Olivier Lopez
email: lopez@ensai.fr

Single Index Regression Models with right censored responses

Keywords: semiparametric regression, dimension reduction, censored regression, Kaplan-Meier estimator, single-index models

In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses.

 through a simulation study.

Introduction

In regression analysis, one investigates on the function m(x) = E[Y | X = x], which is traditionally estimated from independent copies (Y i , X i ) 1≤i≤n ∈ R 1+d . The parametric approach consists of assuming that the function m belongs to some parametric family, that is m (x) = f 0 (θ 0 , x), where f 0 is a known function and θ 0 an unknown finite dimensional parameter. On the other hand, the nonparametric approach requires fewer assumptions on the model, since it consists of estimating m without presuming the shape of the function. However, this approach suffers from the so-called "curse of dimensionality", that is the difficulty to estimate properly the function m when the dimension d is high (in practice, d ≥ 3). To avoid this important drawback of nonparametric approaches, while allowing more flexibility than a purely parametric model, one may use the semi-parametric single-index model (SIM in the following) which states

m (x) = E[Y | X ′ θ 0 = x ′ θ 0 ] = f (x ′ θ 0 ; θ 0 ) ,
where f is an unknown function and θ 0 an unknown finite dimensional parameter. If θ 0 were known, the problem would consist of a nonparametric one, but with the covariates belonging nevertheless to a one-dimensional space.

In this framework, numerous semi-parametric approach have been proposed for rootn consistent estimation of θ 0 . Typically, these approaches can be split into three mains categories : M-estimation [START_REF] Ichimura | Semiparametric least squares (SLS)and weighted SLS estimation of single-index models[END_REF], Sherman, 1994b If the responses of this regression model are randomly right-censored, these approaches clearly need to be adapted, for the random variable Y is not directly observed. The right censoring model states that, instead of observing Y , one observes i.i.d. replications of

T = Y ∧ C, δ = 1 Y ≤C , (1.1)
where C is some "censoring variable", and 1 A denotes the indicator function of the set A.

In this setting, semi-parametric Cox regression model (see e.g. [START_REF] Andersen | Cox's Regression Model for Counting Processes : A Large Sample Study[END_REF] can be seen as a particular case of the SIM model, but allows less flexibility. Moreover, it is still interesting to extend mean-regression models to the censored framework. For this reason, Buckley and James (1978) proposed an estimator of the linear model under random censoring, and [START_REF] Lai | Large sample theory of a modified Buckley-James estimator for regression analysis with censored data[END_REF] and [START_REF] Ritov | Estimation in a Linear Regression Model with Censored Data[END_REF] proved its asymptotic normality. [START_REF] Koul | Regression analysis with randomly right censored data[END_REF] initiated what we may call the "synthetic data" approach, based on transformations of the data. See [START_REF] Leurgans | Linear models, random censoring and synthetic data[END_REF], [START_REF] Zhou | Asymptotic normality of the "synthetic data" regression estimator for censored survival data[END_REF] and [START_REF] Lai | Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression[END_REF]. Zhou (1992a) also proposed a weighted least-square approach, applying weights in the least square criterion in order to compensate the censoring.

These techniques were then used in the nonlinear regression setting, that is when f 0 is known but nonlinear. [START_REF] Stute | Nonlinear censored regression[END_REF] established a connection between the weighted least-square criterion and Kaplan-Meier integrals. Delecroix, Lopez and Patilea (2006) extended the synthetic data approach. Heuchenne and Van Keilegom (2005) modified the Buckley-James' technique for polynomial regression purpose. When it comes to the SIM model under random censoring, [START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF] recently proposed an estimate using an extension of the average derivatives technique of [START_REF] Härdle | Investigating Smooth Multiple Regression by the Method of Average Derivatives[END_REF] and the synthetic data approach of [START_REF] Koul | Regression analysis with randomly right censored data[END_REF].

In this paper, we propose a semi-parametric M-estimator of the SIM model under random censoring. We present a technique that is adapted to both main classes of censored regression techniques (synthetic data and weighted least squares), deriving root-n consistency of our estimate of θ 0 , and then using it to estimate m (x). Another advantage of our technique is that we do not require that the covariates X have a density with respect to Lebesgue's measure (only the linear combinations θ ′ X need to be absolutely continuous), which is an important advantage comparatively with the estimation procedure of [START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF].

