Olivier Lopez
email: olivier.lopez0@upmc.fr.

Valentin Patilea
email: patilea@ensai.fr

Ingrid Van Keilegom
email: ingrid.vankeilegom@uclouvain.be.

Single index regression models in the presence of censoring depending on the covariates

Keywords: Curse-of-dimensionality, Dimension reduction, Multivariate distribution, Right censoring, Semiparametric regression, Survival analysis

Consider a random vector (X ′ , Y ) ′ , where X is d-dimensional and Y is onedimensional. We assume that Y is subject to random right censoring. The aim of this paper is twofold. First we propose a new estimator of the joint distribution of (X ′ , Y ) ′ . This estimator overcomes the common curse-of-dimensionality problem, by using a new dimension reduction technique. Second we assume that the relation between X and Y is given by a single index model, and propose a new estimator of the parameters in this model. The asymptotic properties of all proposed estimators are obtained.

Introduction and model

Consider a random vector (X ′ , Y ) ′ , where X = (X (1) , . . . , X (d) ) ′ is d-dimensional and Y is one-dimensional. We assume that Y is subject to random right censoring, i.e. instead of observing (X ′ , Y ) ′ , we observe the triplet (X ′ , T, δ) ′ , where T = Y ∧ C, δ = I(Y ≤ C), and the random variable C is the censoring variable. Typically, Y is (a transformation of) the survival time (whose range can span the whole real line), and X is a vector of characteristics. The data consist of n i.i.d. replications (X ′ i , T i , δ i ) ′ of (X ′ , T, δ) ′ . Under this setting, the purpose of this paper is twofold. First, we propose a new estimator of the joint distribution F (x, y) = P(X ≤ x, Y ≤ y) of X and Y (where X ≤ x means that X (j) ≤ x (j) for j = 1, . . . , d). Second, we assume that the relation between X and Y is given by a single index mean regression model (as in e.g. [START_REF] Härdle | Optimal smoothing in single index models[END_REF], and we propose new estimators of the parameters under this model. These estimators will be constructed under the following fundamental model assumption on the relation between Y and C, which we impose throughout this paper :

(A0) There exists a function g : R d → R, such that :

(i) Y and C are independent, conditionally on g(X)

(ii) P(Y ≤ C|X, Y ) = P(Y ≤ C|g(X), Y ).
The function g will be unknown in general. When g is known, this assumption has been proposed by Lopez (2007a). The assumption is needed for identifying the model.

In the literature on nonparametric censored regression, alternatives to assumption (A0) have been proposed. There are basically two alternatives, which can be regarded as limiting cases of assumption (A0), and in that sense our assumption is a trade-off between these two. The first alternative has been used by e.g. [START_REF] Akritas | Nearest neighbor estimation of a bivariate distribution under random censoring[END_REF] and [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], among many others. They assume that Y is independent of C, conditionally on X, and propose kernel type estimators of the distribution F (x, y) under this assumption. This assumption is a particular case of (A0) by taking g(X) ≡ X.

Their estimators are however restricted to the case where d = 1. Although they could in principle be extended to higher dimensions, this is not recommended in practice, since they will suffer from the curse-of-dimensionality and higher order kernels will need to be used. The second alternative to assumption (A0) has been proposed by [START_REF] Stute | Consistent estimation under random censorship when covariables are present[END_REF][START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF]. He assumes that Y is independent of C, and that P(Y ≤ C|X, Y ) = P(Y ≤ C|Y ). This is again a particular case of (A0), by taking g(X) ≡ 1. Although his estimator can be used for any d ≥ 1, it has the drawback that it assumes that the censoring variable C depends on X in a very particular way. This type of dependence might hold true when the censoring is purely 'administrative' (censoring at the end of the study), but when the censoring can be caused by other factors (like death due to another disease, change of treatment, ...), then less restrictive assumptions on the censoring mechanism are required.

Our assumption (A0) balances somewhere in between these two extreme assumptions.

By imposing assumption (A0), we propose a new dimension reduction technique, which overcomes the drawbacks of these two classical sets of assumptions, by allowing for d ≥ 1 without assuming the complete independence between Y and C. Note that assumption (A0) holds in the particular case where L(C|X, Y ) = L(C|g(X)). By assuming that the censoring variable depends on X only through a one-dimensional variable g(X), we avoid the curse-of-dimensionality problems which strike regression approaches where X is multivariate and Y is independent of C conditionally on X, and at the same time the dependence of C on X is not too restrictive. A related dimension reduction model assumption for the censoring time has been considered in Section 4 of [START_REF] Li | Dimension reduction for censored regression data[END_REF].

In some cases, the function g will be known exactly from some a priori information.

For example, we might know that the censoring only depends on one component of X, for example g(X) = X (1) . Lopez (2007a) proposed an estimator of the joint distribution F (x, y) when g is supposed to be known. However, in many other cases, g will be unknown and needs to be estimated. Throughout this paper, we will assume that

g ∈ G, where G = {x → λ(θ, x) : θ ∈ Θ}, (1.1)
where λ is a known function, and Θ is a compact parameter set in R k . The true (but unknown) value of θ will be denoted by θ 0 .

Throughout the paper, we will assume that we know some root-n consistent estimator θ of θ 0 , that satisfies the following :

(C0) The estimator θ satisfies :

θ -θ 0 = 1 n n i=1 µ(T i , δ i , X i ) + o P (n -1/2 ), with E[µ(T, δ, X)] = 0 and E[µ(T, δ, X) 2 ] < ∞.
Hence, the set Θ can from now on be taken equal to an arbitrarily small environment of θ 0 .

To illustrate the nature of assumptions (A0) and (C0), consider the function g(x) = θ ′ 0 x, and the case where C follows a Cox regression model given X, in the sense that the conditional hazard h(•|x, y) of C given X = x and Y = y satisfies

h(c|x, y) = h 0 (c) exp(θ ′ 0 x)
for some baseline function h 0 only depending on c. Note that this model assumption on C is not unrealistic, since often the censoring variable C represents itself a lifetime, like the time until a patient dies from a disease other than the disease under study. Under this model, we clearly have L(C|X, Y ) = L(C|θ ′ 0 X), and the estimator θ proposed by [START_REF] Andersen | Cox's regression model for counting processes : A large sample study[END_REF] satisfies condition (C0), with

µ(t, δ, x) = Σ -1 (1 -δ)φ(x, t) -φ(x, u)1 t>u [1 -G(u -|x)] -1 dG(u|x) ,
where the matrix Σ is defined by condition D in [START_REF] Andersen | Cox's regression model for counting processes : A large sample study[END_REF],

φ(x, t) = x - E[Xe θ ′ 0 X (1 -H(t|X))] E[e θ ′ 0 X (1 -H(t|X))]
,

with H(t|x) = P(T ≤ t|X = x) and G(c|x) = P(C ≤ c|X = x). See also [START_REF] Gorgens | Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable[END_REF] for regression models more general than Cox in which L(C|X, Y ) = L(C|θ ′ 0 X). Alternatively, one could also assume that C = r(θ ′ 0 X) + U, where r(•) is given, E(U) = 0, and U is independent of X and Y . For the estimation of θ 0 and the verification of condition (C0) under this model, see e.g. [START_REF] Akritas | The least squares method in heteroscedastic censored regression models[END_REF] and [START_REF] Heuchenne | Polynomial regression with censored data based on preliminary nonparametric estimation[END_REF].

The purpose of this paper is twofold. The first contribution of this paper consists in proposing and studying a new nonparametric estimator of the joint distribution of X and Y under assumption (A0). Under different sets of assumptions on the relation between X, Y and C, this distribution has been the object of study of many papers in the past. See e.g. [START_REF] Akritas | Nearest neighbor estimation of a bivariate distribution under random censoring[END_REF], [START_REF] Stute | Consistent estimation under random censorship when covariables are present[END_REF][START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF], [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], among others. As mentioned before, assumption (A0) allows to avoid the curse-of-dimensionality problem present in some of these contributions, and the heavy assumptions on the relation between C and X, which are present in many others.

The second contribution of this paper is the estimation of a semiparametric single index regression model for the censored response Y given X under assumption (A0).

The proposed estimator is based on a two-step procedure, in which first a preliminary (consistent) estimator is obtained, which is then used to obtain in a second step a more accurate estimator. Both steps heavily rely on the estimator of F (x, y) studied before.

Note that in this second contribution two dimension reduction techniques are used : the first one comes from assumption (A0), which is concerned with the relation between Y and C, and the second one comes from the single index model, which is making an hypothesis on the relation between Y and X.

Single index regression models are now a common semiparametric multivariate explanatory approach, see for instance [START_REF] Delecroix | On semiparametric M-estimation in single-index regression[END_REF] for a review.

However, the literature on single index models with a censored response variable is rather poor. To the best of our knowledge, the only contribution that allows for a general relationship between the censoring variable and the covariates is Li, [START_REF] Li | Dimension reduction for censored regression data[END_REF] and it is based on sliced inverse regression (SIR). However, it is well known that the SIR approach requires a linear conditional expectation condition among the covariates, which may be rather restrictive in some applications, see equation (2.3) in Li, [START_REF] Li | Dimension reduction for censored regression data[END_REF]. Lopez (2008) proposed a semiparametric least squares estimator for the single index regression in the particular case where g(X) ≡ 1 in assumption (A0). A similar procedure was introduced by [START_REF] Wang | Asymptotics for a censored generalized linear model with unknown link function[END_REF] under the stronger assumption that C is independent of (X ′ , Y ) ′ . See also [START_REF] Lu | Randomly censored partially linear single-index models[END_REF]. [START_REF] Lu | Censored multiple regression by the method of average derivatives[END_REF] used the same more restrictive condition to define an average derivative estimator of the index. It is worthwhile to notice that these three contributions involve a Kaplan-Meier estimate of the censoring distribution, while in general assumption (A0) requires a nonparametric estimate of the conditional distribution of C given g(X).

This paper is organized as follows. In the next section the estimators of the joint distribution and of the parameters in the single index model are explained in detail. Section 3 is devoted to the presentation of the asymptotic results of the proposed estimators. Finally, Appendix A contains the assumptions under which the results of Section 3 are valid, while Appendix B contains some technical lemmas and the proofs of the main results.

The estimators

Estimation of the distribution F (x, y)

We first explain how to estimate the joint distribution F (x, y) of X and Y . For an arbitrary value of θ, let

G θ (t|z) = P(C ≤ t|λ(θ, X) = z),
(2.1) and define

Ĝθ (t|z) = 1 -

T i ≤t 1 - w θ in (z) n j=1 w θ jn (z)1 T j ≥T i 1-δ i , (2.2)
where

w θ in (z) = K λ(θ,X i )-z an n j=1 K λ(θ,X j )-z an .
Here, a n is a bandwidth sequence converging to zero as n tends to infinity, and K is a probability density function (kernel). Note that Ĝθ (t|z) reduces to the estimator proposed by [START_REF] Beran | Nonparametric regression with randomly censored survival data[END_REF] when λ(θ, X) is equal to X.

With at hand the estimator θ introduced in condition (C0), and the corresponding estimator ĝ(x) = λ( θ, x) of g(x), we now define the following estimator of F (x, y) :

Fĝ (x, y) = 1 n n i=1 δ i 1 T i ≤y,X i ≤x 1 -Ĝθ (T i -|ĝ(X i )) . (2.3)
Note that this estimator is in the same spirit as the estimator proposed by [START_REF] Stute | Consistent estimation under random censorship when covariables are present[END_REF][START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF], but the denominators of the two estimators are different, because of the different sets of underlying assumptions. See also [START_REF] Fan | Censored regression: local linear application and their applications[END_REF] for a similar weighting scheme in a nonparametric regression framework. Also note that when g would be known, this estimator equals the estimator proposed and studied in Lopez (2007b).

In Section 3.1 we will study the asymptotic properties of the estimator Fĝ (x, y).

