ρ 2 (G -1 m ). • Computation of H: E sup t∈Bm ν n (t) 2 = E fm -f m L 2 2 E fm -f m 2 L 2 c(g) √ D m ρ 2 (G -1 m ) n ∧ g ∞ G -1 m 2 F n ,
by Proposition 4.4. Thus, we take:

H 2 := c(g) √ D m ρ 2 (G -1 m ) n ∧ ( g ∞ ∨ 1) G -1 m 2 F n .
• Computation of v:

sup t∈Bm Var G -1 m (t)(Z 1 ) sup t∈Bm E G -1 m (t)(Z 1 ) 2 = sup t∈Bm R d + G -1 m (t)(x) 2 h(x) dx h ∞ sup t∈Bm G -1 m (t) 2 L 2 g ∞ sup t∈Bm G -1 m × d t m 2 R m = g ∞ ρ 2 (G -1 m ). Thus, v := g ∞ ρ 2 (G -1 m ). We consider two cases. 1. "c(g) √ D m ρ 2 (G -1 m ) ( g ∞ ∨ 1) G -1 m 2
F " case, then we have:

nH 2 v = c(g) g ∞ D m , nH M = c(g) 2 d/2 √ nD -1/4
m c(g) 2 d/2 n 1/4 , since D m n. For δ > 0, Talagrand's inequality gives:

E fm -f m 2 L 2 -2(1 + 2δ)H 2 + g ∞ ρ 2 (G -1 m ) n exp -Kδ c(g) g ∞ D m + D m ρ 2 (G -1 m ) C(δ) 2 n 2 exp - KC(δ) √ 2δ 7 c(g) 2 d/2 n 1/4 .
We control the first term using assumption (A3'):

m∈Mn g ∞ ρ 2 (G -1 m ) n exp -Kδ c(g) g ∞ D m g ∞ K(δ) n ,
where K(δ) is a constant not depending on n. To control the second term, we use that D m ρ 2 (G -1 m ) n when m ∈ M n :

m∈Mn D m ρ 2 (G -1 m ) C(δ) 2 n 2 exp - KC(δ) √ 2δ 7 c(g) 2 d/2 n 1/4
exp -K(δ) n 1/4 = o 1 n , so we have our result. 2. "c(g)

√ D m ρ 2 (G -1 m ) ( g ∞ ∨ 1) G -1 m 2
F " case, then we have:

nH 2 v = G -1 m 2 F ρ 2 (G -1 m ) 1, nH M = √ n ( g ∞ ∨ 1) 1/2 G -1 m F 2 d/2 √ D m ρ(G -1 m ) √ n 2 d/2 D -1/2 m .
Let us show that ρ(G -1 m ) 1. We notice that 2 d/2 b0 is an eigenvalue of G -1 m : indeed, if E 0 is the elementary hypermatrix with 1 in position (0, . . . , 0) and zeros elsewhere, then:

G -1 m × d E 0 = 2 d/2 b 0 E 0 .
Since the operator norm is bigger than the spectral radius, we have: 2 d/2 . We apply Talagrand's inequality with δ = a log n and a > 0 to be chosen later:

ρ(G -1 m ) 2 d/2 b 0 = 1 E e -(Y (
E fm -f m 2 L 2 -2(1 + 2a log n)H 2 + g ∞ ρ 2 (G -1 m ) n 1 n Ka + D m ρ 2 (G -1 m ) C(a log n) 2 n 2 exp - KC(a log n) √ 2a 7 log n 2 d/2 .
We assume that 2 n exp(3/a), such that C(a log n) = 1. Then:

E fm -f m 2 L 2 -2(1 + 2a log n)H 2 + g ∞ ρ 2 (G -1 m ) n 1+Ka + D m ρ 2 (G -1 m ) n 2 exp - K √ 2a 7 × 2 d/2 log n .
We control the first term using that if m ∈ M n then ρ 2 (G -1 m ) n:

m∈Mn g ∞ ρ 2 (G -1 m ) n 1+Ka g ∞ n Ka-1 g ∞ n ,
if a 2/K = 12. To control the second term, we use that if m ∈ M n then D m ρ 2 (G -1 m ) n:

m∈Mn D m ρ 2 (G -1 m ) n 2 exp - K √ 2a 7 × 2 d/2 log n n -K √ 2a 7×2 d/2 1 n if a 2 d ×7 2 2K 2 . Proof of Theorem 5.3. Let m ∈ M n , f m -f 2 L 2 3 f m -fm∧ m 2 L 2 + 3 fm∧ m -fm 2 L 2 + 3 fm -f 2 L 2 . ( 23
)
By definition of A(m) and m, and since κ 1 κ 2 , we get:

f m -fm∧ m 2 L 2 + fm∧ m -fm 2 L 2 A(m) + κ 1 V ( m) + A( m) + κ 1 V (m) 2A(m) + (κ 1 + κ 2 )V (m).
Thus, by taking the expectation in (23) and using Proposition 4.4, we obtain:

E f m -f 2 L 2 6 E[A(m)] + 3(κ 1 + κ 2 ) V (m) + 3 f -f m 2 L 2 + V (m) . ( 24
)
Now, we need to control

E[A(m)]. Let m ∈ M n . fm -fm∧m 2 L 2 3 fm -f m 2 L 2 + 3 f m -f m∧m 2 L 2 + 3 f m∧m -fm∧m 2 L 2 . Thus, A(m) 3 max m ∈Mn fm -f m 2 L 2 - κ 1 6 V (m ) + + max m ∈Mn f m∧m -fm∧m 2 L 2 - κ 1 6 V (m ) + + max m ∈Mn f m -f m∧m 2 L 2 . ( 25
)
We denote these three terms T 1 , T 2 (m) and T 3 (m), in that order.