The paper is organized as follows. In section 2 we present the regression model and our methodology. In section 3, we derive consistency of our semi-parametric estimates in a general form, asymptotic normality is obtained in section 4. A simulation study is presented in 5 to test the validity of our estimate with finite samples. Section 6 is devoted to technical proofs.

Model assumptions and methodology

In the following, we assume that we have the following regression model,

Y = f (θ ′ 0 X; θ 0 ) + ε,
where θ 0 is a vector of first component equal to 1, and

E [ε | X] = 0. The function f is defined in the following way, f (u; θ) = E [Y | X ′ θ = u] .
Considering the censoring model (1.1), we will define the following distribution function,

F (t) = P (Y ≤ t) , G (t) = P (C ≤ t) , H (t) = P (T ≤ t) , F (X,Y ) (x, t) = P (Y ≤ t, X ≤ x) .
In the following, we will assume that inf{t,

F (t) = 1} = inf{t, H(t) = 1}, (2.2)
P(Y = C) = 0. (2.3)
Otherwise, if (2.2) does not hold, since some part of the distribution of Y remains unobserved, consistent estimation requires making additional restrictive assumptions on the law of the residuals. Note that, in this case, our estimators will still be root-n convergent, but not necessary to θ 0 . Concerning (2.3), we use this assumption to avoid dissymetry problems between C and Y.

As a property of conditional expectation, for any function J(•) ≥ 0, we have

θ 0 = arg min θ∈Θ E (Y -f (θ ′ X; θ)) 2 J(X) = arg min θ∈Θ M (θ, f ) (2.4) = arg min θ∈Θ (y -f (θ ′ x; θ)) 2 J(x)dF (X,Y ) (x, y) .
In equation (2.4), of course we can not exactly know θ 0 , since two objects are missing in the definition of M, that is the distribution function F (X,Y ) and the regression function f (θ ′ x; θ). A natural way to proceed consists of estimating these two functions, and then plugging in these estimators into (2.4).

Estimating the distribution function

We already mentioned there are two main approaches for studying regression models in presence of censoring, the Weighted Least Square approach (WLS in the following) and the Synthetic Data approach (SD in the following).

The WLS approach. In the uncensored case, the distribution function F (X,Y ) can be estimated using the empirical distribution. This tool is unavailable under random censoring, since it relies on the (unobserved) (Y i ) 1≤i≤n .Under random censoring, Stute

F (t) = 1 - i:T i ≤t 1 - n j=1 1 δ j =1,T j ≤T i 1 -Ĥ(T i -) δ i
, where Ĥ denotes the empirical distribution function of T . F can be rewritten as

F (y) = W in 1 T i ≤y ,
where W in is the jump at observation i. It is particularly interesting to notice that the jump at observation i is connected to the Kaplan-Meier estimate of G at the same value (see, for example, Satten and Datta, 2000), that is

W in = 1 n δ i 1 -Ĝ (T i -) . (2.5)
Kaplan-Meier estimate is known to be a consistent estimate of F under the two following identifiability assumptions, that is Assumption 1 Y and C are independent.

Assumption 2 P (Y ≤ C | X, Y ) = P (Y ≤ C | Y ) .
A major case for which Assumptions 1-2 hold is the case where C is independent from (Y, X). However, Assumption 2 is more general and covers a significant amount of situations (see [START_REF] Stute | Nonlinear censored regression[END_REF].

The SD approach.

The SD approach consists of considering some alternative variable which has the same conditional expectation as Y . For this, observe that, through elementary calculus, under Assumptions 1-2,

∀φ, E δφ (X, T ) 1 -G (T -) | X = E [φ (X, Y ) | X] . (2.6)
From (2.6), we see that, if we define, accordingly to [START_REF] Koul | Regression analysis with randomly right censored data[END_REF],

Y * = δT 1 -G (T -) , we have E [Y * | X] = E [Y | X]
Ŷ * i = δ i T i 1 -Ĝ (T i -) .
See also [START_REF] Leurgans | Linear models, random censoring and synthetic data[END_REF], [START_REF] Lai | Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression[END_REF] for other kind of transformations.