Estimation of the single index model

We first need to introduce some notations. For θ ∈ Θ, let Z θ = λ(θ, X), and let Z θ ⊂ R be the support of the variable Z θ . We assume that Z θ is compact for all θ ∈ Θ. Also, define H θ (t|z) = P(T ≤ t|Z θ = z) and let τ H θ ,z = inf{t : H θ (t|z) = 1}.

We assume that the following single index mean regression model is valid : for some

β 0 ∈ B ⊂ R d , with, say, first component β (1) 0 = 1, E[Y | X, Y ≤ τ ] = E[Y | β ′ 0 X, Y ≤ τ ] = m(β ′ 0 X), (2.4)
where m is an unknown function, and where τ is some fixed truncation point, satisfying

τ < inf θ∈Θ inf z∈Z θ τ H θ ,z . Let f (t; β) = E[Y |β ′ X = t, Y ≤ τ ]. Then, f (•; β 0 ) = m(•). Also, let B = {1} × B, where B is a compact subset of R d-1
, and denote by X the support of the covariate vector X,

which is a compact subset of R d .
The truncation at τ in model (2.4) is very natural and common in the context of regression with right censored observations, and is caused by the lack of information in the right tail of the conditional distribution of Y given X. See e.g. [START_REF] Akritas | Nearest neighbor estimation of a bivariate distribution under random censoring[END_REF] and [START_REF] Akritas | The least squares method in heteroscedastic censored regression models[END_REF] for similar truncation mechanisms. Note that when

L(Y |X) = L(Y |β ′ 0 X), i.e.
when the whole distribution of Y given X only depends on X via β ′ 0 X, then model (2.4) is satisfied for any value of τ . The estimation of β 0 consists of several steps. We first explain these steps in an informal, intuitive way to outline the main ideas behind the proposed method, and we next work out each of these steps in a rigorous way.

1. Estimate f (t; β) using some nonparametric estimator f (t; β).

2. Construct a preliminary consistent estimator β n of β 0 .

3. Use β n to compute a trimming function. This trimming function avoids technical problems caused by denominators close to zero in the nonparametric estimation of

f (t; β).
4. Construct a second semi-parametric estimator β of β 0 by using the trimming function of the preceding step.

Estimation of f (t; β)

One possible estimator of f (t; β) is

f (t; β) = K( β ′ x-t h )y1 y≤τ d Fĝ (x, y) K( β ′ x-t h )1 y≤τ d Fĝ (x, y) , (2.5)
where h = h n is a second bandwidth sequence, possibly different from the bandwidth a n used to estimate the joint distribution F (x, y), and where K is a kernel function.

However, other estimators may be used, for example [ Fβ (τ |t)] -1 y1 y≤τ d Fβ (y|t), where Fβ (y|t) denotes [START_REF] Beran | Nonparametric regression with randomly censored survival data[END_REF]

estimator of P(Y ≤ y|β ′ X = t).
In what follows, we do not specify the choice of estimator of f (t; β). Instead we will work with a generic estimator f (t; β) that satisfies certain conditions that need to be fulfilled in order to obtain the asymptotic normality of β, and we will prove in Section 3.2 that the estimator in (2.5) satisfies these conditions.

Preliminary estimation of β 0

We assume that we know some set B such that

inf β∈B,x∈B f τ β (β ′ x) = c > 0,
where the function f τ β denotes the density of β ′ X, conditionally on Y ≤ τ . Define the following preliminary trimming function :

J(x) = 1 x∈B . (2.6) Let M(β, f, J) = E[(Y -f (β ′ X; β)) 2 1 Y ≤τ J(X)],
and note that this is minimized as a function of β when β = β 0 . Motivated by this fact, we define the preliminary estimator β n of β 0 by replacing all unknown quantities in M(β, f, J) by appropriate estimators, i.e.

β n = arg min β∈B (y -f (β ′ x; β)) 2 1 y≤τ J(x)d Fĝ (x, y) = arg min β∈B M n (β, f , J).
(2.7)

Note that other criterion functions can be used, based on M or L-estimating functions.

We do not consider them here, since their analysis is very similar to the one for the least squares criterion function.

New trimming function

We will now refine the definition of the trimming function, by using the preliminary estimator β n . Define

J(x) = 1 f τ βn (β ′ n x)>c , (2.8)
so instead of requiring that f τ β (β ′ x) > c for all β, we now only consider β = β n , which will be satisfied for many more x-values, and hence this new function J(x) is trimming much less than the preliminary naive trimming function J(x).

To simplify our discussion, we will directly consider that the true function f τ βn is used in the definition of J. In practice, the trimming function can be estimated by

1 f τ βn (β ′ n x)>c , where f τ β (t) = 1 nb n P(Y ≤ τ ) n i=1 δ i 1 T i ≤τ 1 -Ĝθ (T i -|ĝ(X i )) K β ′ X i -t b n ,
and where b n → 0 is a bandwidth parameter. In applications, c 1 = cP(Y ≤ τ ) can be chosen arbitrarily small by the statistician. Considering f τ βn or f τ βn does not change anything asymptotically speaking, see the arguments in [START_REF] Delecroix | On semiparametric M-estimation in single-index regression[END_REF], see also Step 0 in the proof of Theorem 3.5 below. By similar arguments, the estimator of

β 0 obtained with 1 f τ βn (β ′ n x
)>c is asymptotically equivalent to the 'ideal' estimator obtained with the trimming function

J 0 (x) = 1 f τ β 0 (β ′ 0 x)>c ,
(2.9) as long as β n is a consistent estimator of β 0 . Let us point out that J 0 only depends on β ′ 0 x and, in view of equation (A.14) in the proof of Theorem 3.5, this property will be essential for achieving √ n-asymptotic normality of our estimator β defined below.

Estimation of β 0

With at hand this new trimming function, we can now define a new semi-parametric least squares estimator of β 0 :

β = arg min β∈Bn (y -f (β ′ x; β)) 2 1 y≤τ J(x)d Fĝ (x, y) (2.10) = arg min β∈Bn M n (β, f , J),
where B n is a set shrinking to {β 0 }, which is computed from the preliminary step. The proof of the asymptotic normality of β will be carried out in two steps. We will first show that minimizing M n (β, f, J) is asymptotically equivalent to minimizing M n (β, f, J 0 ). This then brings back the minimization problem to a fully parametric one.

3 Asymptotic properties

Estimation of the distribution F (x, y)

Let us first introduce a few notations. Denote

H(t) = P(T ≤ t), H θ (t|z) = P(T ≤ t|Z θ = z), H θ,0 (t|z) = P(T ≤ t, δ = 0|Z θ = z), and H θ,1 (t|z) = P(T ≤ t, δ = 1|Z θ = z). For any function L(u), let ∇ u L(u) (respectively ∇ 2 u,u L(u)
) denote the vector (respectively matrix) of partial derivatives of order 1 (respectively order 2) of L(u) with respect to u.

In particular, denote by ∇ θ G θ (t|λ(θ, x)) the vector of partial derivatives of the function G θ (t|λ(θ, x)) with respect to all occurrences of θ. Let us point out that, in general, the vector valued function ∇ θ G θ (t|λ(θ, x)) depends on x, and not only on λ(θ, x). Finally, for

any matrix A of dimensions k × ℓ (where k, ℓ ≥ 1) we denote |A| = [trace(A ′ A)] 1/2 .
We further need to introduce two (intermediate) estimators of F (x, y) :

Fg (x, y) = 1 n n i=1 δ i 1 T i ≤y,X i ≤x 1 -G θ 0 (T i -|g(X i )) , (3.1) Fg (x, y) = 1 n n i=1 δ i 1 T i ≤y,X i ≤x 1 -Ĝθ 0 (T i -|g(X i )) . (3.2)
In the following result we consider integrals of the form φ (x, y) d Fg (x, y) with φ belonging to some class of functions F , and we state that this class of integrals is Glivenko-Cantelli and admits an i.i.d. representation uniformly over all φ ∈ F . The proof can be found in Lopez (2007b). For a completely nonparametric estimator of F (x, y) that is not based on model assumption (A0), Sánchez-Sellero, González-Manteiga and Van Keilegom (2005) obtained a similar uniform consistency and convergence result. The assumptions mentioned below can be found in Appendix A.

Theorem 3.1 i) Under Assumptions 1 and 3, for a n → 0 and na n → ∞, and for a class

F satisfying Condition 1, we have sup φ∈F φ (x, y) d[ Fg -F ] (x, y) → a.s. 0. ii) For Z i = λ(θ 0 , X i ), define M i (t) = (1 -δ i )1 T i ≤t - t -∞ 1 T i ≥y dG θ 0 (y|Z i ) 1 -G θ 0 (y -|Z i ) ,
which is a continuous time martingale with respect to the natural filtration

σ({Z i 1 T i ≤t , T i 1 T i ≤t , δ i 1 T i ≤t , i = 1, ..., n}).
Under Assumptions 1-4 and for a class F satisfying Conditions 2-3,

φ (x, y) d[ Fg -Fg ] (x, y) = 1 n n i=1 φ(Z i , s)dM i (s) [1 -F (s-| Z i )][1 -G θ 0 (s | Z i )] + R n (φ) ,
where

sup φ∈F |R n (φ)| = o P (n -1/2
), φ is defined above Condition 3, and

F (s|z) = P(Y ≤ s|Z θ 0 = z).
The following Theorem furnishes the behavior of the difference between integrals with respect to Fĝ and integrals with respect to Fg .

Theorem 3.2 i) Under Assumptions 1, 3 and 5, for a n → 0 and na n → ∞, and for a class F satisfying Condition 1, we have

sup φ∈F φ(x, y)d[ Fĝ -Fg ](x, y) = o P (1).
ii) Under Assumptions 1-3 and 5, for a n → 0 and na 3 n (log n) -1 → ∞, and for a class F whose envelope is as in Condition 1,

φ(x, y)d[ Fĝ -Fg ](x, y) = -E φ(X, Y ){∇ θ G θ 0 (Y -|λ(θ 0 , X))} ′ 1 -G θ 0 (Y -|g(X)) 1 n n i=1 µ(T i , δ i , X i ) + Rn (φ),
where the function µ is defined in (C0), and where sup φ∈F | Rn (φ)| = o P (n -1/2 ).

Estimation of the single index model

We now return to the single index model (2.4) and to the estimators β n and β defined in (2.7) and (2.10). We start with stating the asymptotic consistency of the estimator β n .

Note that the estimator β is by construction consistent, since it is defined on a shrinking neighborhood of β 0 .

Theorem 3.3 Let J be defined as in (2.6). Under Assumptions 1, 3, 5, 7, and 9-(A.1), and for a n → 0 and na n → ∞, we have

sup β∈B |M n (β, f , J) -M(β, f, J )| → 0, in probability. Consequently, β n → β 0 in probability.
The next lemma is an important property in the literature on single index models. In the classical uncensored single index regression model, the property Lemma 3.4 Assume that the derivative ∇ β f (β ′ 0 •; β 0 ) exists and is bounded. Then, for any β 0 satisfying condition (2.4),

E[∇ β f (β ′ 0 X; β 0 )|β ′ 0 X] = 0
E [∇ β f (β ′ 0 X; β 0 )1 Y ≤τ | β ′ 0 X] = 0.
This lemma is crucial for obtaining our i.i.d. representation and the asymptotic normality of β, which we state in the next Theorem. We denote by ∇ β f (β ′ 0 •; β 0 ) the vector of partial derivatives with respect to the last d -1 components of β.