2 actually, we assume later that a 12, so there is no assumptions on n after all.

Bound on E[T 1 ]. We control the maximum by the sum on all terms:

T 1 m ∈Mn fm -f m 2 L 2 - κ 1 6 V (m ) + .
By Lemma 5.1, there exist κ 0 such that for every choice of κ 1 > κ 0 , E[T 1 ] C/n for some constant C depending on κ 1 and g.

f m∧m -fm∧m 2 L 2 - κ 1 6 V (m ∧ m ) + , ( 26
)
since V is non-decreasing in its arguments. For J ⊆ {1, . . . , d} fixed, we introduce the collection:

M n (m, J) := {m ∈ M n | ∀j ∈ J, m j < m j and ∀j / ∈ J, m j = m j }.

Then we notice that {j | m j < m j } = J if and only if m ∧ m ∈ M n (m, J). So (26) becomes:

T 2 (m) max Then we notice that S m ∩ S m,m = S m∧m and that S m ⊆ S m,m . We denote by Π m the projection on S m and Π m,m the projection on S m,m . Then, because Π m is 1-lipschitz,

f m -f m∧m 2 L 2 = Π m (f -Π m,m f ) 2 L 2 f -Π m,m f 2 L 2 f -f m 2 L 2 ,
where the last inequality comes from the definition of the projection on S m,m . So T 3 (m) is controlled by the bias term f -f m 2 L 2 .

End of the proof. We use the bounds on E[T 1 ], E[T 2 (m)] and T 3 (m) to control E[A(m)] in ( 24):

E f -f m 2 L 2 9 f -f m 2 L 2 + 3(1 + κ 1 + κ 2 ) V (m) + 6(1 + 2 d )C n .
Since this bound holds for every m ∈ M n , we get:

E f -f m 2 L 2 max{9, 3(1 + κ 1 + κ 2 )} inf m∈Mn f -f m 2 L 2 + V (m) + 6(1 + 2 d )C n .

  we make R m a Euclidean vector space, defining the addition and the scalar multiplication by [T + U ] k := T k + U k and [λT ] k := λT k , and defining the scalar product and the associated norm by T

( A3 )

 A3 For every b > 0, we have m∈Mn ρ 2 (G -1 m ) e -b √ Dm K(b), with K(b) a positive constant not depending on n. Under assumptions (A1) and (A2), we can apply Proposition 4.4 to control the variance term E fm -f m 2 L 2 by:

4.

  Repeat 500 times: (a) Generate a sample Z i = X i + Y i , i = 1, . . . , n. (b) Compute the Laguerre coefficients vector ĉ20 and compute â20 = G -1 20 ĉ20 . (c) Compute m1 and m2 minimizing: mi = arg min

Figure 1 :

 1 Figure 1: Density estimation when X is a mixture of gamma distributions and when Y ∼ Γ(2, √ 20). The thick blue curve represents the true density and each thin red curve represents the estimation obtained with one sample (total: 200 samples). First line: n = 200, second line: n = 2000. Left: f m1 , right: f m2 .

Figure 2 :

 2 Figure 2: Density estimation when W (1) has a Gamma distribution and W (2) has a Weibull distribution, for a sample of size n = 5000. First column: true density, second column: adaptive estimator f m, third column: max model estimator f(12,12) . The selected model is m = (5, 8).

  (a) × T (b) = T (a * b). Notation. If m ∈ (N * ) d , we denote by T m (a) ∈ R (m,m) the sub-hypermatrix of T (a) constructed by taking only the coefficients [T (a)] ,k for , k m -1. Note that as a linear map on R m , we have T m (a) = T (a) |R m because of the triangular structure.

  k∈N d β k z k is normally convergent on D d and defines a function which is continuous on D d and holomorphic on D d .

Proposition 7. 5 .

 5 If θ ∈ R d , we denote e iθ := (e iθ1 , . . . , e iθ d ). If β ∈ 1 (N d ), then [β k ] k∈N d are the Fourier coefficients of θ → G( 1+e iθ

Fm

  2α where T (ζ) is the Toeplitz hypermatrix contructed from the coefficients ζ j .Proof of Lemma 7.7. We recall the following identity: for z ∈ D and r ∈ N * , we have (1-z) -r = ∞ j=0 r-1+j j z j . Thus, for z ∈ D d and α ∈ (N * ) d , we have:

