Pierre Vandekerkhove
email: pierre.vandek@univ-mlv.fr

Estimation of a semiparametric mixture of regressions model

Keywords: . M-estimator, mixture, regression model, empirical process, semiparametric identifiability, uniform convergence rate AMS 2000 subject classifications. 62G05, 62G20, 62G08; secondary 62E10

We introduce in this paper a new mixture of regressions model which is a generalization of the semiparametric two-component mixture model studied in Bordes et al. (2006b). Namely we consider a two-component mixture of regressions model in which one component is entirely known while the proportion, the slope, the intercept and the error distribution of the other component are unknown. Our model is said to be semiparametric in the sense that the probability density function (pdf) of the error involved in the unknown regression model cannot be modeled adequately by using a parametric density family.

When the pdf's of the errors involved in each regression model are supposed to be zero-symmetric, we propose an estimator of the various (Euclidean and functional) parameters of the model, and establish under mild conditions their almost sure rates of convergence. Finally the implementation and numerical performances of our method are discussed using several simulated datasets and one real hight-density array dataset (ChIP-mix model).

Introduction

Let U = (U i ) i≥1 be a label sequence of independent and identically distributed (iid) random variables according to a Bernoulli distribution with parameter p ∈ (0, 1). We consider an iid sample (Z 1 , . . . , Z n ) where for all i = 1, . . . , n, Z i = (X i , Y i ) is a bivariate random variable defined, relative to U i , as follows:

Y i = a 0 + b 0 X i + ε [0] i , if U i = 0, Y i = a 1 + b 1 X i + ε [1] i , if U i = 1, (1)
where the design sequence X = (X i ) i≥1 , is a sequence of iid random variables with cumulative distribution function (cdf) H and probability density function (pdf) h, when the errors ε [j] = (ε

[j]
i ) i≥1 , j = 0, 1, are respectively sequences of iid random variables with cdf F j and pdf f j . In addition we suppose that the sequences U , X, and ε [j] , j = 0, 1, are mutually independent. This model, called the 2-component mixture of regressions model, belongs to the wide class of mixture of regressions models which has been studied in [START_REF] Zhu | Hypothesis testing in mixture regression models[END_REF]; see also [START_REF] Yau | Finite mixture regression model with random effects: application to neonatal hospital length of stay[END_REF] in a LOS (length of stay) medical problem, Bouveyron and Jacques (2010) for prediction, [START_REF] Young | Mixtures of Regressions with Predictor-Dependent Mixing Proportions[END_REF] in a nonparametric modelling context, or [START_REF] Hunter | Semiparametric Mixtures of Regressions[END_REF] for a very recent work on a semiparametric EM-type algorithm for general mixture of regressions models. Recently Martin-Magniette et al. (2008) introduced this model in microarray analysis, specifically for the two color ChIP-chip experiment. In their work, these authors suppose that the error sequences (ε

[j]
i ) i≥1 , j = 0, 1, are such that ε

[j] i = ε i for all (j, i) ∈ {0, 1} × N * , where N * := N \ {0} and ε i is a Gaussian random variable with mean 0 and variance σ 2 (homoscedaticity with respect to the so-called probe status U i ).

In this work, we propose to weaken this last assumption while completely specifying the regression model under the probe standard condition (the parameter θ [0] := (a 0 , b 0 ) ∈ R 2 and f 0 are supposed to be entirely known). Note that this kind of assumption arises naturally in microarray analysis. See model Bordes et al. (2006) and references [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF], [START_REF] Efron | Size, power and false discovery rate[END_REF], or Bordes et al. (2006) where analytic expression of f 0 , characterizing probe expressivity levels under a certain standard condition, is assumed to be available (generally derived from training data and probabilistic computations). In particular we will suppose that, in model [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF], the distribution of the ε [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF] i is seen as a nuisance parameter (it cannot be modeled using a parametric distribution family), causing model [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF] to be a purely semiparametric model. Note that when θ [0] is known the observations Y i , for i = 1, . . . , n, can be centered according to Y i := Y i -(a 0 + b 0 X i ) which implies the following simplification of model [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF]:

Y i = ε [0] i , if U i = 0, Y i = α + βX i + ε [1] i , if U i = 1, (2)
where α := a 1a 0 and β := b 1b 0 . We suppose that in model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF], which is from now on our model of interest, the Z i = (X i , Y i )'s distribution admits a pdf with respect to the Lebesgue measure on R 2 given by:

g(x, y) = h(x)g Y |X=x (y) = h(x)[pf (y -(α + βx)) + (1 -p)f 0 (y)], (x, y) ∈ R 2 , (3)
where f denotes the unknown pdf of the ε [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF] i , f 0 the known pdf of the ε

[0]
i , h the unknown pdf of the X i , with f and f 0 assumed to be zero-symmetric densities. We will finally denote by ϑ := (p, α, β) ∈ (0, 1) × R 2 the unknown Euclidean parameter of model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model when a component is known[END_REF]. Model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF] corresponds exactly to a contaminated version of the semiparametric additive regression model studied in Cuzick (1992a,b) and more recently in [START_REF] Yu | Semiparametric Regression: Efficiency Gains From Modeling the Nonparametric Part[END_REF]. On the other hand, this work extends for the first time to the bivariate case, the asymptotic study (consistency and convergence rates) of the semiparametric class of mixture models introduced by [START_REF] Hall | Nonparametric estimation of component distributions in a multivariate mixture[END_REF] for R s -valued observations with s ≥ 3. We remind that semiparametric mixture models were also studied in the univariate case, through two specific models:

g(y) = pf (y -µ 1 ) + (1 -p)f (y -µ 2 ), y ∈ R, (4)
where (p, µ 1 , µ 2 ) ∈ (0, 1/2) × R 2 and f , supposed to be even, are unknown, see Bordes et al. (2006), [START_REF] Hunter | Inference for mixtures of symmetric distributions[END_REF], Martin-Magniette et al. (2008), and

g(y) = pf (y -µ) + (1 -p)f 0 (y), y ∈ R, (5)
where (p, µ) ∈ (0, 1) × R and f are unknown, f 0 is known, and the pdfs f and f 0 are supposed to be even, see Bordes et al. (2006), [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF].

The paper is organized as follows. In Section 2 we present an M-estimating method, inspired by Bordes et al (2006a), Bordes et al. (2006b) and [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF], that allows us to estimate the Euclidean and the functional parameters of model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF]. In Section 3 we address the semiparametric identifiability problem in expression [START_REF] Bordes | Semiparametric estimation of a two-component mixture model when a component is known[END_REF] and establish rates of convergence of our estimators. In Section 4 we discuss the performances of our method on simulated examples and also analyze one real high-density array dataset previously considered in Martin-Magniette et al. (2008). Technical results and tedious proofs are relegated to the appendix, Section 5.

Estimating method

In this section we generalize the estimating method proposed by Bordes and Vandekerkhove (2010) for model [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF] and a fixed unknown location parameter µ, to model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model when a component is known[END_REF], where the location parameter, given {X = x}, is equal to α + βx and changes with x due to the non-zero slope parameter β. The main challenge of this paper will consist then in establishing uniform asymptotic results close to those obtained in [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF] in spite of the introduction of this multiplicative parameter. In particular, the difficulty generated by this change relates to the statement of minimal conditions on the functional parameters of the model in order to allow a uniform control on the (bigger) classes of functions we will apply to the (X, Y )-empirical process to get convenient approximation results (see, e.g., the treatment of the terms T (i) 1,n , i = 1, 2, in Section 6.6). In the spirit of Bordes et al. (2006a), Bordes et al. (2006b) and Bordes and Vandekerkhove (2010), we will assume that f and f 0 are both zero-symmetric pdfs. To avoid trivial situations or trivial non-identifiability problems (see Remark in Section 3.1), we will impose p = 1 and θ := (α, β) ∈ Φ ⊂ R×R * , where R * := R\{0}, which implies that the Euclidean parameter ϑ will be assumed to belong to a parametric compact and convex space

Θ := [δ, 1 -δ] × Φ ⊂ (0, 1) × {R × R * } , (6)
where δ ∈ (0, 1). For simplicity, we will endow the spaces R s , s ≥ 1, with the • s norm (for clarity the dimension s is recalled in index) defined for all v = (v 1 , . . . , v s ) by v s = s j=1 |v j | where | • | denotes the absolute value. Let us introduce now the following notation:

θ x := α + βx, (θ, x) ∈ Φ × R.
Following the ideas developed by the authors mentioned above, it is possible to use the symmetry of f to identify the true value of the Euclidean parameter. The original idea of this paper consists of noticing that for θ fixed in Φ, the sample (Y θ 1 , . . . , Y θ n ) obtained by considering the following θ-depending transformation of the original dataset (so-called for simplicity θ-transformation),

Y θ i := Y i -θ X i , i = 1, . . . , n, (7)
is distributed according to

Ψ θ (y) = p * R f (y + (θ -θ * ) x)h(x)dx + (1 -p * ) R f 0 (y + θ x)h(x)dx, (8)
where ϑ * = (p * , α * , β * ) ∈ Θ denotes the true value of the parameter. In addition, when θ = θ * ,

Ψ θ * (y) = p * f (y) + (1 -p * ) R f 0 (y + θ * x)h(x)dx, (9)
which is very close to model (5) studied in Bordes et al. (2006b) where the location µ is known but the proportion p is unknown. These two major remarks are illustrated and discussed on a simulated example in Section 4.

∈ R × Θ f ϑ (y) = 1 p Ψ θ (y) - 1 -p p R f 0 (y + θ x)h(x)dx, (10)
that satisfies under ϑ = ϑ * , and all y ∈ R

f (y) = f ϑ * (y) = f ϑ * (-y) = f (-y). ( 11
)
Note that, on the right hand side of [START_REF] Cuzick | Efficient estimates in semiparametric additive regression models with unknown error distribution[END_REF], the second integral term is unknown but can be estimated pointwise by a standard integration Monte Carlo method, see expression [START_REF] Hohmann | Semiparametric location mixtures with distinct components[END_REF], or a nonparametric Monte Carlo approach, see expression [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF]. The intuition consists now in claiming that, if we vary ϑ over Θ and that we are able to check that f ϑ is zero-symmetric for a certain value of ϑ, then we have reached the true value of the Euclidean parameter. A natural idea to detect this kind of situation, and then to estimate ϑ = (p, θ), is to consider a contrast function based on the comparison between the cdf version of f ϑ (y)

H 1 (y; ϑ) := H 1 (y; p, F θ , J θ ) := 1 p F θ (y) - 1 -p p J θ (y), (y, θ) ∈ R × Θ,
and the cdf version of f ϑ (-y)

H 2 (y; ϑ) := H 2 (y; p, F θ , J θ ) := 1 - 1 p F θ (-y) + 1 -p p J θ (-y), (y, θ) ∈ R × Θ,
where for all (y, θ) ∈ R × Φ,

J θ (y) := y -∞ I θ (z)dz, y ∈ R, with I θ (z) := R f 0 (z + θ x)h(x)dx, z ∈ R,
and

F θ (y) := y -∞ f θ (z)dz.
Notice that for all θ fixed in Φ, J θ (•) and F θ (•) are the cdfs respectively associated to the θ-transformed known component population (the Y i such that U i = 0 in (2)) and the θ-transformed whole data (the Y i such that U i = 0 or U i = 1 in (2)). It is then convenient to define the comparison-function

H(y; ϑ) := H 1 (y; ϑ) -H 2 (y; ϑ), (y, ϑ) ∈ R × Θ. ( 12
)
and to notice that under ϑ * , using the symmetry of f ,