Theorem 3.5 Let φ(x, y) = (y -f (β ′ 0 x; β 0 ))∇ β f (β ′ 0 x; β 0 )1 y≤τ J 0 (x).
Under Assumptions 1-11, we have

β -β0 = Ω -1 φ(x, y)d( Fg (x, y) -F (x, y)) (3.3) + 1 n n i=1 φ(g(X i ), s)dM i (s) [1 -F (s -|g(X i ))][1 -G θ 0 (s|g(X i ))] -E φ(X, Y ){∇ θ G θ 0 (Y -|λ(θ 0 , X))} ′ 1 -G θ 0 (Y -|g(X)) 1 n n i=1 µ(T i , δ i , X i ) + o P (n -1/2 ) = Ω -1 1 n n i=1 η(T i , δ i , X i ) + o P (n -1/2 ),
where the function µ is defined in (C0), and where

Ω = E 1 Y ≤τ J 0 (X)∇ β f (β ′ 0 X; β 0 )∇ β f (β ′ 0 X; β 0 ) ′ .
Hence,

n 1/2 ( β -β0 ) d → N 0, Ω -1 E[η(T, δ, X)η(T, δ, X) ′ ]Ω -1 .
If we wish to estimate the asymptotic variance in (3.5), we see that we need to estimate the variance of Ω -1 η. However, one can estimate consistently Ω by

Ω = 1 n n i=1 1 Y ≤τ J(X i )∇ β f ( β′ X i ; β)∇ β f ( β′ X i ; β) ′ .
Similarly, when it comes to estimate the covariance matrix of η, one can proceed by taking the empirical variance of a random vector (η(T i , δ i , X i )) 1≤i≤n , where η denotes an estimated version of η in which we replaced each unknown quantity by its empirical counterpart (f replaced by f , β 0 by β, F by F ...).

We end this Section with the verification of Assumptions 9-11 for the estimator f (t; β) defined in (2.5). Define the (uncomputable) kernel estimator based on Fg ,

f * (t; β) = K( β ′ x-t h )y1 y≤τ d Fg (x, y) K( β ′ x-t h )1 y≤τ d Fg (x, y) . (3.4)
The advantage of Fg , and hence of f * , is that it is composed of sums of i.i.d. terms. Classical arguments show that f * satisfies Assumptions 9 to 11. This is shown in Proposition 3.6 below. On the other hand, Proposition 3.7 shows that the difference between f and f * is sufficiently small so that f also satisfies these assumptions.

Proposition 3.6 Assume that (i) K is a symmetric density function with compact support, and with two continuous derivatives of bounded variation;

(ii) f (•; β 0 ) ∈ H 0 1 and ∇ β f (β ′ 0 •; β 0 ) ∈ H 0 2
, with H 0 1 and H 0 2 defined in (A.4) and (A.5); (iii) nh 5 (log n) -1/2 → ∞, and nh 8 → 0, Then, f * satisfies Assumptions 9-11.

Proposition 3.7 Under the assumptions of Theorem 3.2, we have

sup β∈B,x∈X |f * (β ′ x; β) -f (β ′ x; β)| = O P ((log n) 1/2 n -1/2 a -1/2 n ), sup β∈B,x∈X |∇ β f * (β ′ x; β) -∇ β f (β ′ x; β)| = O P ((log n) 1/2 h -1 n -1/2 a -1/2 n ), sup β∈B,x∈X |∇ 2 β,β f * (β ′ x; β) -∇ 2 β,β f (β ′ x; β)| = O P ((log n) 1/2 h -2 n -1/2 a -1/2 n ),
where f is the estimator defined in (2.5). Moreover,

∇ β f (β ′ 0 x; β 0 ) = x m1 (β ′ 0 x)+ m2 (β ′ 0 x), with, for j = 1, 2, sup x∈X | mj (β ′ 0 x) -m * j (β ′ 0 x)| = O P ((log n) 1/2 h -1 n -1/2 a -1/2 n ), sup u∈β ′ 0 X | m′ j (u) -m * ′ j (u)| = O P ((log n) 1/2 h -2 n -1/2 a -1/2 n ),
where the functions m * j are defined in (A.15), and where m ′ denotes the derivative of the univariate function β ′ 0 X ∋ u → m(u).

Notice that f ′ (u; β 0 ) = m1 (u) (resp. f * ′ (u; β 0 ) = m * 1 (u)). Combining Propositions 3.6 and 3.7 shows that f satisfies Assumptions 9-11 provided that nh 8 → 0, na n h 4 (log n) -1 → ∞ and ha

-1/2 n (log n) 1/2 → 0. In the case where a n = n -1/[4-δ] for some δ ∈]0, 1[, these con- ditions are satisfied if nh 4 4-δ 3-δ (log n) -4-δ 3-δ → ∞ and nh 8-2δ (log n) 4-δ → 0.

Simulation study

To investigate small sample behaviour of our procedure, we considered to different models to perform a simulation study. In the first model, we consider the regression function

m 1 (β ′ 0 x) = β ′ 0 x -0.5(β ′ 0 x) 2 ,
m 2 (β ′ 0 x) = log (1 + 0.5β ′ 0 x) ,
with β 0 = (1, 0.75, 0.25, -0.5). We consider residuals ε i = Y im j (β ′ 0 X i ) (for j = 1, 2) that are Gaussian variables N (0, 1) independent from X i . The covariates are composed of 4 independent components, following an uniform distribution on [0, 1].

Concerning the censoring variable, we take

C i |X i ∼ E(λ exp θ ′ 0 X i )
, where θ 0 = (0.1, 0.2, -0.1, 0.3), and λ is a parameter that allows us to modify the average proportion of censored responses. The parameter θ 0 is estimated through maximizing Cox pseudo-likelihood, since the regression model on C i is a proportional hazards model.

We consider 10000 replications of this simulation scheme for n = 200. For each simulated sample j, we compute the resulting estimator β(j) of β 0 and compute β(j) -β 0 2 2 . We then deduce an estimator of the mean square error (MSE) E[ ββ 0 2 2 ]. We took a n = 2 for the bandwidth involved in Beran's estimator. Since the procedure is more sensitive to the choice of the second bandwidth h, we consider a set of bandwidths h j = 0.5 + j0.1, for j = 1, ..., 10, and for each sample, we take the bandwidth that gives the lowest value of M n . In Table 1, we compare the MSE of the estimator that we propose to the MSE of an estimator based on Kaplan-Meier weights, that is if we replace Beran's estimator in our approach by a standard Kaplan-Meier estimator. This alternative estimator is the one defined in [START_REF] Lopez | Single-index Regression models with right-censored responses[END_REF]. As for our approach, this estimator puts more weights to the largest uncensored observations caused by censoring. Nevertheless this alternative procedure is not adapted to Assumption (A0) that we use in the present framework. Therefore, the estimator of [START_REF] Lopez | Single-index Regression models with right-censored responses[END_REF] is expected to fail in this simulation setting.

As expected, our estimator based on conditional Kaplan-Meier weighting outperforms the estimator of [START_REF] Lopez | Single-index Regression models with right-censored responses[END_REF] in the different situations we consider. It is also natural to observe that the MSE of our estimator β decreases when the proportion of censoring increases.

Appendix A : Assumptions and conditions

We split the assumptions in three parts, namely those required for the estimation of F (x, y), the estimation of β 0 , and the estimation of f (•; β).

Assumptions needed for the estimation of F (x, y). The asymptotic results related to the estimator Fĝ (x, y) will be valid under the following assumptions and conditions.

Assumption 1 The distribution P(Z θ ≤ z) has three uniformly bounded derivatives for z ∈ Z θ and θ ∈ Θ, and the densities

f Z θ (z) satisfy inf θ∈Θ inf z∈Z θ f Z θ (z) > 0.
For any function J(t | z) we will denote by J c (t | z) the continuous part, and

J d (t | z) = J(t | z) -J c (t | z).
Assumption 2 below has been introduced by [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF] to obtain their asymptotic i.i.d. representation of the conditional Kaplan-Meier estimator.

Assumption 2 (i) Let L(y|z) denote H θ 0 (y|z) or H θ 0 ,0 (y|z). Then, ∇ z L(y|z) and ∇ 2 z,z L(y|z) exist, are continuous with respect to z, and are uniformly bounded as functions of (z, y).

(ii) For some positive nondecreasing bounded (on [-∞; τ ]) functions L 1 , L 2 , L 3 , we have, for all z ∈ Z θ 0 , (v) For all d i , define

|H θ 0 c (t 1 | z) -H θ 0 c (t 2 | z)| ≤ |L 1 (t 1 ) -L 1 (t 2 )| , |∇ z H θ 0 c (t 1 | z) -∇ z H θ 0 c (t 2 | z)| ≤ |L 2 (t 1 ) -L 2 (t 2 )| , |∇ z H θ 0 ,0c (t 1 | z) -∇ z H θ 0 ,0c (t 2 | z)| ≤ |L 3 (t 1 ) -L 3 (t 2 )| ,
s i = sup z∈Z θ 0 |F g (d i -| z) -F g (d i | z)| , s ′ i = sup z∈Z θ 0 |∇ z F g (d i -| z) -∇ z F g (d i | z)| , r i = sup z∈Z θ 0 |G θ 0 (d i -| z) -G θ 0 (d i | z)| , r ′ i = sup z∈Z θ 0 |∇ z G θ 0 (d i -| z) -∇ z G θ 0 (d i | z)| .
Then,

d i ≤τ (s i + s ′ i + r i + r ′ i ) < ∞.
Assumption 3 The kernel K is a symmetric probability density function with compact support, and K has bounded second derivative.

Assumption 4 The bandwidth a n satisfies (log n)n -1 a -3 n → 0 and na 4 n → 0.

Assumption 5 The function (x, t, θ) → G θ (t|λ(θ, x)) is differentiable with respect to θ,

and the vector ∇ θ G θ (t|λ(θ, x)) is uniformly bounded in (x, t, θ).
The class of functions F considered in Section 3.1 should satisfy the following conditions, which are taken over from Lopez (2007b). The conditions make use of concepts from the context of empirical processes, which can be found e.g. in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF].

Condition 1 Let p 0 (x, y, c) = 1 y≤c [1 -G θ 0 (y -|g(x))] -1 . The class p 0 F is P (X,Y,C) -
Glivenko-Cantelli, and has an integrable envelope Φ 0 satisfying Φ 0 (x, y, c) = 0 for y > τ .

Condition 2 The covering number N(ε, F , L 2 (P (X,Y ) )) is bounded by Aε -V for ε > 0 and for some A, V > 0, and F has a square integrable envelope Φ satisfying Φ(x, y) = 0 for y > τ .

Let Z = Z θ 0 = g(X), let F z (x, y) = P(X ≤ x, Y ≤ y|Z = z)
, and for any function φ(x, y), define φ(z, s) = 1 s≤y φ(x, y)dF z (x, y). Let Z θ 0 ,η be the set of all points at a distance at least η > 0 from the complementary of Z θ 0 .

Condition 3 For all φ ∈ F , φ is twice differentiable with respect to z, and

sup s≤τ,z∈Z θ 0 ,η {|∇ z φ(z, s)| + |∇ 2 z,z φ(z, s)|} ≤ M < ∞,
for some constant M not depending on φ. Moreover, Φ is bounded on

Z θ 0 ,η ×] -∞; τ ],
and has bounded partial derivatives with respect to z, where Φ is the envelope function of Condition 2.

The reason for introducing the set Z θ 0 ,η is to prevent us from boundary effects coming from kernel estimators. See Lopez (2007b) for a detailed discussion on this issue.

Assumptions needed for the estimation of β 0 . We next state the additional assumptions needed for the asymptotic results concerning the estimation of the parameters in the single index model.

Assumption 6 There exist 0 < c 0 < c 1 < ∞ and η > 0 such that, for each c ∈ [c 0 , c 1 ] and x ∈ X ,

1 f τ β 0 (β ′ 0 x)>c = 1 =⇒ g(x) ∈ Z θ 0 ,η . Moreover, assume that |f τ β 1 (β ′ 1 x) -f τ β 2 (β ′ 2 x)| ≤ C β 1 -β 2 α ,
for some positive constant C and some α > 0.

Assumption 7 Assumptions needed for the estimation of f (•; β). The last group of assumptions is required for the generic estimator f (•; β). They are verified in Section 3.2 for the estimator defined in (2.5).