  ) e -ik•θ dθ.Therefore, (d k ) are the Fourier coefficients of w -1 |T d . Thus, we have shown thatw -1 |T d ∈ H 2 (T d ) and: w -1 (t) = k∈N d d k t k , with L 2 (T d )-convergenceof the series. By Lemma 7.7, the function (1-z) -α admits a power series expansion on D d given by (1-z) -α = k∈N d ζ k z k . Thus, the power series equality B(z) × (1 -z) -α × w -1 (z) = 1 on the domain D d translates to their coefficients into the relation β * ζ * d = δ 0 , where δ 0 is the element of R N d defined by: δ 0 k = 1 if k = 0, 0 else. Taking the corresponding Toeplitz hypermatrices, we get G × T (ζ) × T (d) = I N d , where I N d is the infinite hypermatrix given by:

  m ∈ M n we have D m D m ρ 2 (G

  Bound on E[T 2 (m)]. We decompose the maximum by looking at which indices j we have m j < m j : | m j <mj }=J

  we did with T 1 , we have for every choice of κ 1 > κ 0 , E[T 2 (m)] 2 d × C/n. Bound on T 3 (m). For m ∈ M n fixed, let J m,m := {j | m j < m j } and let S m,m be the following closed subspace of L 2 (R d + ): S m,m := Span{ϕ k : k ∈ N d and ∀j ∈ J m,m , k j m j -1}.

Proposition 3.2. Let G be the infinite hypermatrix defined in

  (8). For every k ∈ N d , there exists [H k ] k such that for every a, c ∈ N d statisfying the relation (8), we have:

Table 1 :

 1 MISE computation over 500 samples. MISE are multiplied by 10 4 . The column "σ 2

				R 1	139	8.57	113	9.95
				R 2	204	30.4	203	37.2
				R o	10.6	0.93	14.7	1.50
	Γ(20,	√	20)	R 1 R 2	127 80.4	32.1 16.4	407 350	109 58.0
				R o	26.8	3.84	34.7	5.24
				R 1	68.1	7.13	88.7	8.23
	Rayleigh	R 2	42.1	6.44	50.3	9.13
				R o	29.8	4.76	35.0	7.33
				R 1	55.2	4.89	79.0	6.80
	Weibull	R 2	64.3	8.56	70.3	10.5
				R o	30.8	4.84	42.0	6.59
				R 1	122	16.8	184	29.8
	B(4, 5)	R 2	39.6	6.50	53.0	16.5
				R o	30.9	4.52	35.5	5.59
				R 1	286	63.8	717	82.9
	Γ mixture	R 2	162	47.1	595	71.2
				R o	151	34.0	228	49.4

Y = 1/10" corresponds to the case Y i ∼ Γ(2, √ 20) and the column "σ 2 Y = 1/4" to the case Y i ∼ Γ(2, √

8

).

this inequality concerns countable families of functions, but it's not a problem here since νn is continuous on S m * and Bm is separable.

Acknowledgments

The author wants to thank Fabienne Comte and Céline Duval for their helpful advice.

Funding

This work was supported by a grant from Région Île-de-France.

Appendix A. Talagrand's inequality

This concentration inequality is derived from the work of [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF]. A proof can be found in the PhD thesis of Chagny (2013a, chapter 2).

Theorem A.1 (Talagrand's inequality). Let n ∈ N * and let F be a set at most countable of measurable functions on a polish space E. Let ξ 1 , . . . , ξ n be independent random variables with values in E. We consider ν n the centred empirical process defined by:

We assume that there exists three positive constants M , H and v such that:

Then for every δ > 0, we have the following inequality:


Lucie Cormons
email: lucie.cormons@univ-grenoble-alpes.fr

Humbert Fiorino
email: humbert.fiorino@univ-grenoble-alpes.fr

Damien Pellier
email: damien.pellier@univ-grenoble-alpes.fr

Sylvie Pesty
email: sylvie.pesty@univ-grenoble-alpes.fr

Caroline Poulet
email: caroline.poulet@univ-grenoble-alpes.fr

Testing social robot acceptance: what if you could be assessed for dementia by a robot? A pilot study

Keywords: social Human-Robot Interaction (HRI), assistive robotics, acceptability of robots by the elderly, "persona methodology"

It is beneficial to identify and begin treatment of neurocognitive disorders of the elderly as early as possible. In order to help diagnose these disorders, social assistive robots are promising technologies to assist psychologists. To be accepted by the elderly, the robot behaviours must be close enough to the fundamental competences of the psychologists in order not to confuse the patient. This pilot study aims (1) to design a social assistive robot capable of performing a memory evaluation test, (2) to gather opinions on the robot's acceptability with an innovative method (persona) and (3) to identify robot behavioural improvements. We used the "persona methodology" for this pilot study. A panel of students playing the role of a "persona" performed the memory test called RL/RI16, by interacting with the social robot Pepper and then were interviewed about their experience. The robot plays the psychologist role. The interviews and videos analysis showed that the robot is not yet well accepted but the analysis results gave interesting leads to continue.

I. INTRODUCTION

According to data from World Population Prospects 1 , by 2050, one in six people in the world will be over age 65 (16%), up from one in eleven in 2019 (9%). The number of persons aged 80 years or over is projected to triple, from 143 million in 2019 to 426 million in 2050. The more the elderly population increases the more that population is likely to suffer from Major Neurocognitive Disorders (MNDs) like Alzheimer. MNDs affect 47 million people worldwide [START_REF]Rapport mondial sur le vieillissement et la santé[END_REF]. In order to improve patient management, cognitive deficits must be detected early using neuropsychological tests performed by psychologists or neuropsychologists [START_REF] Pasquier | Early diagnosis of dementia: Neuropsychology[END_REF].