H(y; ϑ * ) = 0, y ∈ R.
To avoid numerical integration in the approximation of an empirical contrast function, based on the comparison function H over R, we proceed as follows. Let Q be an instrumental weight probability distribution with pdf q with respect to the Lebesgue measure. We assume that q is strictly positive over R and easy to simulate. Then we consider

d(ϑ) := R H 2 (y, ϑ)dQ(y), (13)
where obviously d(ϑ) ≥ 0 for all ϑ ∈ Θ and d(ϑ

* ) = 0. Let (V 1 , . . . , V n ) be an iid sample from Q. An empirical version d n (•) of d(•) can be obtained by considering d n (ϑ) := 1 n n i=1 H 2 (V i ; p, Fn,θ , Ĵn,θ ), ϑ ∈ Θ, (14)
where Ĵn,θ (y

) := y -∞ În,θ (z)dz, (y, θ) ∈ R × Φ, (15)
with În,θ (z

) := 1 n n i=1 f 0 (z + θ X i ), (z, θ) ∈ R × Φ, (16)
which leads actually to the simple expression for Ĵn,θ (y)

Ĵn,θ (y) := 1 n n i=1 F 0 (y + θ X i ), (y, θ) ∈ R × Φ. (17)
In ( 14), the function Fn,θ denotes a smooth version of the empirical cdf Fn,θ (y

) := 1 n n i=1 1 I Y θ i ≤y , (y, θ) ∈ R × Φ, defined by Fn,θ (y) := y -∞ Ψn,θ (t)dt, (y, θ) ∈ R × Φ, (18)
where Ψn,θ (t

) := 1 nb n n i=1 K t -Y θ i b n , (t, θ) ∈ R × Φ. (19)
In [START_REF] Kui | Two-component Poisson mixture regression modelling of count data with bivariate random effects[END_REF], we assume the standard condition ensuring, for each θ ∈ Φ, the L 1 convergence of Ψn,θ towards Ψ θ defined in [START_REF] Cruz-Medina | Nonparametric estimation in semiparametric univariate mixture models[END_REF] (see [START_REF] Devroye | The equivalence of weak, strong and complete convergence in L 1 for kernel density estimates[END_REF], namely where, for all θ ∈ Θ, Ĩn,θ and Jn,θ are respectively nonparametric estimators of I θ and J θ based on an iid simulated sample (ε

b n → 0, nb n → +∞, ( 20
)
[0] 1 , . . . , ε[0] n ) from f 0 obtained by considering Ĩn,θ (t) := 1 nb n n i=1 K t -(-θ X i + ε[0] i ) b n , (t, θ) ∈ R × Φ. Jn,θ (y) := y -∞ Ĩn,θ (t)dt, (y, θ) ∈ R × Φ,
In this second plug-in step, we consider, for the sake of simplicity in our proofs, the nonparametric estimates ( 22) and ( 22) instead of ( 15) and ( 16). This choice allows us to use similar nonparametric results for both fn,θ and Ĩn,θ (see the proof of Theorem 3.1 ii) and iii)), but the same results should be obtained, at the price of an additional technical lemma, by considering directly the Monte Carlo estimators [START_REF] Hettmansperger | Almost nonparametric inference for repeated measures in mixture models[END_REF] and ( 16).

3 Semiparametric identifiability and consistency

Identifiability

In this section we recall briefly why model ( 3) is identifiable under conditions similar to those established in Bordes et al. (2006b) and summarized below. Let us define

F s := {f ∈ F; R |x| s f (x)dx < +∞} for s ≥ 1
, where F denotes the set of even pdfs. When (f, f 0 ) ∈ F s with s ≥ 2, we denote m := R x 2 f (x)dx and m 0 := R x 2 f 0 (x)dx. In addition we will denote by λ ⊗2 the product Lebesgue measure on R 2 . Definition 3.1 (Identifiability). Let (p 1 , θ 1 , f 1 , h 1 ) and (p 2 , θ 2 , f 2 , h 2 ) denote two sets of parameters for model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model when a component is known[END_REF]. The parameter in model ( 3) is said to be semiparametrically identifiable if

(p 1 , θ 1 , f 1 (y), h 1 (x)) = (p 2 , θ 2 , f 2 (y), h 2 (x)),
for λ ⊗2 -almost all (x, y) ∈ R 2 , whenever we have:

(p 1 f 1 (y -θ 1 x) + (1 -p 1 )f 0 (y)) h 1 (x) = (p 2 f 2 (y -θ 2 x) + (1 -p 2 )f 0 (y)) h 2 (x), ( 22
)
for λ ⊗2 -almost all (x, y) ∈ R 2 . Lemma 3.1 If the Euclidean parameter space Θ is a subset of R × R * , supp(f ) = supp(f 0 ) = R, supp (
h) contains at least two intervals respectively in the neighborhood of 0 and +∞ (or -∞), and the pdfs involved in model ( 3) satisfy (f 0 , f ) ∈ F 3 × F 3 , then the parameter in model ( 3) is identifiable.

Proof. Integrating [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] with respect to y over R, we then obtain that h

1 (•) = h 2 (•) λ-almost everywhere. Let h(x) := h 1 (x) for all x ∈ supp(h) := supp(h 1 ) ∩ supp(h 2 ).
Notice now that, for all x ∈ supp(h), [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] coincides with (5) when we replace the location parameter µ by θ x. In our case, the first three conditional moment equations given {X = x} associated to [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] lead to

   p 1 (θ 1 x) = p 2 (θ 2 x), (1 -p 1 )m 0 + p 1 ((θ 1 x) 2 + m 1 ) = (1 -p 2 )m 0 + p 2 ((θ 2 x) 2 + m 2 ), p 1 (3((θ 1 x)m 1 + (θ 1 x) 3 ) = p 2 (3((θ 2 x)m 2 + (θ 2 x) 3 ). ( 23
)
Replacing in Bordes et al. (2006b) formula [START_REF] Bouveyron | Adaptive mixtures of regressions: Improving predictive inference when population has changed[END_REF] the location µ by θ 1 x and noticing that in our case θ = m 1 and θ 0 = m 0 , the solutions are either, for all x ∈ supp(h),

(p 1 , θ 1 x) = (p 2 , θ 2 x), which implies (p 1 , α 1 , β 1 ) = (p 2 , α 2 , β 2 ), or                p 2 = p 1 2(θ 1 x) 2 3m 1 + (θ 1 x) 2 -3m 0 , θ 2 x = θ 1 x + 3m 1 -(θ 1 x) 2 -3m 0 2θ 1 x , m 2 = m 1 + (m 1 + (θ 1 x) 2 -m 0 )(3m 0 + (θ 1 x) 2 -3m 1 ) 4(θ 1 x) 2 . ( 24
)
Assume that β 1 = 0 and consider the limit when x → +∞ in the first row of [START_REF] Shorack | Empirical Processes with Applications to Statistics[END_REF].

We then necessarily obtain that p 2 = 2p 1 which is only compatible when we take the limit as x → 0, with m 1 = m 0 . Hence if m 1 = m 0 , model ( 3) is always identifiable.

If we assume m 1 = m 0 , the second row of (24) leads to θ 2 x = (θ 1 x)/2. If we introduce this last equation in the third row of [START_REF] Shorack | Empirical Processes with Applications to Statistics[END_REF], we obtain

m 2 -m 1 = 1 4 (θ 1 x) 2 , x ∈ R,
which is impossible when x → +∞ and thus provides us the global identifiability of model [START_REF] Bordes | Semiparametric estimation of a two-component mixture model when a component is known[END_REF].

Remark. In Lemma 3.1 we have considered for simplicity the case where the slope parameter β is supposed to be different than zero. Actually, this condition can be technically relaxed if we allow θ to be equal to (α, 0) with α = 0. In fact, considering the first row of (24) and taking the limit as x → +∞, we obtain

β 2 = β 1 = 0.
To conclude, it is then enough to integrate [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] with respect to x over R which leads to discuss the same condition as in Bordes et al. (2006b, p. 735, eq. 3). Then Proposition 2 in Bordes et al. (2006b) provides an almost everywhere-type identifiability result which unfortunately cannot be strictly compared to the result stated in Lemma 3.1. For this reason we decided to reject θ = (α, 0), α ∈ R * from the sub-parametric space Φ, see (6).

Assumptions and asymptotic results

In the following we provide some general conditions ensuring the consistency of our estimation method as well as semiparametric almost sure rates of convergence.

Regularity conditions (R).

i) The pdfs f and f 0 are strictly positive over R and belong to F 3 .

ii) The pdfs f and f 0 are twice differentiable over R with f (j) ∞ < ∞ and f (j) 0 ∞ < ∞, where f (j) and f (j) 0

denote respectively the j-th order derivatives of f and f 0 , for j = 1, 2.

iii) The pdf

h satisfies R |x| 2 h(x)dx < ∞. iv) For i = 0 or i = 2, R 2 |x| i |F 0 (y + θ * x) -F 0 (y -θ * x)|h(x)dxdy < ∞,
and for i = 1 or i = 3, and all u ∈ R,

lim y→±∞ y i (F 0 (y + u) -F 0 (y -u)) = 0.
v) There exist two collections of functions { i,j } 0≤i≤j≤2 and 0 i,j 0≤i≤j≤2

belonging to L 1 (R 2 ) and such that, for all (x, y) ∈ R 2 and all θ ∈ Θ

|x i f (j) (y + (θ -θ * ) x)|h(x) ≤ i,j (x, y),
and |x i f (j) 0 (y + θ x)|h(x) ≤ 0 i,j (x, y).
For all z ∈ C, let denote respectively z and (z) the conjugate and imaginary part of z. We will also denote f , f0 the Fourier transforms of f , f 0 , and define for all

κ = (κ 1 , κ 2 ) ∈ R 2 , ν κ (t) := e itκ 1 h(κ 2 t)
, where h denotes the Fourier transform of h.

The following conditions mainly ensure the contrast property for the function d defined in [START_REF] Hall | Nonparametric estimation of component distributions in a multivariate mixture[END_REF]. We point out that these conditions are not equivalent, as it is already the case in Bordes and Vandekerkhove (2010, p. 25), to those established to prove the identifiability property in Lemma 3.1.

Contrast condition (C).

i) The density function h is not non-zero symmetric and the three first moments of

X satisfy E(X)(4E(X) 2 + 3E(X 2 )) = -E(X 3 ).
ii) The set of parameters ϑ = (p, θ) = (p, α, β) with p = p * that satisfies

p * (ν θ-θ * (t)) f (t) = (p * -p) (ν θ (t)) f0 (t), t ∈ R, (25)
is empty or does not belong to the parametric space Θ.

iii) The second order moments of f and f 0 , respectively denoted m and m 0 , are assumed to satisfy

m = m 0 + α 3 * + 3α 2 * β * E(X) + 3α * β 2 * E(X 2 ) + β 3 * E(X 3 ) 3(α * + β * E(X))
.

Remarks. In order to control the non-symmetry of h required in (C) i), one can use an adequate 95 %-confidence interval I X,95% for E(X) and check if 0 ∈ I X,95% , which should make suspect that E(X) = 0. If such a situation happens we advice to add a non-null constant c to the X in order to create a likely non-symmetric design data, i.e., 0 / ∈ I X+c,95% . This modification is slope-invariant and generates a new value for the intercept which is then equal to ρ := αβc. Obviously, if (ρ n , βn ) is a strongly consistent estimator of (ρ, β) with a certain rate of convergence, then αn = ρn + c βn inherits the same convergence properties. Note, in addition that if h is symmetric about a non-null constant, then condition (C) i) is satisfied since E(X) and E(X 3 ) have the same sign.