(i) E(|Y | 3 ) < ∞; (ii) E [{f (β ′ X; β) -f (β ′ 0 X; β 0 )} 2 1 Y ≤τ ] = 0 =⇒ β = β 0 ; (iii) β 0 = (1, β′ 0 ) ′ with β0 an interior point of B; (iv) The class {(x, y) → f (β ′ x; β)1 y≤τ : β ∈ B} satisfies
Assumption 9 For all c > 0,

sup β∈B,x∈X | f (β ′ x; β) -f (β ′ x; β)|1 f τ β (β ′ x)>c = o P (1), (A.1) sup β∈B,x∈X |∇ β f (β ′ x; β) -∇ β f (β ′ x; β)|1 f τ β (β ′ x)>c = o P (1), (A.2) sup β∈B,x∈X |∇ 2 β,β f (β ′ x; β) -∇ 2 β,β f (β ′ x; β)|1 f τ β (β ′ x)>c = o P (1). (A.

3)

Assumption 10 There exist Donsker classes H 1 and H 2 such that f (•;

β 0 ) ∈ H 1 and ∇ β f (β ′ 0 •; β 0 ) ∈ H 2
, and such that with probability tending to one, f (•;

β 0 ) ∈ H 1 and ∇ β f (β ′ 0 •; β 0 ) ∈ H 2 .
Lemma A.1 Assume that the class of functions

(u, z, t, δ) → K 0 ψ(α, u) -ψ(α, x) b n ζ α (x, u, z, t, δ) : α ∈ A, x ∈ X (A.7)
is a VC-class of functions for a constant envelope, assume that E[|ξ(T )| 3 ] < ∞, and that

nb 3 n /(log n) → ∞. Then, n -1/2 b -1/2 n [log(1/b n )] -1 ν n G = O P (1),
where • G denotes the uniform norm over all maps in G.

Remark. Note that if K 0 is of bounded variation with compact support, and if ψ(α, x) = α ′ x, then (A.7) holds, see [START_REF] Nolan | U-processes : rates of convergence[END_REF].

Proof of Lemma A.1. As the class G is not necessarily uniformly bounded, introduce a truncation bound M n , and consider the class G n of functions g [START_REF] Einmahl | Uniform in bandwidth consistency of kernel-type function estimators[END_REF] to the class of functions G n . Their condition 1 holds, taking the envelope

(n) α,x,v (x, z, t, δ) = g α,x,v (x, z, t, δ)1 |ξ(t)|≤Mn . We set M n = (nb n ) 1/2 . Apply Proposition 1 in
G(u, z, t, δ) = M n K 0 ∞ , and β = c 1 b 1/2
n , for some c 1 > 0. Condition 2 holds as the class G n is VC. Indeed, the class t → 1 t≤v indexed by v ∈ R is VC (see Example 19.6 in [START_REF] Van Der Vaart | Asymptotic Statistics[END_REF], and hence (A.7) and Lemma 2.14 (ii) in [START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF] yield that G n satisfies their condition 2, whereas condition 3 holds for σ 0 = σ = c 2 b n , for some c 2 > 0. For condition 4, we have

sup g∈Gn g ∞ ≤ M n K 0 ∞ ≤ c 3 √ nσ 2 ,
for some constant c 3 > 0.

Applying Proposition 1 in [START_REF] Einmahl | Uniform in bandwidth consistency of kernel-type function estimators[END_REF], we can deduce that, for any M < ∞ sufficiently large, for all u ′ > 0 and for some A 1 > 0,

P ν n Gn ≥ M(nb n ) 1/2 (log b -1 n ) 1/2 ≤ P ν n Gn ≥ A 1 E sup g∈Gn n i=1 ε i g(W i ) + u ′ ,
where Talagrand's inequality (1994), see also [START_REF] Einmahl | Uniform in bandwidth consistency of kernel-type function estimators[END_REF]. Taking u ′ = nb n log b -1 n , deduce that

W i = (X ′ i , Z i , T i , δ i ) ′ and ε 1 , • • • , ε n is a sequence of independent Rademacher ran- dom variables, independent of W 1 , • • • , W n . Now we can apply
ν n Gn = O P (nb n ) 1/2 (log b -1 n ) 1/2 .
Now, for some c 4 > 0,

ν n G ≤ ν n Gn + c 4 n i=1 |ξ(T i )|1 |ξ(T i )|≥Mn .
The second term on the right hand side is of the order

O P (b -1 n ) = o P ((log n) 1/2 n 1/2 b 1/2 n )
. Indeed, its expectation can be bounded, using Hölder's inequality, by

nE[|ξ(T )| 3 ] 1/3 P(|ξ(T )| ≥ M n ) 2/3 . Using Tchebyshev's inequality, P(|ξ(T )| ≥ M n ) ≤ E |ξ(T )| 3 M 3 n = O(n -3/2 b -3/2 n ).
The second technical lemma shows the consistency of the estimator Ĝθ (t|λ(θ, x)) and its vector of partial derivatives, uniformly in t, θ and x, and it also establishes the rate of convergence of the estimator Ĝθ (t|ĝ(x)), uniformly in t and x. Proof. For the first part, with probability tending to 1, for t ≤ τ, 1 -Ĝ(t|λ(θ, x)) > 0.

Taking the logarithm, one obtains

log(1 -Ĝ(t|λ(θ, x))) = n i=1 (1 -δ i )1 T i ≤t log (1 -W n,i (x, θ)) ,
where

W n,i (x, θ) = W n (X i , T i ; x, θ) = K λ(θ,X i )-λ(θ,x) an n j=1 1 T j ≥T i K λ(θ,X j )-λ(θ,x) an .
A Taylor expansion leads to

log(1 -Ĝ(t|λ(θ, x))) = - n i=1 (1 -δ i )W n,i (x, θ)1 T i ≤t + O P (n -1 a -2 n ),
where the order of the remainder term is uniform in t, θ, x, as

sup i:T i ≤τ sup x,θ |W n,i (x, θ)| = O P (n -1 a -1 n ).
The remainder term is o

P (1) if na 2 n → ∞. Rewrite n i=1 (1 -δ i )1 T i ≤t W n,i (x, θ) = 1 na n n i=1 (1 -δ i )1 T i ≤t K λ(θ, X i ) -λ(θ, x) a n S θ (λ(θ, x), T i ) -1 + 1 na n n i=1 (1 -δ i )1 T i ≤t K λ(θ, X i ) -λ(θ, x) a n × Ŝθ (λ(θ, x), T i ) -S θ (λ(θ, x), T i ) S θ (λ(θ, x), T i ) Ŝθ (λ(θ, x), T i ) ,
where

S θ (λ(θ, x), y) = [1 -H θ (y|λ(θ, x))]f Z θ (λ(θ, x)), Ŝθ (λ(θ, x), y) = 1 na n n j=1 1 T j ≥y K λ(θ, X j ) -λ(θ, x) a n .
Apply Lemma A.1 to obtain the uniform convergence of Ŝθ towards S θ , and to show that sup x,θ∈Θ,t≤τ

1 na n n i=1 (1 -δ i )1 T i ≤t K λ(θ, X i ) -λ(θ, x) a n S θ (λ(θ, x), T i ) -1 - t -∞ dH θ,0 (s|λ(θ, x)) 1 -H θ (s -|λ(θ, x)) = o P (1).
Since S θ is uniformly bounded away from zero for y ≤ τ, see Assumption 1, the result follows from

exp - t -∞ dH θ,0 (s|λ(θ, x)) 1 -H θ (s -|λ(θ, x)) = 1 -G θ (t|λ(θ, x)).
For the gradient, we have

∇ θ Ĝθ (t|λ(θ, x)) = (1 -Ĝθ (t|λ(θ, x))) n i=1 (1 -δ i )1 T i ≤t ∇ θ W n,i (x, θ) 1 -W n,i (x, θ) .
From this, we deduce that the convergence of ∇ θ Ĝθ follows from the convergence of Ĝθ , of Ŝθ and of

1 na 2 n n i=1 (1 -δ i )1 T i ≤t ∇ θ λ(θ, x)K ′ λ(θ, X i ) -λ(θ, x) a n , and
1 na 2 n n i=1 1 T i ≤t ∇ θ λ(θ, x)K ′ λ(θ, X i ) -λ(θ, x) a n .
These two quantities can be studied using Lemma A.1, which shows that their centered versions converge uniformly with rate (na 3 n ) -1/2 log n, while the bias term is of order a 2 n . The third result can be deduced from a Taylor expansion, Assumption 5 and Proposition 4.3 in Van Keilegom and [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF]. Indeed, we can deduce that

sup t≤τ sup x:g(x)∈Z θ 0 ,η | Ĝθ (t|ĝ(x)) -G θ 0 (t|g(x))| ≤ sup t≤τ sup x:g(x)∈Z θ 0 ,η | Ĝθ 0 (t|g(x)) -G θ 0 (t|g(x))| + O P ( θ -θ 0 ).
We are now ready to give the proofs of the main results.

Proof of Theorem 3.2. Part i) of the Theorem can be easily derived by replacing the differentiability condition in Assumption 5 by a uniform continuity condition on G θ with respect to θ, and equation (A.8) in Lemma A.2.

For part ii), a Taylor expansion with respect to θ leads to

φ(x, y)d[ Fĝ -Fg ](x, y) = - 1 n n i=1 δ i φ(X i , T i )∇ θ Ĝθn (T i -|λ(θ n , X i ))( θ -θ 0 ) [1 -Ĝθn (T i -|λ(θ n , X i ))] 2 ,
for some θ n between θ and θ 0 . From the convergence of θ towards θ 0 , it follows that θ n tends to θ 0 . Moreover, applying equation (A.8) and (A.9) in Lemma A.2, we obtain that

φ(x, y)d[ Fĝ -Fg ](x, y) = - 1 n n i=1 δ i φ(X i , T i )∇ θ G θ 0 (T i -|λ(θ 0 , X i ))( θ -θ 0 ) [1 -G θ 0 (T i -|g(X i ))] 2 +R n (φ), = U n (φ) + R n (φ), with sup φ |R n (φ)| ≤ |R n (Φ)| = o P (n -1/2 ), and
U n (φ) = - 1 n n i=1 δ i φ(X i , T i )∇ θ G θ 0 (T i -|λ(θ 0 , X i )) [1 -G θ 0 (T i -|g(X i ))] 2 1 n n j=1 µ(T j , δ j , X j ) + R ′ n (φ), with sup φ |R ′ n (φ)| ≤ |R ′ n (Φ)| = o P (n -1/2 ).
Centering the first sum in U n (φ) and applying a uniform Central Limit Theorem (see e.g. [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF], we obtain the stated representation.

Proof of Theorem 3.3. Consider the difference

|M n (β, f , J) -M n (β, f, J)| ≤ 2 |y|1 y≤τ d Fĝ (x, y) sup x: J(x)=1,β∈B | f (β ′ x; β) -f (β ′ x; β)| + 1 y≤τ | f (β ′ x; β) + f (β ′ x; β)|d Fĝ (x, y) sup x: J(x)=1,β∈B | f (β ′ x; β) -f (β ′ x; β)|.
The first term on the right hand side converges uniformly to zero by Assumption 9 and the law of large numbers for Fĝ (see Theorem 3.1 and Theorem 3.2). The integral in the second term can be bounded by

(1 + o P (1)) × 2Ψ(x)d Fĝ (x, y),
where o P (1) is uniform in β, by Assumption 7 and 9-(A.1). Now we have to show that M n (β, f, J * ) converges to M(β, f, J * ) uniformly in β. For this, apply Theorem 3.1 and Theorem 3.2 using Assumption 7. By usual arguments for proving consistency (see e.g.

Van der Vaart, 1998, Theorem 5.7), the consistency of β n follows.