Back to equation (2.4), the SD approach will first consists of observing that

θ 0 = arg min θ∈Θ E (Y * -f (θ ′ x; θ)) 2 J(X) = M * (θ, f ) (2.7) = arg min θ∈Θ (y * -f (θ ′ x; θ)) 2 J(x)dF * (X,Y * ) (x, y * ) ,
where F * (X,Y * ) (x, y * ) = P (X ≤ x, Y * ≤ y * ) . Note that M * and M are not the same functions. Indeed, Y * happens to have the same conditional expectation as Y (hence M and M * have the same minimizer θ 0 ), but it has not the same law.

Estimating f (θ ′ x; θ)

In the uncensored case, a common non-parametric way to estimate a conditional expectation is to use kernel smoothing. In this case, the Nadaraya-Watson estimate for f

(θ ′ x; θ) is f (θ ′ x; θ) = n i=1 K θ ′ X i -θ ′ x h Y i n i=1 K θ ′ X i -θ ′ x h = yK θ ′ u-θ ′ x h d Femp (u, y) K θ ′ u-θ ′ x h d Femp (u, y) .
We are still facing the same problem of absence of the empirical distribution function.

However, WLS and SD approaches can be used to extend the Nadaraya-Watson estimate to censored regression. In the following, we will only use the SD approach of Koul & al. to estimate the conditional expectation, that is

f (θ ′ x; θ) = n i=1 K θ ′ X i -θ ′ x h Ŷ * i n i=1 K θ ′ X i -θ ′ x h . (2.8)
While using this estimator, we do not have to deal with Kaplan-Meier integrals at the denominator. In fact, the integral at the denominator becomes an integral with respect to the empirical distribution function of X. However, alternative estimates (not necessarily kernel estimates) can still be used, provided that they satisfy some further discussed conditions to achieve asymptotic properties of θ. Therefore we chose to present our results without presuming on the choice of f (θ ′ x; θ), and then to check in the Appendix section that the estimator defined in (2.8) satisfies the proper conditions.

Also observe that, using this kernel estimate, contrary to the average derivative technique of [START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF], we do not need to impose that X has a density with respect to Lebesgue's measure. We only need that the linear combinations θ ′ X do.

The choice of the trimming function J. The reason why we introduced the function J in (2.4) appears in the definition (2.8). To ensure uniform consistency of this estimate, we will need to bound the denominator away from zero. For this, we will need to restrain the integration domain to a set where f θ ′ X (u) is bounded away from zero, f θ ′ X denoting the density of θ ′ X. If we were to know θ 0 , we could consider a set

B 0 = {u : f θ ′ 0 X (u)
≥ c} for some constant c > 0, and use the trimming J(θ ′ 0 X) = 1 θ ′ 0 X∈B 0 . Of course, this ideal trimming can not be computed, since it depends on the unknown parameter θ 0 . [START_REF] Delecroix | On semiparametric Mestimation in single-index regression[END_REF] proposed a way to approximate this trimming from the data. Given some preliminary consistent estimator θ n of θ 0 , they use the following trimming,

J n (θ ′ n X) = 1 fθ ′ n X (θ ′ n X)≥c .
In the following proofs, we will mostly focus on the estimation using the uncomputable trimming J(θ ′ 0 X), and we will show in the appendix section that there is no asymptotic difference in using J n (θ ′ n X) rather than J(θ ′ 0 X).

Estimation of the single-index parameter

Preliminary estimate of θ 0 . For a preliminary estimate, we assume, as in [START_REF] Delecroix | On semiparametric Mestimation in single-index regression[END_REF] that we know some set B such as inf x∈B,θ∈Θ {f θ ′ X (θ ′ x) ≥ c > 0}, and we consider the trimming function J (x) = 1 x∈B . To compute our estimate θ n , we then can use either of the WLS or SD approach. For example, using the WLS approach, let

θ n = arg min θ∈Θ y -f (θ ′ x; θ) 2 J (x)d F(X,Y ) (x, y) = arg min θ∈Θ M p n (θ, f ). (2.9)
Estimation of θ 0 . In view of (2.4) and (2.7), we will define our estimates of θ 0 according to the two regression approaches discussed above,

θW LS = arg min θ∈Θn y -f (θ ′ x; θ) 2 J n (θ ′ n x)d F(X,Y ) (x, y) = arg min θ∈Θn M W LS n θ, f , θSD = arg min θ∈Θn y * -f (θ ′ x; θ) 2 J n (θ ′ n x)d F * (x, y * ) = arg min θ∈Θn M SD n θ, f .
In the definition above, for technical convenience, we restrained our optimization to shrinking neighborhoods Θ n of θ 0 , chosen accordingly to the preliminary estimation by θ n .