Unfortunately, the elderly are often not diagnosed in time [START_REF] Iliffe | Sooner or later? Issues in the early diagnosis of dementia in general practice: a qualitative study[END_REF]. Society must reconsider the health and social 1 https://population.un.org/wpp/ care system to meet the new challenges of an ageing population, especially in terms of major neurocognitive disorder (such as Alzheimer) screening. To fight this issue, a virtual revolution has taken place for clinical purpose including the implementation of social robots within the health care context [START_REF] Walker | Social intelligence: Its history and measurement[END_REF]. Combining also with the shortage of physicians in some parts of the territory, robots may provide important aid to overburdened health-care providers [START_REF] Philip | Virtual human as a new diagnostic tool, a proof of concept study in the field of major depressive disorders[END_REF]. Robots appear to be accepted by patients, especially if they demonstrate social skills [START_REF] Heerink | Studying the acceptance of a robotic agent by elderly users[END_REF][7] [START_REF] Klamer | Acceptance and use of a social robot by elderly users in a domestic environment[END_REF], but the qualities required for robots to be highly accepted by patients are not yet well understood. Furthermore, several studies have examined both companion type robots and service type robots interacting with elderly but few have analysed the acceptability of robots during a more professional task. A lack of studies regarding robots performing more health-oriented tasks is to be noted in the current literature. This knowledge gap motives the current research on acceptability of robots playing the role of a health-care professional.

As the elderly will interact with the robot for a memory problem screening test, which is stressful in itself, particular care must be taken to provide the robot with appropriate socio-communicative skills. The ingredients to promote the acceptance of robots by the elderly are a current hot topic [START_REF] Caulfield | What is Connected Health and why will it change your practice?[END_REF]. For now, recent studies show the importance of robot social behaviour even more so in a health-and eldercare environment. In other words, robots that possess better social skills would be able to communicate more effectively, being judged as more pleasant and therefore would be more acceptable during interactions [START_REF] Heerink | Studying the acceptance of a robotic agent by elderly users[END_REF]. Also, programming a robot to interact with an elderly person requires multiple important factors. Indeed, elderly can require more effort than younger patients in terms of communication due to aging. The elderly are more likely to suffer from hearing loss, vision loss [START_REF] Covelet | Prendre enfin conscience des enjeux des déficits sensoriels des personnes âgées[END_REF], attention loss [START_REF] Salthouse | Selective review of cognitive aging[END_REF] and have more difficulty speaking, especially in louder tones [START_REF] Salthouse | What and when of cognitive aging[END_REF]. Therefore, considering the health care situation, the special needs of the elderly might inform the design of robots intended for such an audience.

The pilot study presented in this paper has three main objectives. First, it aims to create a socially assistive robot capable of performing a memory test based on a real psychologist / elderly person interaction script. Second, it aims to analyse the acceptability of the robot with an innovative methodology (persona). Third, it aims to get participant feedback on robot behavioural improvements, as well as to gain insight regarding the task performance and social cues necessary to increase acceptability.

II. RELATED WORK

A. Robots as a tool for eldercare

A growing number of projects address the development of robots for experiments in eldercare. Most of the time, two kinds of experiments are conducted with the elderly suffering from neurocognitive impairment: some use companion and/or mediation robots like Paro (seal shaped robot) [START_REF] Shibata | Robot therapy: A new approach for mental healthcare of the elderly-a mini-review[END_REF] where the focus is on social acceptance, while others use service type social robots like iCat [START_REF] Heerink | Studying the acceptance of a robotic agent by elderly users[END_REF] [START_REF] Klamer | Acceptance and use of a social robot by elderly users in a domestic environment[END_REF]. The latter studies focused more on acceptance of the robot regarding its basic functionalities, like asking the participant whether they were adhering to their activity plan [START_REF] Heerink | Assessing acceptance of assistive social agent technology by older adults: The almere model[END_REF]. Most of the studies on robotic acceptance showed that the elderly are more comfortable with a more sociable robot and behaviour analysis shows that the elderly are invited to be more expressive by a more sociable robot [START_REF] Heerink | Enjoyment intention to use and actual use of a conversational robot by elderly people[END_REF][15] [START_REF] Heerink | Assessing acceptance of assistive social agent technology by older adults: The almere model[END_REF].

Some other experiments were conducted where the robot takes the place not of a companion, but of a professional. [START_REF] Fasola | Robot exercise instructor: A socially assistive robot system to monitor and encourage physical exercise for the elderly[END_REF] did a study for the elderly suffering from body impairment where the robot was designed to help them in their rehabilitation journey. Participants indicate that they perceived the robot to be moderately intelligent, and to be moderately helpful with the exercises. The robot was seen by the participants to be an entity that they could trust and that could help them, rather than just entertaining them. More recently, a team conducted a study where patients were diagnosed with depression by either a psychiatrist or an Embodied Conversational Agents (ECA) [START_REF] Philip | Virtual human as a new diagnostic tool, a proof of concept study in the field of major depressive disorders[END_REF]. Patients found the face-to-face interview with the ECA very acceptable, suggesting that ECA "can communicate empathy, elicit patient trust, reduce the feeling of being judged by a human and reduce emotional barriers to disclose an affective state" (p.5). With the same idea, [START_REF] Rizzo | SimCoach: An intelligent virtual human system for providing healthcare information and support[END_REF] proposed the SimCoach project involving a virtual agent to identify symptoms of depression, anxiety, and Post-Traumatic Stress Disorder (PTSD). The authors aim to break down barriers to care like stigma or unawareness by providing confidential help, and to promote the initiation to psychological healthcare information and assistance to both military personnel and family members.