Proof of Lemma 3.4. The proof is somewhat similar to the proof of Lemma 5A in [START_REF] Dominitz | Some convergence theory for iterative estimation procedures with an application to semiparametric estimation[END_REF]. First observe that

f (β ′ X; β) = E [Y | β ′ X, Y ≤ τ ] = E [f (β ′ 0 X; β 0 )|β ′ X, Y ≤ τ ] = E [f (β ′ 0 X; β 0 )1 Y ≤τ | β ′ X] P(Y ≤ τ |β ′ X) . Let α(X, β) = β ′ 0 X -β ′ X. Define Γ X (β 1 , β 2 ) = E [f (α(X, β 1 ) + β ′ 2 X; β 0 )1 Y ≤τ |β ′ 2 X] ,
and note that f

(β ′ X; β) = Γ X (β, β)/P(Y ≤ τ |β ′ X). Then, ∇ β 1 Γ X (β 0 , β 0 ) = -f ′ (β ′ 0 X; β 0 )E [XP (Y ≤ τ | X) | β ′ 0 X] , ∇ β 2 Γ X (β 0 , β 0 ) = f ′ (β ′ 0 X; β 0 )XP (Y ≤ τ | β ′ 0 X) + f (β ′ 0 X; β 0 )∇ β h(X, β 0 ),
where h(x, β) = P(Y ≤ τ |β ′ X = β ′ x). It follows that .11) Therefore,

∇ β f (β ′ 0 x; β 0 ) = f ′ (β ′ 0 x; β 0 ) {xP (Y ≤ τ | β ′ 0 X = β ′ 0 x) -E [XP(Y ≤ τ |X) | β ′ 0 X = β ′ 0 x]} P(Y ≤ τ |β ′ 0 X = β ′ 0 x) + ∇ β h(x, β 0 )f (β ′ 0 x; β 0 ) P(Y ≤ τ |β ′ 0 X = β ′ 0 x) - ∇ β h(x, β 0 )f (β ′ 0 x; β 0 )E[1 Y ≤τ |β ′ 0 X = β ′ 0 x] P(Y ≤ τ |β ′ 0 X = β ′ 0 x) 2 := xm 1 (β ′ 0 x) + m 2 (β ′ 0 x). (A
E [∇ β f (β ′ 0 X; β 0 )1 Y ≤τ |β ′ 0 X] = E [f ′ (β ′ 0 X; β 0 ) {XP (Y ≤ τ | β ′ 0 X) -E [XP(Y ≤ τ |X) | β ′ 0 X]} 1 Y ≤τ | β ′ 0 X] P(Y ≤ τ |β ′ 0 X) = 0.
Proof of Theorem 3.5. The proof consists of three steps :

Step 0 : Replace J by J 0 . For any B n a sequence of shrinking neighborhoods of β 0 , sup β∈Bn M n (β, f, J) -M n (β, f , J 0 ) ≤ o P (M n (β, f , J 0 ) + n -1 ).

See [START_REF] Delecroix | On semiparametric M-estimation in single-index regression[END_REF], page 738. Similar arguments apply also when the trimming J is defined with f τ βn (β ′ n x) justifying the practical implementation of the trimming function.

Step 1 : Bring the problem back to the parametric case.

For notational simplicity, we work with ∇ β f instead of ∇ β f . Note that ∇ β f = (0, ∇ ′ β f ) ′ . We will show that, on B n , M n (β, f , J 0 ) = M n (β, f, J 0 ) + o P ββ 0 √ n

+ o P β -β 0 2 + C ′ n ,
where C ′ n does not depend on β. Decompose

M n β, f , J 0 = M n (β, f, J 0 ) - 2 n n i=1 δ i J 0 (X i ) (T i -f (β ′ X i ; β)) 1 T i ≤τ 1 -Ĝθ (T i -|ĝ(X i )) f (β ′ X i ; β) -f (β ′ X i ; β) + 1 n n i=1 δ i J 0 (X i )1 T i ≤τ 1 -Ĝθ (T i -|ĝ(X i )) f (β ′ X i ; β) -f (β ′ X i ; β) 2 = M n (β, f, J 0 ) -2A 1n + B 1n .
Step 1.1 : Study of A 1n .

as .13) To show that this term is negligible, we will use empirical process theory. We have that f ∈ H 1 , where H 1 is the Donsker class defined in Assumption 10, and f ∈ H 1 with probability tending to 1. Consequently, the class of functions

(β -β 0 ) ′ n n i=1 δ i J 0 (X i )1 T i ≤τ ∇ β f (β ′ 0 X i ; β 0 ) 1 -G θ 0 (T i -|g(X i )) f (β ′ 0 X i ; β 0 ) -f (β ′ 0 X i ; β 0 ) +o P ( β -β 0 2 ). (A
H ′ 1 = (y, c, x, t) → 1 y≤c 1 y≤τ ∇ β f (β ′ 0 x; β 0 )J 0 (t)φ(β ′ 0 t) 1 -G θ 0 (y ∧ c -|g(x))
: φ ∈ H 1 is a Donsker class, see Example 2.10.8 in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF]. Furthermore,

for all φ ∈ H 1 , E δJ 0 (X)∇ β f (β ′ 0 X; β 0 )φ(β ′ 0 X)1 T ≤τ 1 -G θ 0 (T -|g(X)) = E [∇ β f (β ′ 0 X; β 0 )φ(β ′ 0 X)J 0 (X)1 Y ≤τ ] = 0, (A.14) since E[∇ β f (β ′ 0 X, β 0 )1 Y ≤τ |β ′ 0
X] = 0 (see Lemma 3.4), and since J 0 (X) is a function of β ′ 0 X alone. Deduce that, since H ′ 1 is a Donsker class, and since f tends uniformly to f, that the first term in (A.13) is of order o P ( ββ 0 n -1/2 ). See the asymptotic equicontinuity of Donsker classes, cf. [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF], Section 2.1.2.

For A 5n , apply a second order Taylor expansion. Using that ∇ 2 β,β f is bounded, and that ∇ 2 β,β f converges uniformly to ∇ 2 β,β f, we obtain

A 5n = (β -β 0 ) ′ n n i=1 δ i J 0 (X i )1 T i ≤τ (T i -f (β ′ 0 X i ; β 0 ))[∇ β f (β ′ 0 X i ; β 0 ) -∇ β f (β ′ 0 X i ; β 0 )] 1 -Ĝθ (T i -|ĝ(X i )) +o P ( β -β 0 2 ).
Proceed as for A 4n to replace Ĝ and ĝ by G and g, using part 3 of Assumption 11. The same arguments as for A 4n can then be used, but considering instead the Donsker class

H ′ 2 = (y, c, x) → 1 y≤c J 0 (x)1 y≤τ (y -f (β ′ 0 x; β 0 ))φ(x) 1 -G θ 0 (y -|g(x)) : φ ∈ H 2 ,
where H 2 is defined in Assumption 10, and observing that, for any function φ,

E δJ 0 (X)φ(X) (Y -f (β ′ 0 X; β 0 )) 1 T ≤τ 1 -G θ 0 (T -|g(X)) = E [E [(Y -f (β ′ 0 X; β 0 ))1 Y ≤τ | X] J 0 (X)φ(X)] = 0,

  plays a major role in proving the asymptotic normality of M-estimators. See Delecroix, Hristache and Patilea (2006). The next lemma shows that in our context, where we have to truncate at τ because of censoring in the data, the analogous truncated version of this property holds true without any further model conditions.

  the last two assumptions implying the same kind for ∇ z H 1c . (iii) The jumps of F g (• | z) and G θ 0 (• | z) are the same for all z ∈ Z θ 0 . Let (d 1 , d 2 , ...) be the atoms of G. (iv) F g (• | z) and G θ 0 (• | z) have two derivatives with respect to z, with the first derivatives uniformly bounded (on [-∞; τ ]). The variation of the functions ∇ z F g (• | z) and ∇ 2 z,z F g (• | z) on [-∞; τ ] is bounded by a constant not depending on z.

  Condition 1 for a continuous integrable envelope Ψ. Assumption 8 The classes {x → ∇ β f (β ′ x; β) : β ∈ B} and {x → ∇ 2 β,β f (β ′ x; β) : β ∈ B} are VC-classes of continuous functions for a uniformly bounded envelope.

Lemma A. 2 |

 2 Under the assumptions of Theorem 3.2, we havesup t≤τ,θ∈Θ,x∈X | Ĝθ (t|λ(θ, x)) -G θ (t|λ(θ, x))| = o P (1), (A.8) sup t≤τ,θ∈Θ,x∈X |∇ θ Ĝθ (t|λ(θ, x)) -∇ θ G θ (t|λ(θ, x))| = o P (1), Ĝθ (t|ĝ(x)) -G θ 0 (t|g(x))| = O P (n -1/2 a -1

Acknowledgements. The third author acknowledges financial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy), and from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement No. 203650.

Typical examples of such kind of Donsker classes are classes of regular functions. Let T = {β ′ 0 x : x ∈ X } ⊂ R and let C 1 ℓ (T , M) = {h : T → R ℓ : sup t∈T {|h(t)| + |h ′ (t)|} ≤ M} for ℓ ≥ 1 and for some M < ∞. Define H 0 1 = C 1 1 (T , M), (A.4)

The class H 0 2 is a Donsker class, which follows from stability properties of Donsker classes (see e.g. Examples 2.10.7 and 2.10.10 in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF]).

Assumption 11 For all c > 0,

where ε n and ε

Appendix B : Technical lemmas and proofs

We start this Appendix with two technical lemmas, needed in the proofs of the main results. The first technical lemma gives a concentration inequality for the convergence rate of semi-parametric estimators.

Let b n be a sequence of real numbers tending to zero, and let {ζ α : α ∈ A} be a family of uniformly bounded functions, where A is a compact subset of R p (with p ≥ 1).

Consider the class of functions

where K 0 , ψ and ξ are fixed functions, X ⊂ R d is a compact set, and t ∈ R, and consider the process (in α, x and v)

Typically, K 0 denotes either a kernel or its derivative of order 1 or 2.

A 1n can be expressed as

A 2n does not depend on β. For A 3n , observe that, for any β ∈ B n , we can replace

) is a bounded function of x and β (Assumption 8, since the class of functions has a bounded envelope), and using the uniform convergence of ∇ β f (β ′ x; β) (Assumption 9), we can obtain from a first order Taylor expansion applied twice (for f (β ′ x; β) and for f (β

For A 4n , first replace Ĝθ with G θ 0 . For this, note that [1 -G θ 0 (T i -|g(X i ))] is bounded away from zero with probability tending to 1 for T i ≤ τ, and that

using part 2 of Assumption 11, and Lemma A.2. A first order Taylor expansion for

x; β 0 ) and property (A.12) lead to

Next, a second order Taylor development shows that the first term above can be rewritten by the definition of our regression model. Deduce that

Step 1.2 : Study of B 1n .

Rewrite B 1n as

Observe that, for any β ∈ B n , we can replace J 0 (X i ) by 1 f τ β (β ′ X i )>c/2 using Assumption 8. Next, by a Taylor expansion and the uniform convergence of ∇ β f , we have that

). The term B 3n does not depend on β. For B 4n , a second order Taylor expansion leads to

Replace Ĝ by G and use Assumption 11, part 1, to conclude.

Step 2 : Study of M n (β, f, J 0 ).

Observe that, on o P (1)-neighborhoods of β 0 , from a Taylor expansion,

and apply Theorem 1 and 2 of [START_REF] Sherman | Maximal inequalities for degenerate U-processes with applications to optimization estimators[END_REF] to conclude.

Proof of Proposition 3.6. The uniform convergence results in Assumptions 9 and 11 can be deduced from studying the uniform convergence rate of the numerator and the denominator in (3.4) (and their derivatives) separately. This is a consequence of Lemma A.1. Since the other terms can be studied in a similar way, we only consider the case of the denominator and its derivatives in (3.4). In each case, the bias part can be dealt with uniformly with classical kernel arguments, and is of order h 2 . For the centered version of f * , the result can be deduced from the study of the uniform convergence rate of empirical processes indexed by some class of functions as the one defined in (A.6), with

,

where j = 0 (resp. 1,2) for f * (resp. ∇ β f * , ∇ 2 β,β f * ), and ξ(T ) = T. The kernel K 0 in (A.6) is either K or K′ or K′′ , and ψ(β, x) = β ′ x. It follows from the conditions on K and from [START_REF] Nolan | U-processes : rates of convergence[END_REF] that the class of functions

is a VC-class of functions. Moreover u → (xu) j (j = 0, 1, 2) is also a VC-class of bounded functions using permanence properties of VC-classes, see Lemma 2.6.18 in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF]. Finally, since 1 -G θ 0 (T -|Z) is bounded away from zero, (A.7) holds. Now applying Lemma A.1, we get

where h 2 comes from the bias term. Hence, Assumption 9 holds if h → 0 and

n h → ∞, and nh 8 → 0. The first part of Assumption 10 follows directly from the uniform convergence of f * .