We also point out that condition (C) ii), which is necessary to get the contrast property for d over Θ, is quite difficult to figure out when nothing is specified on f , f 0 and h. We suggest, in the spirit of conditions C1 and C2 in Hohmann and Holzmann (2012), to consider the sufficient and more intuitive regularity comparisontype criterion for C ii)

∀θ ∈ Φ, (ν θ-θ * (t)) (ν θ (t)) f (t) f0 (t) -→ +∞ or 0, as t → +∞, (26)
which is valid since, according to [START_REF] Schuster | Using the bootstrap in testing symmetry versus asymmetry[END_REF], the term on left hand side of ( 26) is equal to

|p -p * |/p * ∈ [|p -p * |, 1/δ]
which is in contradiction with [START_REF] Titterington | Statistical Analysis of Finite Mixture Distributions[END_REF]. However, condition ( 25) is thoroughly discussed in the Gaussian case as done in the appendix, Section 6.1. We prove, in particular, that there exists at most one spurious solution satisfying ( 25) that has to be removed from the parametric space so it is not detected by our estimation algorithm as shown in Fig. ??.

Kernel and Bandwidth conditions (K).

i) The even kernel density function K is bounded, uniformly continuous, and square integrable, with bounded variation and second order moment.

ii) The even kernel density function K is twice differentiable and its first and second derivatives belong to L 1 (R).

iii) The bandwidth b n satisfies b n 0, nb n → +∞ and

√ nb 2 n = o(1).
We state in the following lemma some basic properties of the discrepancy function d and its estimator d n respectively defined in ( 13) and ( 14). (iii) Under condition (C) iii) we have

d(ϑ * ) = 2 R Ḣ(y, ϑ * ) ḢT (y, ϑ * )dQ(y) > 0.
(iv) Under conditions (R) and (K) i) and iii), for any γ > 0, d n converges to d almost surely with the rate

sup ϑ∈Θ |d n (ϑ) -d(ϑ)| = o a.s. (n -1/2+γ ).
The proof of this lemma is relegated to the appendix, Section 6.6.

We state in the following theorem the asymptotic properties of our Euclidean and functional estimators defined in ( 21), ( 22) and ( 22).

Theorem 3.1 i) If assumptions (R), (C) and (K) are satisfied then

θn -ϑ * 3 = o a.s. (n -1/4+γ ), γ > 0.
ii) The estimator fn of f defined in [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] converges almost surely in the

L 1 sense if n -1/4+γ /b n → 0, for all γ > 0.
iii) For any γ > 0, the estimator Fn of F defined in ( 22) converges uniformly at the following almost sure rate

Fn -F ∞ = O a.s. (n -1/4+γ /b n ) + O a.s. (b 2 n ), γ > 0. ( 27
)
The above rate is optimized by considering b n = n -1/12 , providing the rate of convergence O a.s. (n -1/6+γ ), for all γ > 0.

The proof of this theorem is relegated to the appendix, Section 6.7.

Comment. Points ii) and iii) reveal the intuitive idea that the bandwidth b n must not decrease too fast in order to allow the appropriate positioning of the plug-in centered data in the expression of Ψn, θn .

4 Numerical experiments

Role of the θ-transformation

We propose in this section to highlight the role played by the θ-transformation, see [START_REF] Bowman | Computational aspects of nonparametric smoothing with illustrations from the sm library[END_REF], in our method. For this purpose, we consider an example which corresponds to model [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF]

taking p * = 0.7, α * = 2, β * = 1, ε [j]
1 ∼ N (0, 1), j = 0, 1 and X 1 ∼ N (2, 3). In Fig. 1 we plot successively a simulated data set (X i , Y i ) 1≤i≤n , corresponding to the previous description with n = 200, and the two θ-transformed datasets obtained with θ = (1, 0.5) and θ = θ * = (2, 1). These figures are completed by adding their corresponding 2nd-coordinate sample data histograms. Note that these histograms are empirical estimates of the densities f θ defined in [START_REF] Cruz-Medina | Nonparametric estimation in semiparametric univariate mixture models[END_REF], with θ respectively equal to (0, 0), (1, 0.5) and (2, 1). We see clearly through these three situations how a progressive transformation of the data allows one to reach a tractable situation in the sense that it looks strongly like the semiparametric contamination model [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF] studied in Bordes et al. (2006b) where a known density is mixed with a symmetric unknown density, which corresponds to the behavior observed in the second row, third column histogram in Fig. 1. Loosely speaking the second idea of our method consists in arguing that once θ is close to θ * we are allowed to estimate the proportion p according to a Bordes and Vandekerkhove (2010) type-method which corresponds to the minimization step [START_REF] Maiboroda | Adaptive estimating equations for a location parameter constructed by using observations with admixture[END_REF]. In contrast to this technically satisfying idea, the θtransformation and the choice of the weight distribution Q introduced in (13) are two sources of difficulty which have to be considered carefully. In fact when β * is large and the law of the design data has heavy tails with respect to the tails of f , then the θ * transformation will move the points coming from the F 0 -population and located far from the origin to extremely distant positions, which implies intuitively that the integral type density involved in [START_REF] Bowman | Computational aspects of nonparametric smoothing with illustrations from the sm library[END_REF] should be extremely heavily tailed. Thus in order to capture the information contained in the tails of the θ-transformed data set it is important to sufficiently weight the empirical index of symmetry H 2 (x; p, Fn,θ , Ĵn,θ ) of expression ( 14) for large values of x, which reduces to choosing an instrumental distribution Q with non-negligible tails with respect to F θ * .

Preliminary discussion

The aim of this section is to illustrate graphically, on two-dimensionnal examples, the behavior of the empirical distance d n (p, 0, β) (the parameter α is assumed to be equal to zero) when p and β lie close to the true value of the parameter. For simplicity the parameter will still be denoted ϑ := (p, β), with θ := β and d n (ϑ) := d n (p, 0, β).

The interest of this study is to understand closely the influence of the mixing proportion p and the slope parameter β on the shape of the contrast function d (flatness, sharpness, smoothness, etc.). Our models of interest are denoted M1 and M2 and defined according to (2) as follows

M1: (p * , β * ) = (0.7, 1), V ∼ Q = N (0, 4 2 ), ε [j] ∼ N (0, 1), X ∼ N (2, 3 2 ), M2: (p * , β * ) = (0.3, 1), V ∼ Q = N (0, 2 2 ), ε [j] ∼ N (0, 1), X ∼ N (2, 3 2 ),
where j = 0, 1. In Fig. 2 ∼ N(0, 1), j = 0, 1, X1 ∼ N(2, 3 2 ), with the difference that on the left hand side p * = 0.7, V1 ∼ N(0, 4 2 ), when on the right hand side p * = 0.3 V1 ∼ N (0, 2 2 ).

plots the curve levels of d n evaluated here on a homogeneous 10 × 10 grid of the rectangular design domain. The observation of the first plot in Fig. 2 shows that the graph of d n looks like a sharp valley with a flat trough when β is almost located around β * and p evolves over [0.5,0.8], which induce the feeling that the proportion parameter estimation could be structuraly more instable than the estimation of the regression part factor. The observation of the second plot in Fig. 2 is more unexpected since the graph of d n does not really look like to a contrast function with its high near from p = 0.1 and its very large and flat trough that recovers almot all Θ 2 inducing a strong feeling of miss of robustness of our estimating method in that kind of situation.

To valid these feelings we propose to apply a large sample study on these two toy examples and a third intermediary one obtained with p = 3 and V ∼ Q = N (0, 4 2 ). The results of this study will be summarized in Table 1. Before that, let present the numerical approach used to approximate our M-estimator [START_REF] Hunter | Inference for mixtures of symmetric distributions[END_REF].

Gradient algorithm and tuning parameters. The gradient optimization procedure (programmed with Scilab) used to compute our M-estimator θn = (p n , βn ) is defined as follows: ∼ N(0, 1), j = 0, 1, X1 ∼ N(2, 3 2 ), with the difference that on the left hand side p * = 0.7, V1 ∼ N(0, 4 2 ), when on the right hand side p * = 0.3 V1 ∼ N (0, 2 2 ).

(i) Initialization: ϑ 1 = ϑ * , ϑ 2 = ϑ * + δ; imsart-
plots the curve levels of d n evaluated here on a homogeneous 10 × 10 grid of the rectangular design domain. The observation of the first plot in Fig. 2 shows that the graph of d n looks like a sharp valley with a flat trough when β is almost located around β * and p evolves over [0.5,0.8], which induce the feeling that the proportion parameter estimation could be structuraly more instable than the estimation of the regression part factor. The observation of the second plot in Fig. 2 is more unexpected since the graph of d n does not really look like to a contrast function with its high near from p = 0.1 and its very large and flat trough that recovers almot all Θ 2 inducing a strong feeling of miss of robustness of our estimating method in that kind of situation.

To valid these feelings we propose to apply a large sample study on these two toy examples and a third intermediary one obtained with p = 3 and V ∼ Q = N (0, 4 2 ). The results of this study will be summarized in Table 1. Before that, let present the numerical approach used to approximate our M-estimator [START_REF] Hunter | Inference for mixtures of symmetric distributions[END_REF].

Gradient algorithm and tuning parameters. The gradient optimization procedure (programmed with Scilab) used to compute our M-estimator θn = (p n , βn ) is defined as follows:

(i) Initialization: ϑ 1 = ϑ * , ϑ 2 = ϑ * + δ;
imsart-aos ver. 2010/09/07 file: "AOS mixreg".tex date: September 28, 2010 1 ∼ N (0, 1), j = 0, 1, X 1 ∼ N (2, 3 2 ), with the difference that on the left hand side p * = 0.7, V 1 ∼ N (0, 4 2 ), when on the right hand side p * = 0.3 V 1 ∼ N (0, 2 2 ). uated on a homogeneous 10 × 10 grid of the rectangular domain [0.5, 0.8] × [0.9, 1.1]. Fig. 2 shows that the graph of d n looks like a sharp valley with a flat trough when β is located near β * and p ranges [0.5,0.8]. Even if on this simulated example the argmin of d n is very close to the true value of the parameter, the previous remark suggests that the estimation of the mixing proportion will be less robust than the estimation of slope parameter. The observation of the second plot in Fig. 2 is more unexpected since the graph of d n does not really look like a contrast function, with its high near p = 0.1 and its very large and flat trough that covers most of Θ 2 , suggesting more instability in our estimating method in that kind of situation. To validate these thoughts, we propose applying a large sample study of these two examples and a third intermediary one obtained by considering model M2 with V ∼ N (0, 4 2 ). The results of this study are summarized in Table 1.

To compute our M-estimator θn = (p n , βn ) we programmed a constrained quasi-Newton FBGS optimization procedure, see [START_REF] Nocedal | Numerical Optimization[END_REF] and initialized at the true value of the parameter ϑ * . Note that in practice, intercept and slope parameters deduced from an "imaginary" straight line crossing the two extremities of the cluster likely labeled 1 and a proportion ratio evaluated by guesswork, provide in general a good initial position candidate for the optimization procedure.

The kernel K used to compute ( 19) is a standard Gaussian kernel and the bandwidth b n = 1 + 4p(1p)(4/(3n)) 1/5 (proposed by [START_REF] Bowman | Computational aspects of nonparametric smoothing with illustrations from the sm library[END_REF] for Gaussian distributions and implemented in R), both obviously satisfying condition (K). Two examples of stabilization/convergence of our gradient optimization procedure are illustrated in Fig 2, where the successive positions (until stabilization) of our algorithm are depicted by cross symbols. 1) where the model of interest is (5), with p = 0.7, µ = 3, and f 0 and f are respectively the pdfs corresponding to the N (0, 1) and N (0, (1/2) 2 ) distributions.