B. Measuring robot acceptability for the elderly

Several models exist to measure the acceptability of a new technology or, more specifically, to measure the intention to adopt a new technology [START_REF] Venkatesh | User acceptance of information technology: Toward a unified view[END_REF]. Primarily based on the theory of planned behaviour of [START_REF] Ajzen | The theory of planned behavior[END_REF], the first model to measure acceptance, named the Technology Acceptance Model (TAM), was developed [START_REF] Davis | Perceived usefulness, perceived ease of use, and user acceptance of information technology[END_REF]. Afterwards, [START_REF] Venkatesh | User acceptance of information technology: Toward a unified view[END_REF] offered an overview of these later technology acceptance models and incorporated the most reliable constructs into the Unified Theory of Acceptance and Use of Technology (UTAUT). The UTAUT aims to explain user intentions to use an information system and subsequent usage behaviour. The theory consists of four key constructs, which are performance expectancy, effort expectancy, social influence, and facilitating conditions. The acceptability is moderated by gender, age, experience and voluntariness of use. However, traditional technology acceptance models do not consider a crucial aspect: the social abilities of the robot and its impact on the interaction. Interaction is, unsurprisingly, a very important aspect of the process of acceptability when it comes to social robotics. In order to address these issues, the Almere model was created [START_REF] Heerink | Assessing acceptance of assistive social agent technology by older adults: The almere model[END_REF]. This model is the adaptation and the theoretical extension of the UTAUT by explaining intent to use not only in terms of variables related to functional evaluation like perceived usefulness and perceived ease of use, but also variables that relate to social interaction. Heerink added variables like perceived enjoyment, defined as the feeling of joy or pleasure associated by the user with the use of the robot, and social presence, defined as the experience of sensing a social entity when interacting with the robot. [START_REF] Heerink | Studying the acceptance of a robotic agent by elderly users[END_REF] said that "a user accepts the robot as a conversational partner, finds the robot's social skills credible, sees the robot as an autonomous social being and is more likely to exhibit natural verbal and nonverbal conversational behaviour as well as feeling comfortable in interacting with the robot". In return, the user will demonstrate more conversational engagement by being more expressive [START_REF] Axelrod | Identifying Affectemes: Transcribing Conversational Behaviour[END_REF] and thus we can use behavioural clues as an indication of conversational acceptance [START_REF] Sidner | Engagement during dialogues with robots[END_REF]. At the end, The Almere model counts 13 variables: anxiety, attitude towards technology, facilitating conditions, intention to use, perceived adaptiveness, perceived enjoyment, perceived ease of use, perceived sociability, perceived usefulness, social influence, social presence, trust, use.

Presently, research on robot acceptability can be divided into two main areas: functional acceptance (usefulness and ease of use) and social acceptance (acceptance of the robot as a conversational partner) [START_REF] Heerink | Enjoyment intention to use and actual use of a conversational robot by elderly people[END_REF].

It seems important to open the spectrum of research to include these constructs in the context of robots with professional health care tasks.

III. METHODOLOGY

A. Overall methodology

The robot we used is Pepper, a semi-humanoid robot manufactured by Softbank Robotics. Pepper is 1.20 meters tall and has a screen on his chest that allows transmitting written content.

1) Dialog scripts creation for the Pepper robot

First, we created an audio-visual corpus where a psychologist (N=3) interacts with an elderly person (N=14) for performing the RL/RI16 memory test. The RL/RI16 memory test (for Rappel Libre/Rappel indicé in French) is an adaptation of an English test (Free and Cued Selective Reminding -FCSR test) [START_REF] Van Der Linden | L'épreuve de rappel libre/rappel indicé à 16 items (RL/RI-16)[END_REF]. It is used widely to identify memory loss. It is structured in five major and well-defined steps, which is an undeniable advantage for robot programming. The test is composed of 16 words to be memorized. The words belong to 16 different semantic categories. Usually, a psychologist performs the test using a pencil, a stopwatch and four cardboards on which are written the 16 words to be learned four by four. The test duration is about 20 minutes.

From the corpus transcription (the 14 Psychologist-Elder dialogs), we analysed the psychologists' sociocommunicative (SC) used to perform the RL/RI16. Two major trends of SC competencies were identified: SC that mainly focus on the task and SC that show mainly focus on the relationship with the patient. We thus have divided the psychologists' SC competencies in two main groups: task-oriented (e.g.: to inform, to give the correction, to explain, etc.) or relational-oriented (e.g.: to be sympathetic, to be reassuring, to motivate the patient).