Elementary algebra shows that the gradient of f * can be written as

Using the same arguments as above, these two functions converge uniformly to m 1 (β ′ 0 x) and m 2 (β ′ 0 x) respectively, where ∇ β f (β ′ 0 x; β 0 ) = xm 1 (β ′ 0 x)+m 2 (β ′ 0 x), see equation (A.11), and Assumption 10 follows.

Proof of Proposition 3.7. The result can be deduced from studying the following type of quantities :

,

where j = 0, 1, 2 and k = 0, 1. Using Lemma A.2, equation (A.10), and the fact that

we can deduce that this type of quantities is of the order O P ((log n) 1/2 h -j n -1/2 a -1/2 n ) (j = 0, 1, 2). Hence the convergence rates follow.


From aÖxÖ'l-(are derived aÖx` (adv.) : behind, backwards. aÖx`m (prép. & conj.) (i) : afterü:'hh(adghmc- From fÑaÖ (m.) are derived : fÑa` (adv.

Body parts

Let us see a few examples of the use of those notions naming body parts. 3

aÖxÖ 0- C`r`ehx`s`xhr`hx`j`qfldmhc`f`a`x`mtv`s`x``y`mh`jv`bbhx`qrhqc hmr`+cnlhmx`ihs`tr`xhmtv`s`c`s`jdfnxdc`mh-ZR-T-+14\

In the morning, he took me from my mother's back and put me on the pommel of his saddle, because he was having pity of my mother who was carrying me on her back. 1- Stmc`x`j`l`g`mx`a`hv`hv`x`l`x`ctahf`qhma`+r`hc`x`s`aa`s `q`mr`bdv`x`a`rgha`x`c`f`rjd-ZR-T-+8\ When he left, he did not even turn back to look at the town, but he only thought about leaving the town for good. (lit :aÖrgåaÖxÖ = give it (the) back)

fÑaÖ 2- Hx`khmr`ctjr`hrtj``lr`f`a`È`x`rtj`bdZ---\ZR-T-+3\

All his parents answered together and said [...] (lit. fÑaÖÈ`x` = one front) 3- FÑa`mrÑx`eÖÈå-'@aq`g`l+166a( He lost heart. (lit. his front fell) 4- Rtmr¨fÑaÖf`aÑr-'hc-( They turned eastward. (lit. they put front eastward)

aÖx` = adverb

When preceded by the preposition fi(at), aÖxÖ can be used in a locative sense meaning on the back. The final vowel is shortened and the final result is the adverbial phrase fiaÖx`.

Space

Adverbs

The first use of aÖxÖ`mcfÑaÖwith a meaning that no longer refers to body parts but to space only is in the adverbial derivation. 4 This derivation is characterized by a shortening of the final vowel :aÖxÖ > aÖx` ; fÑaÖ > fÑa`. 6- Z---\ltj`j`l`g`mx`qfhc`-L`hfhc`lltx`m`a`x`c`jv`qhc`a`j`` q`s`xd+jtl` ‡`mmdmr`ahxtrtm`f`a`c`m`rtj`x`me`È``q`s`xd-Ltx`q` c`l`s`ltm`sr`jhx`qrt-ZR-T-+04\ [...] we took the way home. Our chief was behind carrying his bow and arrow, his two younger brothers were in front with their own weapons. We, children and women, we were between them. 7- Rtj`bd+!Vtbdlts`eh-!  Rtj`r`rgh`f`a`rtm`s`hmfhy`qr`g`qrtj`s`gnc`rghj`mhx`j`-ZR-T-+6\ They said "Let's go." They put him in front, they pushed him until they took him to the limit [of the village]. 8- Rtj`r`jdxhma`mjv`m``vtqhm+L`j`tx`xhf`a`+l`g`qahx`jnl`c a`x`ytv`hmc`c`x`g`Ètc`L`j`t-ZR-T-+8\ There, they said good-bye again, Makau went forward, the hunter went back to the place where he had met Makau. 0/-m`f`a`x`xhf`a`+m`a`x`r`hk`a`qh-ZR-T-+4\

The one who is in front goes on; those left behind can only comment. (proverb)

Prepositions

The prepositions are formed by adding the genitive link to the name of the body part, thus forming a noun phrase with its complement.

1-1-0 fÑa`m 00-M`ynm`ctq ‡tr`f`a`ma` ‡nmm`m+m`lh ‡`l`r`x`j`qfl`x`rg`-ZR-T-+18\ I went and kneeled in front of the guest, gave him [the drink], he took [it] and drank.

01- C`y``s`rghr`hl`hfhc`m`x`jhq`mhx`È`tjdmhx`Ènq`mh`f`a`mr``a hr`j`mq` ‡tlhmr`-ZR-T-+22\

As we were leaving, the master called me, took me and put me before him on his camel.

1-1-1 aÖx`m 02- A``ihl`a`r`hm`e`q`ihmftqm`mhmv`mh`atl`ha`msrnqn`a`x`mÈ`jh -ZR-T-+10\

Soon I heard a frightful growling behind the hut.

03-!@&`+x`s`eha`x`mf`qh+`ll`y`hc`vnfnad-!

"No, he went to some villages (lit. behind town), but he will come back tomorrow." 04- C`x`r`j`mh`a`x`mr`x`xhmc`y`hjv`ms`+`ll`c`x`e`q`fv`qshmmm srnqnx`j`l`mhZ---\ZR-T-+10\ He had put me behind him before lying down, but when he started to snore, I was frightened [...]

Orientation

The spatial orientation at work here is obvious, from the orientation of the the human body itself, with the face and the eyes giving a direction. Nevertheless, a diffence between Hausa and European languages like French of English as regards spatial orientation is worth noticing here. 5 If a speaker (S) is looking at two objects (A and B) which he wants to locate, whereas a European speaker would say (a) « B is behind A », or « B est derrière A » (which lexically corresponds to aÖx`m in Hausa), a Hausa speaker would rather use (b

) « Am•fÑa`m@ » (lit. « B. is in front of A », « B est devant A »):
This means that, whereas in European languages, this type of spacial location is speakercentered, in Hausa it is object-centered.

Time

The spatial orientation given by the body is reproduced in the representation of time. The oriention of visual perception is parallelled in the representation of time, which is oriented from the "back" to the "front", with « past » in the back of the speaker and « future » ahead. 6 5 See (HILL 1982) for more details. 6 If this is the case for French, English and Hausa, it is not necessarily so for all languages. We will thus find a rough equivalence between the chest (fÑaÖ) and the future on the one hand, and the back (aÖxÖ) and the past on the other hand.

Adverbs

2-

2-0-2 ÑfÑa`<cÑfÑaÖx`<later on

However, this general symmetry is blurred with the expression of "later on". Both notions can be used, but with different prepositions : fi, at in the case of cÑfÑ, from in the case of aÖx`. 07- Sn+r`he`jhÈ`tjhg`m ‡tqh+jhmr`mg`khmx`qnm+jtkktl`ahmc`x`jdrnx `xhhma``a`qrgha`r`hx`xhjtj`+`ll`hc`m`m`a`qhmr`x`xh`ahmc`x` rn`f`a`a`y`hihc`cha`-ZR-T-+07\ Well, be patient, you know how a child is, the thing he wants to do, if you don't let him do it, he will cry, but if you let him do the thing that he wants to do, later he won't be happy. 08- Hm`rnj`xhl`r`g`khm`qyhjh+j`ktq`c`rgh+j`a`rghfnm`c`vtqhmc`y `hxhfhc`+cnlhmx`m`c`hx`khrtm`ytv`c`f`a`x`-ZR-T-+0/\ I want you to treat him well, to take care of him, to give him a place to farm and build a house, because he has a family that will come later.

Prepositions

2-1-0 fÑa`m

This use of fÑaÖis not common. No example is to be found in S.U. One example is found in Abraham's dictionary. (ABRAHAM 1962) 1/-AÑÑr`mÑaãmcÑx`e•qtÑfÑa`mqÇmm`ma`-'@aq`g`l519168a( It is not known what happened after that day. (lit. in front of that day) MA-cÑfÑqÇmm`m+nqb`mfÑa`'be-dw-mØ05(hroqdedqqdcsnÑfÑa`mqÇmm`m-

2-1-1 aÖx`m 10-A`x`mv`mm`mrtj`y`tm`rtj`xhs`s`Èhg`qZ---\ZR-T-+07\

After this, they sat down and chatted until [.

..] 11-!A`x`mf`hrtv`+f`v`mm`mltstlrtm`mr`L`j`tZ---\ZR-T-+04\

After greetings, here is somebody called Makau [...](beginning of a letter)

12- R`hhmm`s`e`q`sr`ll`mh`q`ms`jn@kk`gx`jhq`xdmhmda`x`ms`rghmrt c`f`J`f`q`-ZR-T-+15\ Then, mother started thinking to herself that maybe god had called me back (e.g. I had died) after their departure from Kagara. 13- X`qnmm`mc`ha`x`ms`rghmjhc`jv`m`s`q`+v`mhÈ`mr`mdx`r`bdrgh-ZR -T-+16\ This boy, nine days after you had left, a sorcerer kidnapped him.

In the following example, notice the translation of aÖx`mby « before » instead of the usual « after ».

14-A`mr`jdf`mhms`a`r`ha`x`mrgdj`qtc`x`v` ZR-T-+07\

I did not see her again before many years.

Conjunctions

In conjunctions, there is no equivalent to fÑaÖ, and instead, the word j•ehmis borrowed from Arabic :

2-2-0 )fÑa`m<j•ehm 15- J`ehmrtrghqx`l`g`q`mrtmxhmhr`-ZR-T-+4\

Before they got ready, the raiders had gone far.

16-Stmj`ehmr`ehx`s`xhL`j`tx`qhf`x`x`rghqx`r`q`hZ---\ZR-T-+0/\

Before the morning had arrived, Makau was already well prepared [...]

2-2-1 aÖx`m

aÖx`m appears regularly as a conjunction meaning « after (that)... », and is used to locate a chronological succession of events. The locating event, or locator (following aÖx`m) is in the non-relative tenses (Perfect and Continuous). The located events are in the relative tenses (Relative Perfect and Relative Continuous). 17- A`x`m`mf`l`v`rgd,v`rgdmm`mctj`+r`h`j`g`È`j`x`m`j`j`vnvt qhmR`qjh-ZR-T-+7\ After they had finished ransacking [the house], they gathered the goods and brought them to the emir. 18- Z---\`ll`l`mynmR`qjha`hs`rghynl`r`a`r`ha`x`m`mxhr`kk`q`y`g`q -ZR-T-+0/\ [...] but the messenger of the emir did not come to his place until (after) they had done the afternoon prayer.