Even if these two models are not strictly comparable, we think that this comparison is very important to better understand the influence of the θ-transformation and the instrumental distribution Q on the performances of our generalized estimation method. From the numerical point of view, we see that the bias of our estimators, for both models, is negligible. However it also appears that the standard deviation associated to (p n , βn ) decreases significantly slower than the standard deviation associated to (p n , μn ) when n grows. The performance summarized in rows 4-6 of Table 1, which corresponds to p = 0.3 (which means that the population that will move far from its initial position due to the θ-transformation will be more important), is very highlightning. We observe that for small n (n = 100, 200) the standard deviations associated to (p n , βn ) are dramatically larger compared to those obtained with p = 0.7. Let denote by std(n, p * , β * , σ V ) the couple of standard deviations calculated in the last column of Table 1 under (n, p * , β * , σ V ). If we compute componentwise the ratios std(n, 0.3, 1, 4)/std(n, 0.7, 1, 4) respectively for n = 100, 200, 500, we obtain approximately (1.54, 1.8), (1.67, 1.44), and (0.95, 1.17), which suggests that when n becomes large the side effect of the θ-transformation vanishes (probably thanks to the size of n, which increases globally the precision of the empirical estimates, and the tails of Q, that allow the algorithm to take these improvements into account efficiently). The performances summarized in rows 7-9 of Table 1, confirms the concerns expressed about model M2. We recall that model M2 is badly affected by the two following drawbacks : smallness of p * (synonymous with important population shifted far by the θ-transformation and existence of a spurious solution) and a smallness of σ V which is then clearly not sufficient to counteract the smallness of p * (and its consequences). In particular we think that, in model M2, the empirical contrast d n is more easily closer to 0 under β * * = β * /2 since as explained in Section 4.1., this value is then significantly smaller than β * . This last remark explains why, in spite of the fact that our algorithms were initialized at the true parameter value, our estimates are strongly biased (attracted quite often by the spurious solution ϑ * * ).

Large sample study

We propose in this section to investigate the numerical performances of our method in more general situations through the four following models: WO (Weakly Overlapped), MO (Moderately Overlapped), SO (Strongly Overlapped), and MIX (MIXture noise) respectively defined, for all i ≥ 1, by: WO: (p * α * , β * ) = (0.7, 2, 1), ε

i ∼ N (0, 1), X i ∼ N (2, 3 2 ), MO: (p * α * , β * ) = (0.7, 2, 1), ε [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF] i ∼ N (0, 2 2 ), X i ∼ N (2, 3 2 ), SO: (p * , α * , β * ) = (0.7, 1, 0.5), ε

[1] i ∼ N (0, 2 2 ), X i ∼ N (1, 2 2 ), MIX: (p * , α * , β * ) = (0.4, 2, 1), ε [1] i ∼ 0.5N (-0.5, 0.5 2 ) + 0.5N (0.5, 0.5 2 ), X i ∼ N (2, 3 2 ),
where for all the above models ε

[0] i ∼ N (0, 1). The characteristics of our models of interest are illustrated in Fig. 3 and the performances of our method for various choices of V 's distribution, i.e., V ∼ N (0, σ 2 V ) with σ V ∈ {2σ, σ, σ/2} where σ denotes the empirical standard deviation of the Y i , are summarized in Table 2. Let us precise that in Table 2, the empirical mean of our estimators is given within parenthesis and the standard deviation is given within brackets. Comments on Table 2. First of all let us remark, as it is generally expected, that uniformly on the proposed choices for σ V , the less the test model is overlapped the best our method performs. Secondly it is important to notice that the best performances of our method are achieved in weakly overlapped situations (models WO and MIX) by taking σ V = σ/2 (smallest choice). Contrarily, it also appears that the best performances for model MO, respectively model SO, are achieved when considering respectively σ V = σ regarding to the estimation of (p * , β * ) or σ V = 2σ regarding to the estimation of α * . These observations actually validate the following intuitive idea: when the populations labelled 0 and 1 are weakly overlapped (small variance of each population with respect to the variance of the design), then the θ-transformation, for θ values near from θ * , places population 1 almost zerosymmetrically and very separately from population 0 (similarly to the situation illustrated in the last plot of Fig. 1). Hence by taking a small variance σ V associated to the weight probability distribution Q, we force our method to be focused on symmetry detection, according to ( 12) and ( 13), on a domain of R containing the most relevant information. On the other hand, when populations 0 and 1 are moderately or strongly overlapped, the previous "favorable" situation is no longer true, which obliges us to increase σ V in order to capture the symmetry information contained in the tails of the comparison-function [START_REF] Efron | Size, power and false discovery rate[END_REF].

Application: NimbleGen high-density array

In

Conclusion

We generalize in this paper the estimation method proposed in Bordes et al. (2006b) and [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF], for semiparametric mixture models where one component is known, to mixture of regressions models with one known component, extending consequently, for the first time, the statistical study of semiparametric mixture models to the bivariate case. We prove in particular that our model is semiparametrically identifiable under a set of conditions close to those suggested in Bordes et al. (2006b), and also propose an M-estimator based on the search of a "symmetry-point" when we apply a so-called θ-transformation to the original dataset.

Regarding our asymptotic results, we prove that our M-estimator ( 21) respectively our functional estimator [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF], converges towards ϑ * , respectively the cumulative distribution F , at the rate o a.s. (n -1/4+γ ) and O a.s. (n -1/6+γ ), for all γ > 0. Let us emphasize that the conditions stated to prove these results are generally satisfied in the Gaussian framework, nevertheless we also prove that there exists, in very particular cases, a spurious solution to our M-estimation problem which can be avoided or rejected by applying some explicit procedures discussed in this paper. We conclude the presentation of our method by discussing, on very simple toy examples, its numerical sensitivity to factors such as: the sample size n, the weight distribution Q involved in [START_REF] Hall | Nonparametric estimation of component distributions in a multivariate mixture[END_REF], the existence of a spurious solution etc. Following the conclusions coming from this preliminary numerical study, we carry out a large sample study in order to highlight the influence of the weight distribution Q (sharpness and tails) on the performances of our method through different kind of situations, i.e., from weakly overlapped to strongly overlapped simulated datasets. The main conclusion we learn from this study concerning the choice of the distribution Q can be summarized as follows: "in weakly overlapped situations let favour a distribution which probability mass is concentrated in a neighborhood of zero, when in strongly overlapped situations rather favour a heavily tailed distribution". Finally we conclude our study by considering a real data set extracted from the NimbleGen high-density array dataset previously analyzed by Martin-Magniette et al. (2008). It turns out from this study that the Euclidean estimator provided by our method is very close to the estimator found in Martin-Magniette et al. (2008), when our respective functional estimators of the pdf f differ slightly. In fact, we observe that our functional estimator places an extra bump on the left side of 0, which symptom brings up the possibility of having, in reality, a more complex link-structure between the ouptut Y and the design X than it is supposed in (1). This last remark indicates that it would be very challenging, in a futur work, to propose a semiparametric approach adapted to a more versatile class of mixture of regressions models, and also having the crucial ability to deal with very large datasets like the more and more popular high-density arrays.

6 Appendix

Conditions (R) and (C) in the Gaussian Case

In this section we discuss conditions (R) and (C) when the true underlying model is a contaminated Gaussian regression model with Gaussian design, i.e., f , f 0 , h are respectively the pdfs of the N (0, m), N (0, m 0 ), and N (E(X), σ 2 h ) distributions.

Comments on condition (R). Conditions (R) i-iii) are standard and easy to verify in the above model. On the other hand, it is interesting to show how conditions (R) iv-v) arise naturally in this case.

Condition (R) iv). We show for simplicity that the first condition in (R) iv) (the same kind of proof works also for the second one) holds when i = 0, θ * = (α * , β * ) ∈ R + * 2 and m 0 = 1. We write the decomposition

|F 0 (y + θ * x) -F 0 (y -θ * x)| = |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I y>1-θ * x, x<-α * β * + |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I y>1+θ * x, x≥-α * β * + |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I y<-1+θ * x, x<-α * β * + |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I y<-1-θ * x, x≥-α * β * + |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I -1+θ * x≤y≤1-θ * x, x<-α * β * + |F 0 (y + θ * x) -F 0 (y -θ * x)|1 I -1-θ * x≤y≤1+θ * x, x≥-α * β *
.

Consider the first term on the right hand side of the above decomposition (the three following terms being treated in entirely same way). For all y > 1θ * x with x < -α * /β * we have yθ * x > y + θ * x > 1. Since for t > 0 large enough, the following inequality is valid

exp(-t 2 ) √ 2π 1 t - 1 t 3 ≤ 1 -F 0 (t) ≤ exp(-t 2 ) t √ 2π ,
we have in particular that for all t > 1, 0 ≤ 1 -F 0 (t) ≤ exp(-t 2 )/ √ 2π. Hence it follows that for y > 1θ * x with x < -α * /β * :

|F 0 (y + θ * x) -F 0 (y -θ * x)| ≤ exp(-(y + θ * x) 2 ) + exp(-(y -θ * x) 2 ) √ 2π ,
which proves that this first term is h(x)dxdy integrable. Let us now sum the last two terms of the above decomposition and notice that

|F 0 (y + θ * x) -F 0 (y -θ * x)| × 1 I -1+θ * x≤y≤1-θ * x, x<-α β * + 1 I -1-θ * x≤y≤1+θ * x, x≥-α β * ≤ 21 I -1-|θ * x|≤y≤1+|θ * x| .
We thus prove that this sum of terms is also h(x)dxdy integrable.

Condition (R) v). We consider for simplicity the construction of the bounding function

0 1,1 when (α, β) ∈ Φ = [α, α] × [β, β], with (α, β) ∈ R + * 2 and m = m 0 = 1. Notice first that for all (x, y) ∈ R 2 |f (1) 0 (y + θ x)| ≤ |y| + α + β|x| √ 2π exp - (y + α + βx) 2 2 . (28)
Secondly, noticing that

   0 ≤ y + βx ≤ y + α + βx if x ≥ 0, y ≥ 0, y + α + βx ≤ y + α + βx ≤ y + α + βx if x ≥ 0, y ≤ 0, y + α + βx ≤ y + α + βx ≤ y + α + βx if x ≤ 0, y ∈ R,
we then obtain that for all (x, y) ∈ R 2 and all θ ∈ Φ:

exp - (y + α + βx) 2 2 ≤ exp - (y + βx) 2 2 1 I x≥0,y≥0 + exp - min(|y + α + βx|, |y + α + βx|) 2 2 1 I x≥0,y≤0 + exp - min(|y + α + βx|, |y + α + βx|) 2 2 1 I x≤0,y∈R ≤ B Φ (x, y),
where

B Φ (x, y) := exp - (y + βx) 2 2 + exp - (y + α + βx) 2 2 + exp - (y + α + βx) 2 2 + exp - (y + α + βx) 2 2 + exp - (y + α + βx) 2 2 .
Since the product B Φ (x, y)h(x) ∝ B Φ (x, y) exp(-x 2 /2) has a Gaussian tail, and thus admit any kind of moment, we can propose according to [START_REF] Yau | Finite mixture regression model with random effects: application to neonatal hospital length of stay[END_REF] and (R) v), Comments on condition (C) ii). In the whole Gaussian case, i.e., when h(•) = N E(X),σ 2 h (•), f 0 (•) = N 0,m 0 (•) and f = N 0,m (•), expression of (25) becomes:

p * sin((α -α * ) + (β -β * )E(X))t) exp - t 2 2 (σ 2 h (β -β * ) 2 + m) = (p * -p) sin((α + βE(X))t) exp - t 2 2 (σ 2 h β 2 + m 0 ) . (29)
When p = p * we already know that condition (C) i) is satisfied (see Remarks on condition (C) in Section 3.2). When p = p * , we denote for convenience ξ := α + βE(X), ξ * := α * +β * E(X), Σ β-β * := σ 2 h (β -β * ) 2 +m, and Σ β := σ 2 h β 2 +m 0 . Taking the first and third order derivative of (29) at point t = 0 we get the conditions

p * ξ * = pξ, and (p -p * )ξ(3Σ β-β * + ξ 2 ) + p * (ξ -ξ * )(3Σ β + (ξ -ξ * ) 2 ) = 0. ( 30
)
Introducing the first relation in [START_REF] Yu | Semiparametric Regression: Efficiency Gains From Modeling the Nonparametric Part[END_REF] into the second one, we obtain

p -p * p ξ * p * 3[Σ β -Σ β-β * ] + 2p * p -p 2 p 2 ξ 2 * = 0. ( 31
)
Now we observe that, to ensure the validity of expression [START_REF] Young | Mixtures of Regressions with Predictor-Dependent Mixing Proportions[END_REF], the factors multiplied by the sin terms on both sides of (29) must be, at least, equivalent as t → ∞. This last remark implies that Σ β = Σ β-β * , or equivalently β = β * /2 + (m 0m)/2β * σ 2 h , and thus [START_REF] Zhu | Hypothesis testing in mixture regression models[END_REF] leads to p = 2p * .