For the pilot study we present in this paper, we used the most task-oriented dialog to create a prototypical script-dialog to be implemented. We decided to choose for the robot the task-oriented script in order to have a strong base to be implemented that could be compared later with a relation-oriented script more difficult to implement because it requires more behavioural repertoires, laughs, different tones that are more complex to implement on the present version of Pepper. Moreover, as the task-oriented script consists in a tighter basis, it will allow us to identify what behaviours are missing and what are mandatory. By time economy for the participants, we only perform the three main steps of the RL/RI16 test and limited the test to eight words to be memorized (mean time = 5min 45; S.D.=0.28 sec).

2) Innovative methodology (persona)

It is always difficult to recruit people for robotic experiments, especially fragile participants (e.g.: the elderly). However, even if the right participants are difficult to recruit, there is still a crucial need to coconstruct with those users. We therefore choose to use "the persona methodology" for the robotic experimentation that is a reasonable compromise because personas represent possible users that designers can refer to when the targeted population is not available. This methodology consists in creating fictive persons with attributes, social and psychological characteristic to represent a targeted population. A persona therefore represents a user with a name, a face, a personal history and values, which gives him a certain humanity [START_REF] Bornet | La méthode des personas : Principes, intérêts et limites[END_REF]. Personas should not be stereotypical, but complex enough and realistic to be effective [START_REF] Bornet | La méthode des personas : Principes, intérêts et limites[END_REF]. These "cognitive instruments" helps to think and guide decisions about the design of multiple products and services. Even though the "persona methodology" does not physically connect designers and users, it makes available representations of future users.

By following the "persona methodology", two personas were created. The first one represented Pierre, a 75 years old man who lives in a retired house. Pierre is lonely and has a lack of autonomy; he suffered from minor neurocognitive issues, especially memory loss and communication difficulties (lack of concentration and lack of fluency). The second persona was Mireille, an active and dynamic 69 years old elderly woman, with no specific issue. A document with the person picture, name, story, characteristics were handed to the participants.

B. Procedure 1) Pre-test

A pre-test has been performed to verify if both the two scripts-dialogs were correctly identified as either taskoriented or relation-oriented. Two actors were recorded for the task-oriented script, one actor was playing the role of the psychologist, the other one was playing the role of a patient. Then we recorded the same two persons acting the relationoriented script. An online questionnaire was created with video segments. Nineteen students of our university replied on a Lickert-scale (1 to 5 points) with two extremities from "relational-oriented" to "task-oriented". After making sure that the task-oriented was correctly identified by participants, the script was programmed in Pepper.

2) Test

For two days, seven students from our university came to the laboratory. The experimenter presented the persona to the participant and asked him/her to feel like he/she was the persona, to understand the persona specificity and needs. The consign were strictly assigned to the participant, and the experimentation precautionary make sure that the participant understood his role. After the signature of legal documents (informed consent form, image rights form), the participant was invited to seat in front of Pepper. Two engineers were asked to stay in the room in case of issues to give some instruction to the participant (speak loud enough, clearly, repeat if it's necessary, speak when the eyes of Pepper are light up, etc.). The cameras were tuned on and the dialog Human-Pepper started. Pepper ran the RL/RI16 test2 asking questions to the participant and waiting for answers (see figure 1). After the interaction with Pepper the participants replied to a semi-structured and audio-recorded interviews (mean time = 21,43min). The interview focused on questions to gather the perceived pros and cons to the robot usage, on both the robot abilities to interact and to its abilities to perform the test, on what changes could be applied to make the robot more efficient to interact with the elderly, and on what emotions the interactions triggered. Finally, some additional socio-demographic questions were asked. The questions find a theoretical support on the Almere model [START_REF] Heerink | Assessing acceptance of assistive social agent technology by older adults: The almere model[END_REF]. The participants had to reply accordingly to their persona conditions (Pierre, Mireille, no persona).

An intentional choice was made regarding the few numbers of participants. Qualitative data were privileged to gather an important set of feedbacks and strong ergonomic recommendations for further robot programming. These types of analysis do not require a huge sample because it privileges the quality over the quantity. Also, data analysis (transcription, content analysis and video analysis) is a timeconsuming process. 3) Data analysis Both the interviews and the video of the interaction were analysed. The interviews were transcribed, and a content analysis was performed. To performed the content analysis, we used the 13 variables of the Almere Model as a grid analysis to see if the verbatim would fit with the model [START_REF] Heerink | Assessing acceptance of assistive social agent technology by older adults: The almere model[END_REF]. The videos were analysed via an analysis grid inspired from the list of social behaviours of [START_REF] De Ruyter | Assessing the effects of building social intelligence in a robotic interface for the home[END_REF] like nodding head, shaking head; verbal and non-verbal greeting; moving away from the robot; moving forwards the robot; smiling; laughing; Showing irritation; looking surprised. We decided to add two categories that quantify Pepper difficulties to interact and the necessity for the participant to speak loudly to be correctly understand by Pepper.

IV. RESULTS AND DISCUSSION

1) Pre-test

Nineteen participants from 18 and 38 years old were asked to judge video segment (task-oriented vs relationaloriented) of actors doing the RL/RI16 test. Participants correctly answered and recognised that the segment videos were task-oriented or relational-oriented (63.2% of correct answer for the relational-oriented video segment; 97.4% of correct answer for the task-oriented video segment).