Estimation of the regression function

With at hand a root-n consistent estimate of θ 0 , it is possible to estimate the regression function by using θ and some estimate f . For example, using f defined in (2.8) will lead to

f θ′ x; θ = n i=1 K θ ′ X i -θ′ x h Ŷ * i n i=1 K θ′ X i -θ′ x h .
3 Consistent estimation of θ 0

In this section, we prove consistency of θ n where θ n is defined in (2.9). As a consequence, θ is consistent since it is obtained from minimization other a shrinking neighborhood of θ 0 . We will need two kinds of assumptions to achieve consistency : general assumptions on the regression model including identifiability assumptions for θ 0 , and assumptions on f .

Identifiability assumptions for θ 0 and assumptions on the regression model.

Assumption 3 EY 2 < ∞. Assumption 4 If M(θ 1 , f ) = M(θ 0 , f ), then θ 1 = θ 0 .
Assumption 5 Θ and X = Supp(X) are compact subsets of R d and f is continuous with respect to x and θ. Furthermore, assume that |f

(θ ′ 1 x; θ 1 )-f (θ ′ 2 x; θ 2 ) | ≤ θ 1 -θ 2 γ Φ (X),
for a bounded function Φ (X), and for some γ > 0.

Assumption 3 is implicitly needed in order to define M, while Assumption 4 ensures the identification of the parameter θ 0 . On the other hand, Assumption 5 states that the class of functions F = {f (θ ′ .; θ) , θ ∈ Θ} is sufficiently regular to allow it to satisfy an uniform law of large numbers property. More precisely, Assumption 5 ensures that this class is Euclidean for a bounded envelope, according to [START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF].

Observe that the condition that Φ is bounded can be weakened, by replacing it by a moment assumption on Φ. However, this condition is quite natural in a context where we will assume that the covariates are bounded random vectors, and this will simplify our discussion. Moreover, it implies that f is a bounded function of θ and x.

Assumptions on f .

Assumption 6 For all function g, define, for c > 0,

g ∞ = sup θ∈Θ,x |g(θ ′ x; θ)|1 f θ (θ ′ x)>c/2 . Assume that f -f ∞ = o P (1).
See section 6 for more details to see that the kernel estimator (2.8) satisfies this assumption under some additional integrability assumptions on the variable Y.

Theorem 3.1 Under Assumptions 3 to 6, we have

sup θ∈Θ M n θ, f -M ∞ (θ, f ) = o P (1)
.

As an immediate corollary, in a probability sense, θ n → θ 0 .

Proof.

Step 1 : replacing f by f. Observe that, since the integration domain is restricted to the set B,

M n (θ, f ) -M n (θ, f ) ≤ f -f ∞ ×[ f + f ∞ d F(X,Y ) (x, y) + 2 |yd F(X,Y ) (x, y)|]. Now using Assumption 6, deduce that sup θ∈Θ |M n (θ, f ) -M n (θ, f)| = o P (1).
Step 2 :

M n (θ, f ). Showing that sup θ∈Θ |M n (θ, f ) -M(θ, f )| = o P (1)
can then be done in the same way as in a nonlinear regression model such as in [START_REF] Stute | Nonlinear censored regression[END_REF]. See the proof of Theorem 1.1 in Stute (1999).

Asymptotic normality

As in the uncensored case, we will show that, asymptotically speaking, our estimators behave as if the true family of functions f were known. Hence studying the asymptotic normality of our estimates reduces to study asymptotic properties of estimators in a parametric censored nonlinear regression model, such as those studied by [START_REF] Stute | Nonlinear censored regression[END_REF] and Delecroix & al. (2008). We first recall some elements about the case "f known"

(which corresponds to a nonlinear regression setting), and then show that, under some additional conditions on f and on the model, our estimation of θ 0 is asymptotically equivalent to the one performed in this unreachable parametric model.

The case f known

This case can be studied using the results of [START_REF] Stute | Nonlinear censored regression[END_REF] for the WLS approach, or the results of Delecroix & al. (2008) for the SD approach. We recall some assumptions under which the asymptotic normality of the corresponding estimators is obtained.