(32)

Using now the first relation in ( 29) and (32), we then obtain

α = α * /2 + E(X)(m 0 - m)/2β * σ 2 h .
The consequences of the previous comments can be presented as follows:

Discussion (CG): i) If p * > 1/
2 then the set of parameters ϑ ∈ Θ satisfying condition ( 29) is always empty, since p = 2p * > 1 is not an admissible solution.

ii) If p * ≤ 1/2 and if, for example, E(X) = 0 or m 0 = m, then ϑ * * = (2p * , α * /2, β * /2).

In such a case it would be crucial to build a conveniently constrained parametric space (most of the time a plot of the dataset helps in building reasonable constraints on the intercept and slope parameter spaces) expecting that it contains ϑ * but not ϑ * * .

iii) More generaly one can expect that when the shape of the sample data (see e.g. Fig. 1) suggest that (m 0m)/2β * σ 2 h is negligible with respect to α * and β * , which occurs when m 0 is close to m or/and σ 2 h β * is very large, then the solution proposed in ii) is loosely speaking still valid. Otherwise, if the situation is too much tricky to define visually a convenient parametric space and that d n has several local minima (which should make suspect the existence of various spurious solutions) one may select the arg min of d n that better recreate the density g over a compact set C large enough of R 2 . Namely if ĝn,ϑ is the ϑ-plugged version of (3) and ĝn is a kernel density estimate of g, we select the ϑ, among the finite set of arg min values, which minimizes C |ĝ n,ϑĝn |dλ ⊗2 . Actually such a methodolgy is asymptotically consistent since model ( 3) is known to be identifiable over R 2 under conditions established in Lemma 3.1.

Explicit formula of H(•, ϑ) and its derivatives

In this section all the expressions are valid for all (ϑ, y) ∈ Θ×R, and the computation of the various derivative functions (under the integral sign) are all allowed according to the Lebesgue Differentiation Theorem and condition (R). According to [START_REF] Cruz-Medina | Nonparametric estimation in semiparametric univariate mixture models[END_REF] and [START_REF] Cuzick | Efficient estimates in semiparametric additive regression models with unknown error distribution[END_REF], we have

H(y, ϑ) = p * p y -∞ R f (z + (θ -θ * ) x)h(x)dxdz + -y -∞ R f (z + (θ -θ * ) x)h(x)dxdz + p -p * p y -∞ R f 0 (z + θ x)h(x)dxdz + -y -∞ R f 0 (z + θ x)h(x)dxdz -1.
For simplicity we introduce

F θ (y) = y -∞ R f (z + (θ -θ * ) x)h(x)dxdz, F θ 0 (y) = y -∞ R f 0 (z + θ x)h(x)dxdz, which leads to ∂ ∂p H(y, ϑ) = - p * p 2 (F θ (y) + F θ (-y)) -(F θ 0 (y) + F 0 θ(-y)) .
Let us denote

Ḟ α (y) := ∂ ∂α F θ (y) = y -∞ R ḟ (z + (θ -θ * ) x)h(x)dxdz, Ḟ α 0 (y) := ∂ ∂α F θ 0 (y) = y -∞ R ḟ0 (z + θ x)h(x)dxdz,
and for

Ḟ β (y) := ∂ ∂β F θ(y) = y -∞ R x ḟ (z + (θ -θ * ) x)h(x)dxdz, Ḟ β 0 (y) := ∂ ∂β F 0 θ(y) = y -∞ R x ḟ0 (z + θ x)h(x)dxdz, we obtain ∂ ∂α H(y, ϑ) = p * p Ḟ α (y) + Ḟ α (-y) + p -p * p Ḟ α 0 (y) + Ḟ α 0 (-y) . ∂ ∂β H(y, ϑ) = p * p Ḟ β (y) + Ḟ β (-y) + p -p * p Ḟ β 0 (y) + Ḟ β 0 (-y) .
At point ϑ = ϑ * the Hessian matrix of H(•, ϑ) defined in (36) is obtained by considering

Ḣ(y, ϑ * ) =     1 p F θ * 0 (y) + F θ * 0 (-y) -1 2f (y) 2f (y)E(X)     . (33) Let us denote now F β,α (y) = ∂ 2 ∂β∂α F θ (y) = y -∞ R x f (z + (θ -θ * ) x)h(x)dxdz, F β,α 0 (y) = ∂ 2 ∂β∂α F θ 0 (y) = y -∞ R x f0 (z + θ x)h(x)dxdz, F α,α (y) = ∂ 2 ∂α 2 F θ (y) = y -∞ R f (z + (θ -θ * ) x)h(x)dxdz, F β,β 0 (y) = ∂ 2 ∂β 2 F θ 0 (y) = y -∞ R x 2 f0 (z + θ x)h(x)dxdz.
We then obtain

∂ 2 ∂p 2 H(y, ϑ) = p * p 3 (F θ (y) + F θ (-y)) -(F θ 0 (y) + F 0 θ(-y)) , ∂ 2 ∂u∂p H(y, ϑ) = - p * p 2 ( Ḟ u (y) + Ḟ u (-y)) -( Ḟ u 0 (y) + Ḟ u 0 (-y)) , u = α, β, ∂ 2 ∂u∂v H(y, ϑ) = p * p F u,v (y) + F u,v (-y) + p -p * p F u,v 0 (y) + F u,v 0 (-y) , u = α, β, v = α, β.

Boundedness

Boundedness of Ψ θ (•) and Ψ θ (•). If f and f 0 are supposed to be bounded over R then we clearly have from [START_REF] Cruz-Medina | Nonparametric estimation in semiparametric univariate mixture models[END_REF] that

|Ψ θ (y)| ≤ f ∞ + f 0 ∞ , (θ, y) ∈ Φ × R.
The same kind of argument holds to prove boundedness of ḟθ (•) when (R) ii) is supposed.

Boundedness of H(•, ϑ). Since for all θ ∈ Φ the functions F θ (•) and J θ (•) are both cdfs, we thus have, since δ ≤ p ≤ 1δ, from expression (12):

H(y, ϑ) ≤ 4 δ + 1, (y, ϑ) ∈ R × Φ. ( 34
)
6.4 Integrable Lipschitz property of Ψ θ (•)

From ( 8), for all (y, θ, θ ) ∈ R × Φ 2 we have

|Ψ θ (y) -f Ψ θ (y)| ≤ p * R |f (y + (θ -θ * ) x) -f (y + (θ -θ * ) x)|h(x)dx +(1 -p * ) R |f 0 (y + θ x) -f 0 (y + θ x)|h(x)dx. ( 35
)
Consider for simplicity the first integral term on the right hand side of (35) (the same argument holding for the second term). According to the Mean Value Theorem there exists, for all (x, y) ∈ R 2 and (θ, θ ) ∈ Φ 2 , a value γ := γ(x, y, θ, θ ) belonging to the line segment with extremities y + (θθ * ) x and y + (θθ * ) x, or equivalently a θ := θ(x, y, θ, θ ) belonging to the line segment with extremities θ and θ such that

γ = y + ( θ -θ * ) x and |f (y + (θ -θ * ) x) -f (y + (θ -θ * ) x)| = | ḟ (γ)(α -α + (β -β )x)| = | ḟ (y + ( θ -θ * ) x)(α -α + (β -β )x)| ≤ sup θ∈Φ | ḟ (y + (θ -θ * ) x)|(|α -α | + |β -β ||x|).
From condition (R) ii) there thus exists a nonnegative constant c such that

R |Ψ θ (y) -Ψ θ (y)|dy ≤ j=0,1 (|α -α | 1-j + |β -β | j ) R×R |x| j (sup θ∈Φ | ḟ (z + (θ -θ * ) x)| + sup θ∈Φ | ḟ0 (z + θ x)|)h(x)dxdz ≤ c θ -θ 2 .

Uniform Lipschitz property of H(•, ϑ)

Let us write

H(y, ϑ) -H(y, ϑ ) = 1 p (F θ (y) -F θ (y) + F θ (-y) -F θ (-y)) + p -p pp (F θ (y) + F θ (y)) + 1 -p p (J θ (y) -J θ (y) + J θ (-y) -J θ (-y)) + p -p pp (J θ (y) + J θ (-y)).
To prove the uniform Lipschitz property of H(•, ϑ) we need to prove it for J θ (•) and F θ (•). We begin with the simplest term J θ (•). According again to the mean value theorem, for all y ∈ R, all (x, z) ∈ R 2 with z ≤ y, and all (θ, θ ) ∈ Φ 2 there exists θ := θ(x, z, θ, θ ) belonging to the line segment with extremities θ and θ such that

|J θ (y) -J θ (y)| ≤ y -∞ R |f 0 (z + θ x) -f 0 (z + (θ x)|h(x)dxdz = y -∞ R | ḟ0 (z + θ x)(|α -α | + |θ -θ ||x|)h(x)dxdz, ≤ y -∞ R sup θ∈Φ | ḟ0 (z + θ x)h(x)dxdz|α -α | + y -∞ R |x| sup θ∈Φ | ḟ0 (z + ux)h(x)dxdz|β -β | ≤ c θ -θ 2 ,
where c denotes a nonnegative constant arising from condition (R) ii). Using the same kind of argument we prove that there exists a nonnegative contant c such that for all (y, θ, θ

) ∈ R × Φ 2 |F θ (y) -F θ (y)| ≤ c θ -θ 2 .
In conclusion, for all y ∈ R, there exists a nonnegative constant c such that for all (y, ϑ, ϑ

) ∈ R × Θ 2 |H(y, ϑ) -H(y, ϑ )| ≤ 2 δ (c + c ) θ -θ 2 + 4 |p -p | δ 2 ≤ c ϑ -ϑ 3 .
6.6 Proof of Lemma 3.2 i) From boundedness and the uniform Lipschitz property of H(•, ϑ), along with the integrability and the integrable Lipschitz property of f θ (•) proved in Sections 6.3, 6.4 and 6.5, there exists a nonnegative constant c such that for all (ϑ, ϑ

) ∈ Θ 2 R H 2 (y, ϑ)dQ(y) - R H 2 (y, ϑ )dQ(y) ≤ R H(y, ϑ) + H(y, ϑ ) H(y, ϑ) -H(y, ϑ ) q(y)dy ≤ c ϑ -ϑ 3 ,
which concludes the proof of i).

ii) To clarify the similarity between the semiparametric contamination model (5) studied in Bordes et al. (2006b) and the contaminated regression model (3), we can say that f θ (•) plays the role of g(•µ) and that I θ (•) plays the role of f 0 (•µ).

If ϑ = ϑ * then d(ϑ) = 0. To prove the converse we notice that d(ϑ) = 0, which implies, since H 1 (•, ϑ) and H 2 (•, ϑ) are continuous and q > 0 over R, that

H 1 (•; ϑ) = H 2 (•; ϑ) which leads, for almost all y ∈ R, to Ψ θ (y) -(1 -p)I θ (y) = Ψ θ (-y) -(1 -p)I θ (-y).
Using formula (8), we obtain

p * R f (y + (θ -θ * ) x)h(x)dx + (p -p * )I θ (y) = p 0 R f (-y + (θ -θ * ) x)h(x)dx + (p -p * )I θ (-y), y ∈ R.
Considering the Fourier transform of the previous equality, using Fubini's Theorem, and noticing that f and f0 are real-valued functions, it follows that

p * e -it(α-α * ) f (t) h((β -β * )t) + (p -p * )e -itα f0 (t) h(βt) = p * e it(α-α * ) f (t) h((β -β * )t) + (p -p * )e itα f0 (t) h(βt), t ∈ R.
Using the notation introduced for the writing of condition (C), the previous equation becomes [START_REF] Schuster | Using the bootstrap in testing symmetry versus asymmetry[END_REF].