2) Test

Seven non computer scientist students (N = 4 women, mean age = 21,4, S.D.=0.92) interacted with Pepper. Two students had to play the role of the persona n°1 (Mireille), two had the persona n°2 (Pierre Legrand), and three had no persona at all.

Video analysis:

Most of the participants greeted sympathetically Pepper at the beginning and at the end of the session (4 out of 7). While Pepper was giving the instructions, 5 participants expressed their understanding by nodding (mean of 2 times each). Regarding the interpersonal distance between Pepper and the participant, the video analysis showed that 4 participants reduced the distance between them and the Pepper by moving forwards it, 3 participants had a neutral position (not forward either backward). These results show quite a lot of social behaviours especially by nodding. No one moved away from Pepper.

The robot having some bugs due to its sensors (mainly it doesn't correctly ear the participants, consequently it repeats itself many times) we analysed the persons' reactions. They mostly react by smiling or laughing. In cases of repeated bugs in short times, participants show some irritation or surprise by frowning or raising their eyebrows. Surprisingly, even though the task was easy (mainly 8 words to be remembered from pictures) and the participants were young, highly educated students (mostly 4 th years of study), they did not succeed in completing the memory task correctly. The mean score of words correctly recalled were 5.4/8. In the interviews, participants never declare that they did poorly at the task, but they said that interacting with the robot ask them a certain cognitive effort. The fact they had to focus more on interacting with the robot (waiting for Pepper's eyes light up before answering, speaking out clearly, etc.) requires cognitive effort and it potentially explains why they did poorly at the memory task. These findings suggest that the memory test is yet a tricky task to be performed by a robot as the Pepper robot. This has impacted the robot interactional performance and costs cognitive effort for the participants and reduce their own test performance. It's clear that a robot performing a memory test must be of high quality, hearing correctly, answering without error and the dialogue must be fluid.

Content analysis:

When we ask the participants' opinion on having a test being done by a robot, we can observe both strengths and weaknesses. On the one hand, Pepper suffers from critics especially because of its technical defects that have negative impact on test performance. Participants reported maintaining a high level of concentration when talking to Pepper because they must focus both on Pepper's eyes colour to know when to talk (4 times) and on speaking loudly because of Pepper poor hearing capacities (8 times). Theses cognitive efforts are likely to impact participants test performance (especially on a memory task where focus is particularly needed). To this point, one participant declares "I was focused on how the robot was reacting to know when to talk, so I would forget the words". Furthermore, the participants judge that the robot is missing three different human aspects that limit the robot usage. First, the lack of communication behaviours (10 times) (e.g.: "I need to put on more effort to understand Pepper"), second, the lack of adaptability and behavioural repertoire (5 times) (e.g. "Pepper always say the same things") and finally, the lack of emotional behaviours (4 times) ("Pepper isn't warm enough"). In summary, participants complain mainly that the robot can't "understand" the person, can't personalize his speech, and doesn't show any reactions nor emotions. Also, participants criticized six times the robot's lack of relational and pedagogic attributes. It's partly due to the task-oriented script we implemented which definitively lacks of relational abilities. Five times participants will point out the restricted behavioural repertoire even if we gave Pepper a repertoire of various responses. Participants expressed that the elderly are likely to ask questions or have atypical behaviours which will challenge the robot. The last negative point is the robot's poor possibility to express emotions. The participants clearly express the need of capturing the interlocutor's emotions to feel comfortable within an interaction (e.g., "with a human being I understand, at least I try to understand, what he thinks through his gaze, his facial expressions, the tone used, but with a robot it is always the same"). Participants explain that the robot seems not understanding participants' social signal in order to adapt its behaviour. They wanted the robot understands if they need help, motivation, or if they are thinking and they need time to focus, etc. Participants wanted to know if they are right or wrong by looking at the robot's expressions. In other words, it seems that they want feedback on their own performance during the interaction. For example, participants repeatedly expressed that the robot should cue their correctness during the task quickly.

In contrast, participants exposed some advantages like they would feel comfortable because they won't feel judge by the robot Pepper so they would feel more comfortable making mistakes (4 times) and they would have no time pressure to reply to the exercise (5 times). One participant underlies that having a robot doing the test may improve test standardization. Then, participants perceived the sociability aspects of Pepper that helped test performance, like humans' interactional attributes that make him friendly (e.g., head and body movements) (7 times). Pepper is also judged as possessing both technical ("it is clear") (3 times) and social psychologist skills ("he looks kind which put you in good conditions to complete the task" or "it makes you want to do the test.") (2 times). The ergonomic aspects were very much appreciated, Pepper was complimented 7 times as being attractive with a great, nice, and functional screen. Seven times participants alluded to what Heerink, Kröse, Evers, and Wielinga (2010) called the perceived sociability that is the "the perceived ability of the system to perform sociable behaviour" (p.4). The perceived sociability positively impacts the interaction quality because participants felt no time pressure to reply to the robot and that the robot had a friendly tone.