2/-A`x`ms`ehs`j`È`mr`hj`j`s`s`bhj`+Z---\  ZaÖx`msÖeås`---\

Not long after she had gone, her grand-mother died, [...] 20- V`s`q`m`v`idmfnrghmk`&`r`qa`x`mRg`hgtTl`qx`a`c`j`q`stf``k l`ihq`mr`++r`hv`mh`kl`ihqhmr`x`s`la`xdrghbdv`+Z---\ZR-T-+0\ One day, in the afternoon, after S.U. had taught his pupils, , one pupil asked him, [...] 21- A`x`mv`mm`mx` ‡`qd+bhjhmjx`jjx`v`mg`khs`qdc`tv`s``fhc`mL`j `t+r`hq`mm`mZ---\ZR-T-+2\ After this was finished, peacefully with mother in Makau's house, when one day [...] aÖx`mcÑ is a variant of aÖx`m-The locator (following aÖx`mcÑ) is in the relative tenses (Relative Perfect and Relative Imperfect) 22-A`x`mc``j`f`l`ghq`ctj`L`j`trghf`fhc`cnlhmjv`ms`-ZR-T-+8\

After they had finished talking, Makau went home to his bed. 23-A`x`mc``j`ihl``lr`l`s`+Z---\ZR-T-+0/\ After a while, then he answered her, [...] 24-A`x`mc`lts`mdmm`mrtj`f`l`s`Èhrtj`gts`+s`rgh+Z---\ZR-T-+03\

After those people had finished talking and had rested, they got up, [...] This use may nevertheless look strange if we consider that a « natural » derivation from body parts, would be for fÑa`'m(to refer to the future and aÖx`'m(to the past, if we follow the anthropomorphic orientation. For this type of chronological succession, where one is, as it were, « facing the future », fÑa`m is the only choice in so far as conjunctions are concerned. This means that this anthropomorphic explanation does not hold here.

Cognitive considerations

An explanation may be found if we examine a metaphorical use of the word aÖxÖ, (back) meaning in this case the end, the completion.

25-MÖf`aÖx`m`

hjß-'@aq`g`l+81`( I've finished the work (lit. I've seen the back of work)

So, the point of view is not anthropomorphic, but objective. What is considered here is the completion of the event, and the other events are located in relation to that completion. The point of view is aspectual, and not temporal. Now, we can explain what happens in ( 18) and ( 19) where ÑfÑa``mccÑfÑaÖx`are both translated as later on: 07- Sn+r`he`jhÈ`tjhg`m ‡tqh+jhmr`mg`khmx`qnm+jtkktl`ahmc`x`jdrnx `xhhma``a`qrgha`r`hx`xhjtj`+`ll`hc`m`m`a`qhmr`x`xh`ahmc`x` rn`f`a`a`y`hihc`cha`-ZR-T-+07\ Well, be patient, you know how a child is, the thing he wants to do, if you don't let him do it, he will cry, but if you let him do the thing that he wants to do, later he won't be happy. 08- Hm`rnj`xhl`r`g`khm`qyhjh+j`ktq`c`rgh+j`a`rghfnm`c`vtqhmc`y `hxhfhc`+cnlhmx`m`c`hx`khrtm`ytv`c`f`a`x`-ZR-T-+0/\ I want you to treat him well, to take care of him, to give him a place to farm and build a house, because he has a family that will come later.

In (19), the point of view is objective and aspectual. The arrival of the family is located in relation to the actions of the father (farm, build a house). Whereas in (18), « he won't be happy » is located in relation to the speaker. The point of view is anthropomorphic and temporal, which explains the use of ÑfÑa`.

  B is behind A » (a) French : « B est derrière A » (a) Hausa : « Am•fÑa`m@ » (b)

  0-0 cÑfÑfÑa`<m`mfÑa` = in the future 05 M`mfÑa`+jñqh ‡ÑrgÖqâÈÖjåmjñ  In the future, you should sweep your room! 2-0-1 ÑaÖx`<in the past, previously 06-ÑaãmcÑltjÑf`x•ÑaÖx`---(Gouffé 79) What we said previously...

4ter, route des Gardes -92190 MEUDON -FRANCE -caron@cnrs-bellevue.fr 1 May be we can iclude ytv•(towards) although it is derived from the verb yÚ.iõ(venir/partir).3 The Hausa examples are taken from a short story, Shehu Umar (hence S.U.), written by Sir Abubakar Tafawa êalewa. It relates the story of a small Hausa boy who, captured and sold a slave to the Arabs, becomes a renowned marabout.

Remarquons que dans les exemples (3) et (5), les emplois métaphoriques du nom des parties du corps s'appuient sur l'orientation intrinsèque du corps humain.


Joseph Chazalon
email: joseph.chazalon@irisa.fr

Bertrand Co Üasnon
email: bertrand.couasnon@irisa.fr

Aurélie Lemaitre
email: aurelie.lemaitre@irisa.fr

Iterative Analysis of Pages in Document Collections for Efficient User Interaction

Keywords: document analysis, degraded documents, document sets, iterative analysis, user interaction

The analysis of sets of degraded documents, like historical ones, is error-prone and requires human help to achieve acceptable quality levels. However, human interaction raises 3 main issues when processing important amounts of pages: none of the user or the system should wait for work; information provided by a human operator should not be restricted to local isolated corrections, but rather produce durable changes in the system; the ability to interact with a human operator should not increase the complexity of document models nor duplicate them between analysis and human interaction processes. To solve those issues, we propose an iterative approach, based on a special mechanism called visual memory, to reintegrate external information during page analysis. So as to demonstrate the interest for existing systems, we explain how we adapted a (rule-based) page analysis tool to enable, in this iterative approach, a delayed interaction with a human operator based on an adaptation of error recovery principles for compilers and the well-known exception handling mechanism. We validated our iterative approach on sales registers from the 18th century.

I. INTRODUCTION

Degraded documents (like historical ones) prevent automated analysis from achieving sufficient quality level, and for many tasks, human intervention is necessary to bridge the semantic gap between machine and man. We identified 3 requirements for this kind of interaction.

As sets of documents we process are often made of tens of thousands of pages, interacting with a human must not make the human operator wait for work nor block the system on each uncertainty which may arise (Req. 1: Viability).

Structure and content recognition are difficult in those documents, and existing systems use sophisticated document models to extract structured data. Then, interaction must not increase model complexity with synchronization considerations, nor fragment document knowledge (Req. 2: Simplicity), in particular between analysis and human interface processes to model-check answers supplied by a human.

When ran, the analysis system we use produces lots of errors which are often similar or linked, as document sets are organized in collections presenting stabilities in page layouts and contents. For example, in neighbor pages of the tabular document in Fig. 1, words written in a particular column belong to the same lexicon. Error corrections are also often similar or linked, like when, in a sequence of numbers, correcting a number provides some contextual knowledge for the others. Therefore, the interaction should be durable, in the sense of [START_REF] Nagy | Adaptive and interactive approaches to document analysis[END_REF], and permit the propagation of corrections, the reprocessing of previous results, and even system improvement, instead of only local and isolated corrections (Req. 3: Durability).

We first consider, in Sec. II, how existing approaches of interactive analysis comply with those requirements. Sec. III presents our contribution to satisfy those needs: an iterative analysis approach for which we exhibit the minimal architecture required. We focus on how to turn existing isolated page analyzers into interactive ones suitable for the processing of document collections, using a visual memory mechanism. Sec. IV details the modifications we made to an existing (rule-based) page analysis tool to enable, in this iterative framework, an asynchronous machine-initiated interaction with a human operator. This implementation relies on broadly available features easily adaptable to existing page analyzers, and was tested on production tasks.

II. RELATED WORK

The simplest way to improve analysis results with human interaction is post-processing error correction. PerfectDoc [START_REF] Yacoub | PerfectDoc: a ground truthing environment for complex documents[END_REF] is a post-processing tool which enables bulk correction of document structure and character recognition. While the human operator does not wait for tasks, the system does not validate human information nor it uses it to correct previous interpretations. Therefore, we consider the human and the system should cooperate during the analysis phase, not after.

Another interesting approach is DocMining [START_REF] Clavier | DocMining: A cooperative platform for heterogeneous document interpretation according to user-defined scenarios[END_REF], a very general framework for document processing where document processing units are successively invoked by a scheduling module following a predefined scenario. A persistent structure is associated to each document where each processing unit consumes and produces elements. DocMining answers several parts of our problem: the central storage module enables a delayed interaction between components; the persistent structure associated to each image permits a simple information fusion; and human interaction can be easily be implemented as a processing units. However, as the persistent structure stores the state of the analysis in this data-driven approach, there is no reprocessing of each page. It limits human interaction to a local substitution to automated processes, and forbids advanced cooperative behaviors like model validation, correction of previous analysis stages, or guiding the analysis. Therefore an iterative analysis where pages are reprocessed with new external information is also necessary.

The smartFIX system [START_REF] Klein | smartFIX: An adaptive system for document analysis and understanding[END_REF] goes a step further in that direction: not only it uses a central repository to enable a delayed interaction, but is also proposes iterative recognition, optimization and control of documents using explicit model definitions: when human operators correct results, they are automatically checked against those models. However, as analysis modules process documents independently, this approach may be somehow limited in the case of collections of old documents. The lack of some strategy at a collection level could prevent it from making use of collection properties, like local word repetitions, by grouping similar problems before validation to increase human operator efficiency. Furthermore, as few details are given about information reintegration in analysis processes, it is hard to know whether human interaction could be more elaborated than the labeling of suspicious fields.

While existing approaches do not comply with all the requirements for a viable, simple and durable interaction when processing document collections, they provide interesting insights. A central database can store information about pages to transmit between processes and enable an asynchronous interaction which avoids them to wait (Req. 1). To remain maintainable, page analyzers have to carry the whole document model and reprocess each page to reintegrate or validate external information. The human interface can then be homogeneously integrated as a page analyzer, without duplicating the document model (Req. 2). This iterative analysis fully makes use of human information, as an efficient interaction relies on the ability to recompute previous results (Req. 3). Furthermore, a practical system also needs: i) a strategy module to coordinate the processes; and ii) a mean to reintegrate external information in page analyzers.

Page Analyzer

III. ITERATIVE ANALYSIS OF DOCUMENT PAGES

We detail here the necessary components (visible in Fig. 2) we identified, then we explain their iterative behavior from a page analyzer's point of view. Finally, we present the interaction models that this iterative approach enables.

A. Required Components and Features 1) Strategy Module:

The strategy module is responsible for: i) coordinating the processes; and ii) providing them with appropriate data and gather their results. It acts both as a scheduler following a particular strategy, and a mediator between processes backed by a central database. The behavior of the strategy module should ensure that each page-level process is invoked with enough data to progress through each of its iterative invocations during the whole analysis. Its abstraction level also enables, while we will not present those elements, to: i) make use of collection context; ii) increase human interaction efficiency: for example, by grouping similar questions as we proposed in [START_REF] Guichard | Exploiting Collection Level for Improving Assisted Handwritten Words Transcription of Historical Documents[END_REF]; and iii) improve the system.

2) Visual Memory: In order to reintegrate external information during the analysis of pages, we use a visual memory. It is the support of information exchange and usage between processes. To each page, we associate a persistent dataset transmitted to each process when invoked by the strategy module. Each process can change its content. Every data it contains has a shape and a position in its image's referential (thus the "visual"). We note M I (t) the visual memory related to the image I at time t.

3) Central Database: The central database stores the associated visual memories for each page in the whole collection. It enables: i) an asynchronous interaction between processes; and ii) an improvement of human interaction efficiency using properties of document collections, like by grouping questions about similar fields to speed up their manual correction. It provides a collection-centric view of the information stored which can progressively grow.

4) Page Analyzer:

The page analyzer uses a document model to try to extract relevant information from an image. In an iterative analysis scheme, it is invoked with a image reference I and the visual memory M I (t), and returns M I (t + 1) to the strategy module when done. Each iteration reprocesses the whole image and enables: i) a validation against the document model (without duplicating it) of external information; and ii) the production of new structured results, based on (integrating) external information.

5) Human Interface: In this presentation, we consider the human interface as a page analyzer. It shows an image and associated memory to the human operator who then proposes new information. In a more general approach, it would have more important links with the strategy module, and would consider elements from different pages.

B. Iterative Analysis Behavior

1) Global Information Flow: Fig. 2 shows a single step (repeated as long as needed) of information exchange (for a given page) between a page analyzer and a user interface in a simple strategy which triggers each component one after another for each page. The communication with the central database is omitted for brevity. At time t, the strategy modules first invokes the page analyzer on the image I, with the memory M I (t). The strategy module gathers M I (t + 1) after the analysis, and transfers it to the human interface, which, in turn, produces M I (t + 2) and sends it back.