Suppose that p = p * and take the first and third order derivative of (25) at point t = 0. We then obtain α

-α * + (β -β * )E(X) = 0 and 3m[α -α * + (β -β * )E(X)] + (α -α * ) 3 + 3(α -α * ) 2 (β -β * )E(X) + 3(α -α * )(β -β * ) 2 E(X 2 ) + (β -β * ) 3 E(X 3 ) = 0, which leads to (β -β * )(4E(X) 3 + 3E(X)E(X 2 ) + E(X 3 )) = 0, and thus implies that θ = θ * if 4E(X) 3 + 3E(X)E(X 2 ) + E(X 3 ) = 0. Suppose now that p = p * , then condition (C) ii) requires that ϑ = ϑ * . iii) First we have d(ϑ * ) = 2 R Ḧ(y, ϑ * )H(y, ϑ * ) + Ḣ(y, ϑ * ) ḢT (y, ϑ * ) q(y)dy = 2 R Ḣ(y, ϑ * ) ḢT (y, ϑ * )q(y)dy, (36)
according to [START_REF] Cuzick | Semiparametric additive regression[END_REF] and the fact that H(•, ϑ * ) = 0 on R. Let v be a vector in R 3 . We have

v T d(ϑ * )v = 2 R v T Ḣ(y, ϑ * ) 2 q(y)dy ≥ 0.
It follows that d(ϑ * ) is a positive 3 × 3 real valued matrix. Let us show that it is also definite. If v ∈ R 3 is a non-null column vector such that v T d(θ * )v = 0, then v T Ḣ(y, ϑ * ) = 0 for almost all y ∈ R. According to (33) in the appendix, we have to discuss the proportionality of f and F θ * 0 (•) + F θ * 0 (-•) -1. Because f 0 is an even density, we have from Fubini's theorem

f (y) = R [F 0 (y + θ * x) -F 0 (y -θ * x)]h(x)dx R 2 [F 0 (y + θ * x) -F 0 (y -θ * x)]h(x)dxdy . ( 37
)
Using integration by parts and assumption (R) iv), the denominator of the right hand side of (37) can be expressed as follows Convergence rate of T

1,n . For simplicity we will suppose that proj 2 (Φ) ⊂ [0, A], where A is a nonnegative real number and for all (x, y) ∈ R 2 , proj 2 : (x, y) → y. Let us introduce P X n = n -1 n i=1 δ X i the empirical measure associated to the iid sample (X 1 , . . . , X n ) with common probability distribution P X with pdf and cdf respectively denoted by h and H). We use the functional notation P f = f dP . Notice now that, according to expression [START_REF] Hunter | Semiparametric Mixtures of Regressions[END_REF], we have for all y ∈ R:, Ĵn,θ (y) -J θ (y) = 1 n n i=1 F 0 (y + α + βX i ) -E(F 0 (y + α + βX))

= (P X n -P X )F 0 (y + α + β •).

Let us consider the class of functions

F 0 = {x → F 0 (u + βx); u ∈ R, β ∈ [0, A]} .
Since (P X n -P X )F 0 (u + β •) = (P X n -P X )F 0 (y + β(• ∨ 0)) + (P X n -P X )F 0 (y + β(• ∧ 0)), it is enough to study the empirical process indexed by the classes of functions

F + 0 = {x → F 0 (u + β(x ∨ 0)); u ∈ R, β ∈ [0, A]} , F - 0 = {x → F 0 (u + β(x ∧ 0)); u ∈ R, β ∈ [0, A]} .
For simplicity we denote Γ y,α (x) = F 0 (y + α(x ∨ 0)) and only consider the class F + 0 , the class F - 0 being treated in a entirely same way. Since F 0 is a cdf, for β 1 ≤ β ≤ β 2 and u 1 ≤ u ≤ u 2 we have

Γ u 1 ,β 1 (x) ≤ Γ u,β (x) ≤ Γ u 2 ,β 2 (x),
x ∈ R, and, since F 0 is supposed to be Lipschitz,

0 ≤ Γ u 2 ,β 2 (x) -Γ u 1 ,β 1 (x) ≤ C(u 2 -u 1 + (β 2 -β 1 )(x ∨ 0)).
Let us consider now ε > 0, and (u ε , u ε ) ∈ R 2 such that F 0 (u ε ) ≥ 1ε, and F 0 (u ε ) ≤ ε.

Note that u ε and u ε do not depend on β. For all N ∈ N, define

u ε = u 1,ε ≤ u 2,ε ≤ . . . ≤ u N,ε = u ε ,
recalling that E(F n,θ (y)) = F θ (y). The first term on the right hand side of (42) satisfies Uniform almost sure rate of convergence of T 2,n . Considering for all i ≥ 0, the random variable W i (ϑ) := H 2 (V i ; ϑ), where ϑ ∈ Θ, we see that where the right hand term is the supremum of an empirical process indexed by a class of Lipschitz bounded functions, which is known to be o a.s. (n -1/2+γ ) for all γ > 0, see Bordes et al. (1996b) for details, which concludes the proof.

6.7 Proof of Theorem 3.1

i) The proof follows entirely the proof of Theorem 3.1 in [START_REF] Bordes | Semiparametric Two-Component Mixture Model with a Known Component: an Asymptotically Normal Estimator[END_REF] and uses the technical results proved in Lemma 3.2.

ii) Consider the following decomposition:

| fn -f | = 1 pn
Consider K a kernel density satisfying (K) ii). We propose to study in a generic way the difference of kernels involved on the right hand side of the above expression. For all (w, z) ∈ R 2 , and letting h := (zw)/b, we write the second-order Taylor expansion with integral remainder term:

K t -w b -K t -z b = h K t -z b + h 2 2 1 0 (1 -u) K t -
|X i | + |X i | 2 , (48)

1 .

 1 Isolating f in (9) and replacing ϑ * = (p * , θ * ) by ϑ = (p, θ) one can define a new parametric class of functions F Θ := {f ϑ : ϑ ∈ Θ} where for all (y, ϑ)

Lemma 3 . 2

 32 (i) Under conditions (R) the function d is Lipschitz over Θ. (ii) Under conditions (C) i) and ii) the function d is a contrast function, i.e., for all ϑ ∈ Θ, d(ϑ) ≥ 0 and d(ϑ) = 0 if and only if ϑ = ϑ * .

Figure 1 :

 1 Figure 1: First row: respectively plot of an original data (X i , Y i ) 1≤i≤n according to model (3) with n = 200, plot of a wrong θ-tranformation (θ = θ * ), plot of the true θ * -tranformation. Second row: respectively histograms of the corresponding first row 2nd-coordinate sample data.

Fig 2 .

 2 Fig 2. Plot of (p, β) → dn(p, 0, β) with n = 100, β * = 1, ε [j] 1

Fig 2 .

 2 Fig 2. Plot of (p, β) → dn(p, 0, β) with n = 100, β * = 1, ε [j] 1

Figure 2 :

 2 Figure 2: Plot of (p, β) → d n (p, 0, β) with n = 100, β * = 1, ε [j]

10 Figure 3 :

 103 Figure 3: Plots of original data (X i , Y i ) 1≤i≤n with n = 200, generated respectively from models WO, MO, SO and MIX.

Figure 4 :Figure 5 :

 45 Figure 4: Respectively the plots of an extracted data set of size n = 30000 coming from the original NimbleGen high-density array dataset considered in Martin-Magniette et al. (2008), its centered version and the histogram of the design distribution.

0 1 , 1

 11 (x, y) = |x|(|y| + α + β|x|)/ √ 2πB Φ (x, y) exp(-x 2 /2) as a candidate for the uniformly bounding function satisfying condition (R) v).

R 2 [ 2 R||

 22 F 0 (y + θ * x) -F 0 (yθ * x)]h(x)dxdy = R [y(F 0 (y + θ * x) -F 0 (yθ * x)] ∞ -∞ h(x)dx -R R y(f 0 (y + θ * x)f 0 (yθ * x))dyh(x)dx = (α * + β * x)h(x)dx = 2(α * + β * E(X)). Ĵn,θ (y) -J θ (y)|, Fn,θ (y) -F θ (y)|.

Fn,θ (k 2 2 .

 2 y) -E( Fn,θ (y)) = R K(yu)d(F n,θ -E(F n,θ ))(u) = R (F n,θ -E(F n,θ ))(u)dK(yu) = R (F n,θ -E(F n,θ ))(ys)dK(s). Thus, if we denote ∆ n,θ (y) := F n,θ (y) -E(F n,θ (y)) = F n,θ (y) -F θ (y), we obtain Fn,θ (y) -E( Fn,θ (y)) -[F n,θ (y) -E(F n,θ (y))] ≤ R (∆ n,θ (ys) -∆ n,θ (y))dK(s) ≤ sup (θ,y)∈Φ×R |∆ n,θ (y)| K T V .The last bias-term on the right hand side of (42) can be studied using the R 2n bound in Shorack and Wellner (1986, p. 766, eq. e), which establishes that for each θ ∈ Φsup y∈R |E( Fn,θ ) -F θ |(y) ≤ ḟθ ∞ (43) If K is replaced by K n (•) = K(•/b n ) and we let k 2,n := R x 2 dK n (x) = b 2 n k 2 , then (41-43) lead to lim sup n→∞ n log log(n) sup (θ,y)∈Φ×R (F n,θ -F θ )(y) < ∞ a.s.,(44)whenever lim sup(n/ log log(n)) 1/2 k 2,n < ∞ which holds when lim supn→∞ n log log(n) b 2 n < ∞,and sup θ∈Φ ḟθ ∞ < ∞ which has been proved in Section 5.3 under Condition (R) ii).

W

  i (ϑ) -E(W i (ϑ)) ,

r u b du where r u := ( 1 -

 1 u)z + uw. Using the symmetry of K and Fubini'dt (47) Replacing w, z respectively by the Y θn i and Y θ * i , and b by b n in (47) we then obtain from (46) the following bound for the L 1 error:Ψn, θn (t) -Ψn,θ * (t) L 1 ≤ C θ nθ * 2 b n × 1 n n i=1

  we plot the mapping (p, β) → d n (p, β) obtained from an M1-sample, respectively M2-sample, of size n = 100, where (p, β) ∈ Θ 1 = [0.5, 0.8] × [0.9, 1.1], respectively (p, β) ∈ Θ 2 = [0.1, 0.6] × [0.6, 1.4]. Notice that according to discussion (CG) at the end of Section 5.1, since p * < 1/2 and m = m 0 , model M2 is not necessarily consistently estimated since the parameter space Θ 2 contains the spurious solution ϑ * * = (2p * , β * /2) (situation we voluntary want to investigate from the numerical point of view). Fig. 2 which plots the level curves of d n eval-16 P. VANDEKERKHOVE the parasite solution ϑ * * = (2p * , β * /2), which is voluntarily the case here. In practice, Fig. 2 is obtained using the Scilab contour2d function which

	1.10
			0.000271			0.00953
						0.00721
	1.05
						0.0049
	1.00
					0.00258
			0.00721 0.00953	0.0049
	0.95	0.0188	0.0118 0.0142
		0.0165
	0.0211 0.0234 0.0257 0.0304 0.028 0.0327
	0.035 0.0373
	0.90
	0.50	0.55	0.60	0.65	0.70	0.75	0.80

  aos ver. 2010/09/07 file: "AOS mixreg".tex date: September 28, 2010 16 P. VANDEKERKHOVE the parasite solution ϑ * * = (2p * , β * /2), which is voluntarily the case here. In practice, Fig. 2 is obtained using the Scilab contour2d function which

	1.4
					0.099
	1.3
	1.2
	1.1
				0.000604
	1.0		0.099 0.197 0.296
	0.9	0.394 0.493 0.591 0.689 0.788 0.886 1.08 1.18 0.984
	0.8	1.28 1.38 1.48 1.67 1.57
	0.7	1.77 1.87 1.97 2.07 2.17 2.26 2.36 2.46 2.56 2.66 2.76 2.85 2.95 3.05 3.15 3.25 3.35
	0.6	3.54 3.64 3.74
	0.5
		0.1	0.2	0.3	0.4	0.5	0.6	0.7

  , applied to the score function ḋn := ∂ ∂p d n , ∂ ∂β d n T

Table 1 :

 1 Mean and Std. Dev. of 100 estimates of (p

	deviation

* , β * ). n (p * , β * , σ V ) Empirical means Standard

Table 1 .