Assumptions on the model. We denote by ∇ θ f (x; θ) the vector of partial derivatives of (x, θ) → f (θ ′ x; θ) with respect to θ, and ∇ 2 θ f the corresponding Hessian matrix.

Assumption 7 f (θ ′ x; θ) is twice continuously differentiable with respect to θ, and ∇ θ f and ∇ 2 θ f are bounded as functions of x and θ.

Assumptions on the censoring. We need some additional integrability condition.

We first need a moment assumption which is related to the fact that we need to have

E[Y * 4 ] < ∞. Assumption 8 y 4 dF (y) [1 -G(y-)] 3 < ∞.
Actually x is not involved in Assumption 8 as it is assumed to be bounded. Furthermore, in the case f known, this assumption can be weakened, but it will be needed in the case f unknown to obtain uniform consistency rate for f. The following assumption is used in [START_REF] Stute | The central limit theorem under random censorship[END_REF][START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF] to achieve asymptotic normality of Kaplan-Meier integrals.

Assumption 9 Let C(y) = y -∞ dG(s) {1 -H(s)}{1 -G(s)} .
Assume that

yC 1/2 (y)dF (X,Y ) (x, y) < ∞.
See [START_REF] Stute | The central limit theorem under random censorship[END_REF] for a full discussion on this kind of assumption. Using our kernel estimator for estimating the conditional expectation will lead us to a slightly stronger assumption (see the appendix section), which is Assumption 10 For some ε > 0,

yC 1/2+ε (y)[1 -G(y-)] -1 dF (X,Y ) (x, y) < ∞.
In the following, we will use the (slightly) stronger Assumption 10 since it may simplify some proofs (see Lemma 6. The following Theorem can be deduced from the proof of Theorem 1.2 in [START_REF] Stute | Nonlinear censored regression[END_REF] and of Theorem 4 in Delecroix & al. (2008). However, to make this article self-contained, a short proof of this result is postponed at section 6.1 of the appendix.

Theorem 4.1 Define

ψ(y, T, δ) = (1 -δ) 1 T >y 1 -H (T -) - 1 T >y,y>v dG (v) [1 -H (v)] 2
and let

U W LS = δ (T -f (θ ′ 0 X; θ 0 )) 1 -G (T -) + {y -f (θ ′ 0 x; θ 0 )}V (y, T, δ)dF (X,Y ) (x, y) , U SD = δT 1 -G (T -) -f (θ ′ 0 X; θ 0 ) + yV (y, T, δ)dF (X,Y ) (x,
M n (θ, f ) = M ∞ (θ, f ) + O P θ -θ 0 √ n + o P θ -θ 0 2 + R n , (4.10)
M n (θ, f ) = 1 2 (θ -θ 0 ) ′ V (θ -θ 0 ) + (θ -θ 0 ) ′ W n √ n + o P n -1 + R n , (4.11)
where R n does not depend on θ, where

V = E ∇ θ f (X; θ 0 ) ∇ θ f (X; θ 0 ) ′ ,
and where W n =⇒ N (0, W ), for W = W W LS and W = W SD in the W LS-case and SD-case respectively.

In view of Theorem 1 and 2 of Sherman (1994), (4.10) states that, in the case where

f is known, | θ -θ 0 | = O P n -1/2
, while (4.11) gives the asymptotic law of θ, showing

that n 1/2 ( θ -θ 0 ) =⇒ N (0, V -1 W V -1
), in both WLS and SD cases.

The case f unknown

As f is unknown in the SIM model, we need to add some conditions about the rate of convergence of f.

Assumptions on f . If we evaluate the function ∇ θ f (x; θ) at the point (x, θ 0 ), a direct adaptation of Lemma A.5 of [START_REF] Dominitz | Some convergence theory for iterative estimation procedures[END_REF] shows that

∇ θ f (x; θ 0 ) = f ′ (θ ′ 0 x){x -E[X | θ ′ 0 X = θ ′ 0 x]}, (4.12)
where f ′ denotes the derivative with respect to t of the function f (t; θ 0 ).

Assumption 11

We assume that the function f (t; θ 0 ) is continuously derivable with respect to t, its derivative is denoted as f ′ and is bounded.

We will also assume some regularity on the model.