2) External Information Fusion with Visual Memory: Information fusion between the current image I and external information, M I , can be possible by: i) placing all data in the image referential, with a shape and a position; ii) ensuring that data, whenever it is extracted from the image or external, is available at any moment during the analysis; iii) providing equivalent access operations to each kind of data (read, create, delete at least).

We recompute M I at the end of each step of this iterative analysis.

C. Available Interaction Models for Page Analyzers

The iterative analysis we propose enables two complementary types of efficient user interaction for page analyzers.

1) Directed Interaction: Like for the machine-initiated concept of [START_REF] Nagy | Adaptive and interactive approaches to document analysis[END_REF], data to be analyzed by the human operator is determined automatically by an algorithm. We use the term directed to indicate the underlying question/answer semantic. It requires the automated system to detect errors and explicitly ask for resolutions. The human operator has no choice on the information to provide, and the system can more easily identify which of its subparts is to be corrected.

2) Spontaneous Interaction: This model is a bit different from the human-initiated concept of [START_REF] Nagy | Adaptive and interactive approaches to document analysis[END_REF]: the human operator is free to provide or alter any information he wants for a given page, but it may not produce changes in the automated results, as the system may not validate or even consider those elements. Even if spontaneous interaction is necessarily limited to several information types by the system, the difficulty is, for the operator, to provide useful elements, and for the system to use them. On the other side, no automated error detection is required. An example of such interaction is, with the document of Fig. 1, when a human operator notices that the system did not locate a number, and decides to provide its position. The analyzer could then use this new information and integrate it in the structured result.

Interaction at the Strategy Level: As this paper focuses on how to adapt a page analyzer, we do not detail interaction capabilities of the strategy module, inside which: i) interaction can be made more efficient using machine learning, or grouping similar requests as we proposed in [START_REF] Guichard | Exploiting Collection Level for Improving Assisted Handwritten Words Transcription of Historical Documents[END_REF]; and ii) a "director" user could choose the processes to invoke in a manual scenario, like for the user-driven mode of [START_REF] Bapst | A scenario model advocating user-driven adaptive document recognition systems[END_REF].

IV. IMPLEMENTATION OF A PAGE ANALYZER BASED ON ERROR DETECTION, CORRECTION AND RECOVERY

We present here how we turned a rule-based page analyzer, based on the DMOS-P approach [START_REF] Coüasnon | Dealing with noise in DMOS, a generic method for structured document recognition: An example on a complete grammar[END_REF], into an iterative process ready for a efficient directed interaction.

A. DMOS-P Framework and Language DMOS-P [START_REF] Coüasnon | Dealing with noise in DMOS, a generic method for structured document recognition: An example on a complete grammar[END_REF] is a concept-driven grammatical document analysis method. It uses a bi-dimensional extension of Definite Clause Grammars, called Enhanced Position Formalism (EPF), to describe pages. The following example illustrates its syntax. To recognize A, we try, at the top of the image, to recognize B, and if it fails, we try, at the bottom, to recognize C. Like attribute grammars, rules have input and output parameters indicated by "+" and "-" signs, as illustrated by: recognizeNumber(+NumPos, -Value) ::= callClassifier(+NumPos, -Value).

B. Communication With Other Components 1) Visual Memory Data Structures:

To enable the communication with the human, the visual memory carries (and locates in the image referential) questions and answers, which are represented with the following containers

Q(T ext, AnswerArea, DataT ype) A(Data)
T ext gives information about the problem and DataT ype indicates the expected type for the content of the answer, which is stored in its Data attribute. AnswerArea indicates the area inside which the answer can be located.

2) How Questions Are Answered: Answering the questions is done outside of the page analyzer, in the humanmachine interface. At time t, it loads the image I, the associated visual memory M I (t), and displays the questions. The human operator answers the questions, which are deleted, and locates each answer inside its acceptation zone AnswerArea. At the end, M I (t + 1) contains only current and previous answers, kept for later use.

3) Challenges: The implementation has to: i) detect errors and ask questions; ii) ensure that answers are used by the analyzer to make progress instead of asking the same questions forever; and iii) make as much progress as possible in independent parts of the analysis if a problem arises.

C. DMOS-P Extensions for Directed Interaction

Three new EPF operators are provided to add the required semantics in page descriptions. Their implementation is an adaptation of error recovery principles for compilers. In logical and functional languages, higher order and continuation easily enable such approach, and in imperative ones, exceptions (and annotations) can be used.

1) Asking Questions (Error Detection): Like one raises an exception, we indicate a problem was detected and ask for external information with the operator raiseQuestion(+Text, +Zone, +DataType)

When called, it 1) adds a question in the visual memory M I , with shape and position defined by +Zone to locate the issue; and 2) continues the analysis just after the latest invocation of catchQuestion. The AnswerArea attribute of the question is automatically defined as the search position during the last invocation of getAnswerOrTry.

2) Using Answer (Error Correction): Like with a try keyword in many languages, we can identify the part of the analysis impacted by a given problem with the operator getAnswerOrTry(+DataType, -Result, +Rule)

where +DataType indicates the type of the content of acceptable answers, and +Rule is a rule which has a unique output parameter whose type is +DataType. Therefore, the rule and the answer can provide elements of the same type. When called, it looks at the current search position for an answer A(Data) in M I where the type of Data is +DataType. If it exists, the value of -Result is Data. Otherwise, it invokes the rule +Rule and the value of the output parameter of +Rule is used as value for -Result.

To enable interaction on structural elements, a special case returns the zone of the answer if the content of the answer as no value (meaning it is just a marker).

3) Continuing the Analysis (Error Recovery): Like with a catch keyword, we indicate an upper bound in an analysis branch where analysis can be safely continued when a given problem arises with the operator

catchQuestion(+Rule)
where +Rule is a rule which may have any parameter. When called, it invokes the rule +Rule and catches any interaction request raised with raiseQuestion in that rule. If no question is raised, the output parameters of +Rule are well defined, otherwise they are left uninstantiated.

D. Usage Example

This example aims at locating and recognizing the numbers contained in the leftmost column of the document shown in Fig. 1, using the trivial dialogue strategy of Fig. 2 Furthermore, the invocation of the start rule is automatically changed to: i) read M I (t); ii) invoke start and catch any question; and iii) write M I (t + 1).

3) Interaction Dialogue: For an image I where the left column cannot be automatically located, and where 2 numbers over 4 are not recognized, a strategy with a simple dialogue between the page analyzer and the human interface (like in Fig. 2) produces the following behavior: At t 0 , the analysis starts.

MI (t0) = ∅

From t 0 to t 1 , the page analyzer tries to locate the left column and fails: it asks a question.

MI (t1) =

{P a → Q("Where is the column?", P a, colT )}

where P a is a zone covering the whole page area.

From t 1 to t 2 , the human operator answers the question and provides the location of the column.

MI (t2) = {Ca → A(lCol)}
where Ca is the zone where the column is located.

From t 2 to t 3 , the page analyzer try to locate the left column and uses the answer. It starts locating and reading numbers, but fails to read two of them.

MI (t3) = {Ca → A(lCol); N a1 → Q("What is this number?", N a1, numT ); N a2 → Q("What is this number?", N a2, numT )}
where N a 1 and N a 2 are the location of the 2 numbers. From t 3 to t 4 , the human operator labels the 2 numbers.

MI (t4) = {Ca → A(lCol); N a1 → A(num(V1)); N a2 → A(num(V2))}
where V 1 and V 2 are the values of the 2 numbers. From t 4 to t 5 , the page analyzer reprocesses the whole page and finishes the analysis using the answers. It produces a consistent structured interpretation.

4) Implementation Advantages: This implementation of a directed interaction allows to clearly separate the documentrelated semantic (which part of the model is impacted by an error, which parts are independent) from the iterative error correction based on the visual memory. Apart from enabling another implementation (like synchronous answering), it also permits to make use of human knowledge at various levels in the analysis: field (lexical), presentation and structure (syntactical), and function (semantical).

E. Application to Documents from the 18 th Century

The implementation of the iterative analysis we presented is one of the pillars enabling the assisted transcription of handwritten words from sales registers of the 18 th century described in [START_REF] Guichard | Exploiting Collection Level for Improving Assisted Handwritten Words Transcription of Historical Documents[END_REF]. In the tests we conduced, 70 documents (1206 words) were processed, and extracted words were grouped by visual similarity in clusters, which were manually annotated if necessary. The iterative analysis permitted to reintegrate and validate external information (extracted using collection context) within a unique page model. This global approach diminished the human labeling of words from 21% (manual labeling of all suspicious elements) to 15% (28% gain) for an overall recognition rate of 80%.

V. CONCLUSION

We presented a solution to the problem of efficiency for the necessary interaction during the analysis of document collections. We showed that: asynchronous exchanges avoids the user or the system to wait (viability); keeping a unique page model, separated from temporal issues, inside the page analyzer facilitates its use and enables an homogeneous integration of human information (simplicity); and iterative analysis permits reprocessing and other durable modifications of the analysis (durability).

We proposed an iterative approach for the analysis of document pages, using a visual memory to permit an asynchronous information exchange between analysis components, which enables two interaction models: directed and spontaneous. We implemented and tested the directed model on a transcription task in historical documents presented in [START_REF] Guichard | Exploiting Collection Level for Improving Assisted Handwritten Words Transcription of Historical Documents[END_REF], where the iterative approach permitted to make use, during the page analysis, of information produced using collection-level information. This implementation is based on error recovery principles, very similar to exceptions in their usage, and may be quite easily adapted to turn existing isolated page analyzers into interactive ones.

Figure 1 .

 1 Figure 1. Excerpt from a page of a tabular document (18 th century).

1 :

 1 process(I, M I (t)) 2: M I (t + 1) 3: process(I, M I (t + 1)) 4: M I (t + 2)

Figure 2 .

 2 Figure 2. A single step (repeated as many times as needed during the iterative analysis) of a simple dialogue strategy between a page analyzer and a human interface, showing the implied components of the architecture and the visual memory M I associated to the image I.

A

  ::= AT(top) && B. % clause 1, tried 1st A ::= AT(bottom) && C. % tried if 1 fails

  . 1) Description without Interaction: A simple noninteractive description first locates the left column, and inside this column, locates and recognizes each number. For clarity, we only detail start and readAllNumbers. Useful memory elements are lCol (of type colT) which indicates the left column position, and num(Value) (of type numT) which associates a value to a number located in the memory. Modified elements are indicated in bold.

	start() ::= AT(allPage) &&
	locateLeftCol(-ColPos) &&
	AT(+ColPos) &&
	readAllNumbers().
	readAllNumbers() ::=
	locateNumber(-NumPos) &&
	recognizeNumber(+NumPos, -Value) &&
	% use Value...
	AT(under) &&
	% loop until no more numbers...
	2) Description with Directed Interaction: start() ::= AT(allPage) &&
	getAnswerOrTry(colT, -ColPos,
	locateLeftCol2(-ColPos)) &&
	AT(+ColPos) &&
	readAllNumbers().
	readAllNumbers() ::=
	locateNumber(-NumPos) &&
	catchQuestion(
	getAnswerOrTry(
	numT, -Value,
	recognizeNumber2(+NumPos,-Value)
	))
	% use Value (may be uninstantiated)
	AT(under) &&
	% we still can read other numbers...
	locateLeftCol2(-ColPos) ::=
	locateLeftCol(-ColPos).
	% If automated version fails, ask.
	locateLeftCol2(-ColPos) ::=
	raiseQuestion("Where is the column?",
	allPage, colT).
	recognizeNumber2(+NumPos,-Value) ::=
	recognizeNumber(+NumPos,-Value).
	recognizeNumber2(+NumPos,-Value) ::=
	raiseQuestion("What is this number?",
	+NumPos, numT).

ACKNOWLEDGMENT

This work has been done in cooperation with the Archives départementales des Yvelines in France, with the support of the Conseil Général des Yvelines.