 1 First of all, it is interesting to compare the performances summarized in rows 1-3 of Table1to those obtained inBordes and Vandekerkhove (2010, p. 35, Table

Table 2 :

 2 Mean and Std. Dev. of 200 estimates of (p * , α * , β * ).

	Model	n	2σ	σ	σ/2
	WO	100 (0.700,1.995,1.006) (0.696,2.001,1.007) (0.704,1.990,1.003)
			[0.065,0.089,0.074] [0.061,0.097,0.081] [0.065,0.081,0.072]
	WO	300 (0.701,1.998,1.005) (0.696,1.995,0.999) (0.704,1.996,0.994)
			[0.034,0.063,0.045] [0.035,0.050,0.042] [0.031,0.034,0.038]
	MO	100 (0.718,1.962,0.971) (0.732,1.975,0.956) (0.726,1.980,0.957)
			[0.083,0.148,0.179] [0.082,0.134,0.178] [0.088,0.165, 0.171]
	MO	300 (0.704,2.022,0.966) (0.707,1.983,0.994) (0.717,1.986,0.978)
			[0.050,0.080,0.116] [0.045,0.074,0.099] [0.056,0.072,0.114]
	SO	100 (0.766,0.958,0.444) (0.748,0.956,0.465) (0.728,0.961,0.471)
			[0.104,0.200,0.136] [0.099,0.219,0.128] [0.098,0.211,0.134]
	SO	300 (0.748,0.970,0.472) (0.729,0.999,0.471) (0.723,0.998,0.474)
			[0.079,0.136,0.084] [0.069,0.156,0.083] [0.080,0.190,0.111]
	MIX 100 (0.415,1.984,0.983) (0.408,1.964,1.016) (0.409,1.988,1.011)
			[0.070,0.133,0.187] [0.073,0.172,0.213] [0.062,0.189,0.174]
	MIX 300 (0.403,1.988,1.009) (0.407,1.974,1.013) (0.407,1.999,0.989)
			[0.043,0.115,0.133] [0.039,0.158,0.138] [0.037,0.075,0.094]

  Ψn, θn -Ψn, θn -Ψn,θ * | + | Ψn,θ * -Ψ θ * | + Ψ θ *

	= ≤	1 p * ( Ψn, θn -Ψn,θ * ) + Ψ θ * -1 -pn 1 pn pn ( Ĩn, θn -Ĩn,θ * ) + 1 -pn pn Ĩn, θn -1 pn Ψn,θ * -1 Ψ θ * 1 -p * p * p * 1 -pn pn Ĩn,θ * -1 -p * I θ * I θ * p * pn -1 | 1 pn -p * 1 + 1 -pn pn | Ĩn, 1 nb n n i=1 K t -Y θn i b n -K t -Y θ * i b n	.

θn -Ĩn,θ * | + | Ĩn,θ * -I θ * | +I θ * 1pn pn -1p * p * .

(45)

It is now enough to study the behavior of | Ψn, θn -Ψn,θ * | and | Ĩn, θn -Ĩn,θ * |. For all t ∈ R, we have | Ψn, θn (t) -Ψn,θ * (t)| ≤

Acknowledgments. The author thanks the referees for their helpful and constructive comments. He is also very gratefull to Philippe Barbe, David Hunter and Marie-Laure Martin Magniettes for their help and good advice during the writing of this manuscript.

for our method. Let us recall the value of these estimated parameters respectively obtained under the homoscedastic (Ho) and heteroscedastic (He) assumption: pHo = 0.26, âHo 0 = 1.47, bHo 0 = 0.82, âHo 1 = -0.47, bHo 1 = 1.16, σHo 0 = 0.42, σHo 1 = 0.42, pHe = 0.32, âHe 0 = 1.48, bHe 0 = 0.81, âHe 1 = -0.29, bHe 1 = 1.13, σHe 0 = 0.56, σHo 1 = 0.80.

Considering that (â Ho 0 , bHo 0 ) (a 0 , b 0 ) is a good candidate to recenter our dataset according to Y = Y -(a 0 + b 0 X) (this fact is assessed in Fig. 4), our model of interest [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF] is then obtained by considering (α, β) = (a1 -âM 0 , b1 -bM 0 ) (so-called centered version). We consider in addition that the pdf f 0 is Gaussian with mean 0 and standard deviation σ0 = 0.686, this value corresponding to the empirical standard deviation of the Y coming from couples (X, Y ) such that X > 14 (we consider that upon this threshold the size of the population coming from label 1 is negligible with respect to the size of the population coming from label 0). To estimate ϑ * = (p * , α * , β * ), we initialize our algorithm at θM = (p M 0 , âM 1 -âM 0 , bM 1 -bM 0 ) = (0.26, -1.94, 0.34), which provides the following estimation: θ = (p, α, β) = (0.31, -1.95, 0.34).

In the left side of Fig. 5 we plot our regularized estimation of the density f

where fn is defined in [START_REF] Martin-Magniette | ChIPmix: mixture model of regressions for two-color ChIP-chip analysis[END_REF] and compare it to the estimation proposed by Martin-Magniette et al. (2008). It is interesting to observe that this two estimations are surprisingly close (except the small bump located between -4 and -2 which insinuate that our method has detected a 2-mixture structure for f ). In the right side of Fig. 5 we suggest, in order to check the relevancy of the different methods, to reconstruct the function Ψ θ * defined in (9) by considering a kernel density estimate based on the Y -(α + βX) (free from mixture model structure), respectively the plug-in estimation obtained by our method and the method of Martin-Magniette et al. (2008), when we substitute (ϑ * , f 0 , f ) in [START_REF] Cuzick | Semiparametric additive regression[END_REF] respectively by ( θ, f N (0,σ 0 ) , fn ) and ( θHe , f N (0,σ He 0 ) , f N (0,σ He 1 ) ) (the best reconstruction is actually obtained under the heteroscedastic assumption), the integral term being estimated adequately by using [START_REF] Hohmann | Semiparametric location mixtures with distinct components[END_REF]. We observe in that case that the semiparametric approach allows a better fitting than the purely parametric approach, except between -3 and -1 where both methods have difficulties in fitting the shape of the target nonparametric estimator. This point should suggest that the true underlying model is in reality more complex than a simple 2-component mixture of regressions model. It would be for example interesting to investigate a generalization of model [START_REF] Benjamini | Controlling the false discovery rate: a pratical and powerful approach to multiple testing[END_REF] where the design itself depends on the random label U , i.e., X := X [i] if U = i, for i = 0, 1, allowing here the possibility to model X [0] by a mixture of Gaussian distributions and X [1] by a simple Gaussian distribution, as it is suggested by the two figures in the center and the right side of Graph. 4 (we can actually observe two small clusters along the first axis and a more spread one on top).

If we calculate now the second-order moment of f we obtain

Using integration by parts and assumption (R) iv), the numerator of the right handside of (37) can be expressed as follows

where

Uniform almost sure rate of convergence of T 1,n . Note first that from boundedness of H(•; ϑ, Fn,θ , Ĵn,θ ) and H(•; ϑ, F θ , J θ ) given by (34), there exist nonnegative constants C and C such that

and consider N (ε) the smallest integer such that u i,εu i-1,ε ≤ ε for i = 2, . . . , N (ε). We denote by • the integer part function. For all ε small enough we clearly have

Let us now define α i,ε = ε(i -1), i = 1, . . . , M (ε), where M (ε) = A + 1 /ε and thus α M (ε),ε > A. Observe in addition that

.

Hence the expression

)) -covering of F + 0 in the L 2 (P X )-norm sense. Using the standard notation N [] (•) (see van der Vaart and Wellner, 1996) the covering number of the class F + 0 is bounded as follows

Thus if there exist constants C and V such that

we get N (ε)M (ε) ≤ C/ε V +2 which allows us to use Theorem 2.14.9, p. 246 in van der Vaart and Wellner (1996) since their Condition (2.14.7), p. 245 is then satisfied after replacing their constant V by V + 2. Let us discuss condition (38). For ε small enough this condition is true if y ε ≤ C/ε V and u ε ≥ -C/ε V . Denoting by F ← 0 the quantile function of F 0 , condition (38) becomes

We consider for simplicity the first condition in (39) (the second one being treated in the same way); it is equivalent to F 0 (C/ε V ) ≥ 1ε, and taking t = C/ε V this condition turns into

Thus it suffices to have

Finally, using the symmetry of f 0 , condition (38) holds if

which is ensured by condition (R) vi). In conclusion if (40) is satisfied and E(X 2 1 ) < ∞ then, according to Theorem 2.14.16, p. 248 in van der Vaart and Wellner (1996), we obtain

Convergence rate of T

(2)

and

Let K be a kernel satifying (K). The K-regularized versions of F θ and F n,θ are

Let us denote by P X,Y n the empirical measure

and by P X,Y the law of (X 1 , Y 1 ). The set of functions for which (x, y) → ax + by + c being a 3-dimensionnal vector space, Corollary 2.5 in [START_REF] Kuelbs | Log Log Laws for Empirical Measures Ann[END_REF] shows that the class of sets

is a Strassen log-log class, which implies that a.s.

Since C contains the class

Now if we replace F n,θ by its regularized version Fn,θ the approximation is controlled as follows,

the same kind of bound being available for Ĩn, θn (t) -Ĩn,θ * (t) L 1 . In conclusion, according to the decomposition (45), point i) of Theorem 3.1, the respective L 1 a.s. convergence of Ψn,θ * and Ĩn,θ * towards Ψ θ * and I θ * under (20), we get from (48) and the strong law of large numbers that fnf L 1 → 0 almost surely as n → ∞ whenever n -1/4+γ /b n = o(1).

iii) The proof uses an integrated version of decomposition (45) and the fact that, for all y ∈ R, the approximation | Fn, θn -Fn,θ * |(y) is controlled by

the last term on the right hand side of above inequality being independent from y.

The same bound holds for | Ĩn, θn -Ĩn,θ * |(y) by a similar argument. To conclude, it is enough to use (49) and Corollary 1 p. 766 in [START_REF] Shorack | Empirical Processes with Applications to Statistics[END_REF], which allows us to control the terms Fn,θ * -F θ * ∞ and Ĩn,θ * -I θ * ∞ , to obtain [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes: With Applications to Statistics[END_REF]. The rate on the right hand side of ( 27) is optimized by considering b n = n -1/12 which then turns into O a.s. (n -1/6+γ ), for all γ > 0.


R Boucekkine
email: raouf.boucekkine@univ-amu.fr

Natali Hritonenko
email: nahritonenko@pvamu.edu

Yuri Yatsenko
email: yyatsenko@hbu.edu

On the optimal control of the vintage capital growth model with endogenous labour supply *

Keywords: Optimal control, integral equations with delays and advances, vintage capital, endogenous labor supply JEL numbers: C61, C62, 040

We prove that the introduction of endogenous indivisible labor supply into the vintage capital growth model does not rule out the turnpike and optimal permanent regime properties, notably the nonmonotonicity properties of optimal paths, inherent in this model.