By looking deeper into the results, we observed that participants are simultaneously impacted by the technical defects and appreciative of the sociable behaviours of the robot. They referred to the defects that impair the relationship 7 times. For example, participants had to speak loudly and repeat their statements several times before the robot understood. A concern was made on the fact that the robot can't reply to questions which can be an important source of confusion in the case a patient does not understand the instruction. This concern is increased for people suffering from mid-cognitive impairment because due to their condition, they would need more help and more explanation than someone with no disorder (Pierre-condition group).

Interacting with a robot isn't a common thing. It can be a source of negative emotions for some participants (more precisely, uncomfortable (3 out of 7 participants), stressed (3 out of 7), and fearful (1 out of 7) even though 3 participants described having perceived it to be friendly, funny and pleasant. Finally, all the participants questioned the ability of a robot to perform a memory test to an elderly person. Three times, participants suggest that the robot could be an accessory to a psychologist by helping him not to make mistakes.

Finally, if we analyse the results per group, the Mireillecondition and no persona condition reveal quite the same results in the overall content analysis. The Pierre-condition group referred to the sociable behaviours (7 times). For example, it seemed sympathetic (especially his tone), the participants felt they had the time to reply, and the instructions were clear. The Pierre-condition group were particularly satisfied with the screen which augmented the audio information provided by Pepper's speech with a visual aid (an image illustrates the word to be memorized, like in the real test). They expressed that they read the words properly and without difficulties. Furthermore, they were amused by the robot character as new and intriguing.

V. CONCLUSION AND PERSPECTIVES

In this pilot study we have presented the design and evaluation of a socially assistive robot that can interact with an elderly person and engage the person in a memory test.

We defined a task-oriented script dialog implemented in the robot based on real psychologists' dialogs in situation of performing the RL/RI16 test. A feasibility study was conducted with the innovative "persona methodology". The results do not validate the acceptability of the assistive robot in being able to perform a memory test to the elderly. Nevertheless, the pilot study gave us a lot of points to work on. The content analysis reflects that even if participants felt it was fun or pleasant to be assessed by a robot, the robot suffer from technical defects decreasing its acceptability. The major positive factor was that participants did not feel judged by the robot and felt they could take more time to reply or make mistakes. The control group and the Mireillegroup condition gave quite similar answers. This result leads to think that an effort in terms of programming a warm, attentive robot with a large set of behavioural repertoires should be made especially when the patients are suffering cognitive impairment. The main fear for the three groups was that the robot would not be able to reply to the questions or understand that the patient has trouble understanding the test instructions like it might be the case if the elderly is suffering from mid-cognitive impairment. Participants recognize qualities to Pepper, but we can observe that the positive sentences are often punctuated with a "but" that is generally introducing a negative opinion (e.g.: it will never replace a human). Pepper needs ergonomic improvements specially to interact with an elderly person who might suffer from sensorial and cognitive impairment. Regarding improvement, participants uttered the need for turn-taking management, and for speech adaptation to the user, especially if the user possesses particularity (low cognitive level). The communication behaviour repertoire should be extended even for standardized tests (e.g.: RL/RI16). The robot hearing capacity needs to be improved. Finally, relational skills and positive non-verbal expressions in reaction to what the person is saying, need to be added. This last wish seems to show that people need feedback from the robot on their own behaviour to adapt themselves to the robot ("with a human, if I am wrong, I would have seen it on the psychologist face. With the robot, I had no information on how I was doing"). The video analysis showed that people were optimistic and patient with the robot even though the robot was not interacting in an optimal way. We are convinced that the relation-oriented script dialog, which includes positive feedbacks to the elderly, nod of the head, encouragements, etc. will already be much better. This pilot study also wanted to challenge a new way of testing acceptability with the "persona methodology". [START_REF] Tapus | The grand challenges in socially assistive robotics[END_REF] underlies that the field is young and that method for measuring acceptability of social assistive robots as to be developed and open to new methodology. This "persona methodology" allowed us to conduct our study even though the robot was imperfect and elderly persons not available, and gave us important points to work on.

In conclusion, new technologies will be required in years to support psychologists who screen for Major Neurocognitive Disorders (MNDs). Therefore, research and especially the collaborative work between informatic science and psychology should continue to address the issues raised in this pilot study. Such robotic solutions could be very useful in helping psychologists to diagnose MNDs diseases as early as possible. The growing interested in electronic-health (E-health) solutions brings up a main issue coming along with it: the robotic acceptance. In our case, we need to focus on how to provide a quality interaction between elderly that potentially suffer from mid-cognitive impairment and a robot. Starting by the analysis of real interaction allows the programmers to better understand the interactional needs required to perform the task efficiently and promote acceptability. But being assessed for dementia by a robot also raise considerable ethic consideration and this issue will undoubtedly need to be addressed.

Fig. 1 .

 1 Fig.1. Pepper and a participant

An interaction sample: https://www.youtube.com/watch?v=OHSQXxAEkJc

ACKNOWLEDGMENT

This research was supported by the French funded ANR project: "Embodying socio-communicative behaviours in humanoid robots" (ANR-14-CE27-0014). This research has been also supervised by Michel Dubois, Professor in Work and Organizational Psychology at LIP/PC2S, UGA who passed away in July 2018. This work could not have been done without his extreme kindness and expertise.