Assumption 12 u → f (u; θ 0 ) where u ranges over θ ′ 0 X is assumed to belong to some Donsker class of functions F .

In our minds, F will be the class C 1 (θ ′ 0 X , M), that is the class of functions φ defined on θ ′ 0 X and being one time differentiable with φ ∞ + φ ′ ∞ ≤ M (see section 2.7 in Van der Vaart and Wellner, 1996). It is important not to impose to much regularity on the regression model, since, as we will see it in Assumption 13, f will also be required to belong to this class with probability tending to one.

Assumptions on f .

Assumption 13 With probability tending to one, u → f(u; θ 0 ) ∈ F where F is defined in Assumption 12. Furthermore,

∇ θ f -∇ θ f ∞ = o P (1), (4.13)
and, defining

W * i = δ i n -1 [1 -G(T i -)] -1 , sup θ∈Θn n i=1 W * i J(θ ′ 0 X i )[T i -f (θ ′ 0 X i ; θ 0 )][∇ θ f (X i ; θ 0 ) -∇ θ f (X i ; θ 0 )] =o P (n -1/2 ), (4.14)
sup

θ∈Θn n i=1 W * i J(θ ′ 0 X i )( f (θ ′ 0 X i ; θ 0 ) -f * (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ)) =o P (n -1/2 ). (4.15)
We can now enounce our asymptotic normality theorem.

Theorem 4.2 Under Assumptions 3 to 13, we have

√ n θW LS -θ 0 ⇒ N 0, V -1 W W LS V -1 , √ n θSD -θ 0 ⇒ N 0, V -1 W SD V -1 .
Proof. First apply Proposition 6.9 to obtain that J n (θ ′ n X i ) can be replaced by J(θ ′ 0 X i ) or by 1 f θ (θ ′ X i )>c/2 , plus some arbitrary small terms which will not be mentioned in the following. Moreover, we consider θ ∈ Θ n which is an o P (1)-neighborhood of θ 0 .

Proof for the WLS approach. Using the representation (2.5) of the Kaplan-Meier weights,

M n θ, f = M n (θ, f ) - 2 n n i=1 δ i J(θ ′ 0 X i ) (T i -f (θ ′ X i ; θ)) 1 -Ĝ (T i -) × f (θ ′ X i ; θ) -f (θ ′ X i ; θ) + 1 n n i=1 δ i J(θ ′ 0 X i ) 1 -Ĝ (T i -) f (θ ′ X i ; θ) -f (θ ′ X i ; θ) 2 = M n (θ, f ) -2A 1n + B 1n .
First decompose A 1n into four terms,

A 1n = 1 n n i=1 δ i J(θ ′ 0 X i ) (T i -f (θ ′ 0 X i ; θ 0 )) 1 -Ĝ (T i -) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) + δ i J(θ ′ 0 X i ) (f (θ ′ 0 X i ; θ 0 ) -f (θ ′ X i ; θ)) 1 -Ĝ (T i -) × f (θ ′ X i ; θ) -f (θ ′ X i ; θ) -f (θ ′ 0 X i ; θ 0 ) + f (θ ′ 0 X i ; θ 0 ) + δ i J(θ ′ 0 X i ) (f (θ ′ 0 X i ; θ 0 ) -f (θ ′ X i ; θ)) 1 -Ĝ (T i -) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) + δ i J(θ ′ 0 X i ) (T i -f (θ ′ 0 X i ; θ 0 )) 1 -Ĝ (T i -) × f (θ ′ X i ; θ) -f (θ ′ X i ; θ) -f (θ ′ 0 X i ; θ 0 ) + f (θ ′ 0 X i ; θ 0 ) = A 2n + A 3n + A 4n + A 5n .
A 2n does not depend on θ.

For A 3n , use Assumption 5 to bound f (θ ′ 0 X; θ 0 )f (θ ′ X; θ) by M × θθ 0 (for some constant M > 0) using a Taylor expansion. Using a Taylor expansion, the bracket in A 3n can be rewritten as (θθ 0 ) ′ [∇ θ f(X; θ) -∇ θ f (X; θ)] for some θ ∈ Θ n . Moreover, using Proposition 6.9, we can replace J(θ ′ 0 X) by 1 {f θ ( θ′ X)>c/2} . Hence we have

A 3n ≤ M θ -θ 0 2 sup θ∈Θ,x∈X |∇ θ f (x; θ) -∇ θ f (x; θ)| d F(X,Y ) (x, y).
The uniform consistency of ∇ θ f in Assumption 13 shows that A 3n = o P ( θθ 0 2 ).