Background

Vintage capital growth models feature non-homogenous capital stocks with a preeminent economic role for the age structure of these stocks (much beyond physical depreciation). The importance of this class of models, notably for the understanding of the dynamics of investment and the diffusion of technological progress, has been recognized from the 60s (see [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF]. However, the optimal control of the associated dynamic systems is highly sophisticated: the latter involve mostly delayed and advanced differential and integral equations with endogenous delays and advances, a class of infinite dimensional equations. As a result, research on vintage capital models has been blocked for decades. Recent developments in the analysis of functional differential equations have allowed this literature to resume in the mid 90s (see Boucekkine et al., 2010, for a complete overview of the literature of age-structured models). In this note, we relax one of the traditional ingredients of vintage capital growth models (just like neoclassical growth models indeed), the exogeneity of labor supply. We do it following the indivisible labor framework of [START_REF] Hansen | Indivisible labor and the business cycles[END_REF], which has the virtue to allow for some further analytical work (due to linearity) in addition to its very known desirable economic properties. We essentially show that this extension does not rule out the non-monotonicity properties of the solution paths, usually referred to as replacement echoes, because they are linked to the optimal replacement of obsolete capital goods. This is specially true for permanent optimal regimes or turnpikes, which have received particular attention in the related optimization literature (see [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF]Hritonenko and[START_REF] Hritonenko | Turnpike properties of optimal delay in integral dynamic models[END_REF]. The main results are given in Section 3 after the specification of the optimization program in Section 2.

The optimization problem 2.1 Generic optimal control problem

We seek to maximize:

max c,n,i,T ∞ 0 u (c(t), n(t)) e -ρt dt, (1)
subject to:

y(t) = t t-T (t) i(z) dz (2) t t-T (t) i(z)e -γz dz = n(t), (3)
y(t) = c(t) + i(t), (4)
and inequality constraints:

0 ≤ i(t) ≤ y(t), 0 ≤ n(t) ≤ 1, T (t) ≥ 0, T (t) ≤ 1, (5)
with initial investment profile, i 0 (t), t < 0 given. The problem consider the maximization of an intertemporal welfare function with an instantaneous utility function u(., .) depending on consumption, c(t) and time spent working, n(t), out of a unitary time endowment. The utility function is standard: u c (., .) > 0, u cc (., .) ≤ 0, u n (., .) < 0 and u nn (., .) ≤ 0. Equations ( 2) and (3) are respectively the production function and the equilibrium condition in the labor market. Indeed, in line with [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF], the production function is Leontief: every unit of capital of any vintage z produces one unit of output and consumes e -γz units of labor, γ > 0 being the rate of labor-saving technical progress. T (t) is the age of the oldest machine still in use in t or scrapping time. Equation ( 4) is the resource constraint of the economy, inequalities (5) are standard constraints: in particular, the apparently peculiar condition T (t) ≤ 1 allows to prevent the re-use of a machine already scrapped (see [START_REF] Malcomson | Replacement and the rental value of capital equipment subject to obsolescence[END_REF]. Finally because the model involves integral constraints with (endogenous) delays, the optimization problem involved is infinite dimensional as use to be most age-structured optimal control problems (see Boucekkine et al., 2010, for a wide set of examples). As a consequence, the initialization of the problem is functional and not pointwise.

Specification

The problem extends [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] by including endogenous labor supply, n(t). In the traditional formulations of the vintage capital model (see [START_REF] Hritonenko | Modeling and Optimization of the Lifetime of Technologies[END_REF], for an exhaustive review), labour supply is exogenous equal to given population size. We shall restrict our attention to the framework considered in [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] with linear utility in consumption. It is shown that in this case, and whatever the initial function i 0 (t), t < 0,1 , the economy converges at finite time to a permanent optimal regime where: i) the scrapping time is constant, T (t) = T > 0, and ii) solution paths show periodicity in a certain sense (replacement echoes).

Indeed, suppose that the scrapping time is constant and labor is exogenous and fixed to any constant, then differentiating equation (3) yields

m(t) = m(t -T ),
for t big enough (but finite), with m(t) detrended investment, that is m(t) = i(t) e -γt . The same conclusion is reached in [START_REF] Hritonenko | Turnpike properties of optimal delay in integral dynamic models[END_REF] in a variety of vintage capital models. 2 We shall study the robustness of this claim when labour supply is no longer exogenous but is the result of an economic decision, that is when n(t) is a control variable. To this end, we introduce endogenous indivisible labor supply following the influential work of [START_REF] Hansen | Indivisible labor and the business cycles[END_REF], a highly frequent specification in macroeconomic theory. This amounts to setting:

u(c, n) = c -θ n,
where θ ≥ 0. The linearity in n is a fundamental feature of indivisible labor supply as it allows to generate realistic figures for the volatility of employment. 3 The standard case is θ = 0 (and n(t) = 1, ∀t ≥ 0). Within this framework, we investigate whether the permanent regime properties identified in the model without endogenous labor supply, notably the occurrence of replacement echoes, do also hold. This amounts to studying whether a permanent optimal regime with a (time-varying) interior solution 0 < n(t) < 1 can emerge. If only corner solutions for n(t) are sustainable along optimal permanent regimes, 4 then we would prove that introducing endogenous labor supply in the usual way into the vintage capital growth models does not break down the main property of its optimal paths, that's replacement echoes.

Optimality properties

To ease the exposition, we eliminate the control c(t) thanks to the resource constraint (4), we replace the control i(t) by detrended investment, m(t) = i(t) e -γt , and we omit the inequality constraints (5) at the minute. By inverting the integration variables following a technique suggested by [START_REF] Malcomson | Replacement and the rental value of capital equipment subject to obsolescence[END_REF], one gets the following Lagrangian once some irrelevant 2 Nonlinearities, for example through nonlinear utility functions as in [START_REF] Boucekkine | Creative destruction, investment volatility and the average age of capital[END_REF] do not "kill" replacement echoes, at least in the short and medium run.

3 In other words, strict concavity with respect to n has the very undesirable feature to deliver too smooth employment paths compared to observed ones, see [START_REF] Hansen | Indivisible labor and the business cycles[END_REF].

4 Needless to say, only the corner solutions n(t) = 1 are of interest, the corner n(t) = 0 displaying zero output (and zero consumption) can be easily ruled out.

terms (dated before t = 0) are taken out: ρ(z-t) dz -e γt e -ρt dt

L = ∞ 0 m(t) t+J(t) t λ(z)e γt -ω(z) e -
+ ∞ 0 y(t)(1 -λ(t))e -ρt dt + ∞ 0 n(t) (ω(t) -θ) dt,
where λ(t) and ω(t) are the Lagrange multipliers associated to the constraints (2) and ( 3) respectively, and J(t), which can be interpreted as expected lifetime of capita goods of vintage t, is given by:

J(t) = T (t + J(t)) . (6)
First-order conditions for interior maximizers y(t) and n(t) are respectively:

λ(t) = 1, (7)
ω(t) = θ. (8)
With respect to the traditional vintage capital framework, ( 8) is a new interior condition equalizing the shadow wage, ω(t), and the disutility of work, θ. Of course, if ω(t) > θ (Resp. ω(t) < θ), one gets the optimality condition for the corner solution n(t) = 1 (Resp. n(t) = 0). As to optimal (interior) investment, using the optimality conditions ( 7)-( 8) derived above, one identifies the optimal interior rule:

Ω(t) ≡ t+J(t) t e γt -θ e (ρ-γ)t e -γz dz = 1, (9)
Ω(t) being the quasi-rents delivered by a marginal investment made at t over the lifetime of the capital good bought. Of course, m(t) = 0 (Resp. m(t) = e -γt y(t)) if Ω < 1 (Resp. Ω(t) > 1). We can now state the first proposition of the paper: Proposition 3.1 The problem cannot admit an interior solution for t large enough.

Proof: Derive the optimality condition with respect to J(t); it is given by:

λ (t + J(t)) = ω (t + J(t)) ,
which by ( 7) and ( 6) implies

w(t) = e γ(t-T (t)) . ( 10
)
If n is an optimal (interior) control, then condition (8) should hold implying:

θ = e γ(t-T (t)) . (11)
This is impossible for T (t) < ∞ at large t (and for any non-empty interior time interval). For interior optimality conditions ( 8) and ( 10) to be compatible at large t, the optimal scrapping time should check:

T (t) = 1,
by differentiation of the optimality condition ( 11). This property is compatible with the "non-reuse" condition in (5). It implies that T (t) = t + k, k a constant on some time interval with non-empty interior. Precisely, one gets k = -ln(θ) γ . However, the latter property is incompatible with the optimal (interior) investment rule, and in particular with the associated definition of capital goods' lifetime (6).5

It follows from the previous proposition that in contrast to [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] and [START_REF] Hritonenko | Turnpike properties of optimal delay in integral dynamic models[END_REF], the interior solution of the optimal control problem cannot be a permanent regime or a turnpike. Does this mean that replacement echoes do no longer govern the dynamics of the vintage capital growth model with indivisible labor supply? The answer is no. Indeed, though the interior solution is not sustainable, the introduction of endogenous labor supply yields qualitatively the same permanent regimes (or turnpikes) as in the traditional vintage models quoted above. This is stated as proposition hereafter. Proposition 3.2 If γ < 1, the vintage capital model with indivisible labor supply delivers the same optimal permanent regime as in [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF]. In particular, along this regime, (i) scrapping time is constant, T (t) = T , (ii) investment is periodic: m(t) = m(t -T ), and (iii) n(t) = 1.

Proof: First of all, one can readily see that the three properties (i) to (iii) are compatible. Indeed, if n(t) = 1, one has to ensure that w(t) > θ, which by ( 11) and (i) implies: e γ(t-T ) > θ. The latter obviously holds at large t. Properties (i) and (ii) hold under n(t) = 1 because the problem is equivalent to [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] when labor supply is fixed. In such a case, only the interior solution Ω(t) = 1 is optimal at large t, ultimately yielding properties (i) and (ii). Note that this is the unique possible permanent regime. Two alternative regimes may sound reasonable. First, n(t) = 1 and m(t) corner may be put forward. But it has been already proved by [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] that when n(t) = 1, the corner regimes m(t) = 0 and m(t) = e -γt y(t) can only be transitory regimes occurring on finite time length intervals. The last possibility is n(t) interior and m(t) corner. In such a case, w(t) = θ but Ω(t) = 1. Forget about the trivial solution Ω(t) < 1 and consider the corner solution Ω(t) > 1 implying m(t) = e -γt y(t).6 By (11), we conclude that t -T (t) is a constant, and by (2) once m(t) is replaced by e -γt y(t), we get that y(t) is an exponential function, ae t , with a > 0. This is in contradiction with the equilibrium condition in the labor market, (3), once m(t) replaced by e -γt y(t) = ae (1-γt) under γ < 1 since n(t) is bounded and t -T (t) is a constant.7 .

This is true under some mild technical conditions, see[START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF].

Trivially, T (t) = t + k and J(t) = T (t + J(t)) are incompatible on time intervals with non-empty interior.

Ω(t) < 1 implies m(t) = 0 and therefore y(t) = 0 and finally n(t) = 0 for large t. We disregard this trivial solution.

The condition γ < 1 is a very weak requirement, γ is the rate of technological progress, a few percentage points. Notice also that a more general production function, y(t) = α t t-T (t) i(z) dz with α > 0 the capital coefficient of the Leontief technology would have delivered the equally weak condition α > γ to rule out the solution n(t) interior and Ω(t) > 1 as a permanent regime.

* This paper was written when Natali Hritonenko and Yuri Yatsenko were visiting GRE-QAM, Aix-Marseille School of Economics. Boucekkine acknowledges the financial support from the Belgian ARC contract 09/14-018 on sustainability, Hritonenko acknowledges the support of NSF DMS 1009197.