For A 4n , use a second order Taylor expansion and the uniform consistency of f to obtain

A 4n = 1 n n i=1 δ i J(θ ′ 0 X i ) (θ -θ 0 ) 1 -Ĝ (T i -) ′ ∇ θ f (X i ; θ 0 ) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) (4.16) +o P ( θ -θ 0 2 ), (4.17)
In the first term, first replace G by Ĝ. Using Lemma 6.2 ii) with η = 1, this introduces a remainder term which is bounded by

O P ( θ -θ 0 n -1/2 ) f -f ∞ n n i=1 δ i J(θ ′ 0 X i )C 1/2+ε (T i -) 1 -G (T i -) ,
where we also used the boundedness of ∇ θ f. Using the uniform consistency of f shows that replacing Ĝ by G in (4.16) only arises an o P ( θθ 0 n -1/2 ) term. Now, we will use the regularity assumption (12) on f (•; θ 0 ). If the class F is Donsker, the class of function

F ′ = (δ, T, X) → δJ(θ ′ 0 X)[1 -G(T i -)] -1 ∇ θ f (X i ; θ 0 )F (θ ′ 0 X i )
is Donsker, from a stability property of Donsker classes (see e.g. [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes[END_REF]. The notation

F (θ ′ 0 X i ) is used to mention that the functions in F are evaluated at θ ′ 0 X i . Furthermore, for all φ ∈ F ′ , E[φ(T i , δ i , X i )] = 0, since E δ i ∇ θ f (X i ; θ 0 ) 1 -G(T i -) |θ ′ 0 X i = E [∇ θ f (X i ; θ 0 )|θ ′ 0 X i ] = 0,
from (4.12). Hence, using the fact that f (•; θ 0 ) ∈ F with probability tending to one, and the asymptotic equicontinuity property of Donsker classes for F ′ (see Van der Vaart and Wellner, 1996), we obtain

1 n n i=1 δ i J(θ ′ 0 X i ) (θ -θ 0 ) 1 -G (T i -) ′ ∇ θ f (X i ; θ 0 ) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) = o P ( θ -θ 0 n -1/2 ),
and finally, A 4n = o P ( θθ 0 n -1/2 ).

Similarly, for A 5n , a Taylor expansion yields

A 5n = (θ -θ 0 ) ′ n n i=1 δ i J(θ ′ 0 X i ) (T i -f (θ ′ 0 X i ; θ 0 )) 1 -Ĝ (T i -) × ∇ θ f X i ; θ -∇ θ f X i ; θ = (θ -θ 0 ) ′ n n i=1 δ i J(θ ′ 0 X i ) (T i -f (θ ′ 0 X i ; θ 0 )) 1 -G (T i -) × ∇ θ f X i ; θ -∇ θ f X i ; θ + o P ( θ -θ 0 n -1/2 ),
where, as for A 4n , we replaced Ĝ by G by using Lemma 6.2 ii) and the uniform consistency of ∇ θ f. Now We then obtain A 5n = o P ( θθ 0 n -1/2 ) + o P ( θθ 0 2 ) using condition 4.15 in Assumption 13.

For B 1n , write

B 1n = 1 n n i=1 δ i J(θ ′ 0 X i ) 1 -Ĝ (T i -) × f (θ ′ X i ; θ) -f (θ ′ X i ; θ) -f (θ ′ 0 X i ; θ 0 ) + f (θ ′ 0 X i ; θ 0 ) 2 + δ i J(θ ′ 0 X i ) 1 -Ĝ (T i -) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) + δ i J(θ ′ 0 X i ) 1 -Ĝ (T i -) f (θ ′ 0 X i ; θ 0 ) -f (θ ′ 0 X i ; θ 0 ) × f (θ ′ X i ; θ) -f (θ ′ X i ; θ) -f (θ ′ 0 X i ; θ 0 ) + f (θ ′ 0 X i ; θ 0 )
Using a second order Taylor expansion and arguments similar to those used for A 3n , we obtain that the first term is of order o P ( θθ 0 2 ). The second term does not depend on θ. For the third, a first order Taylor expansion shows that it is bounded by

θ -θ 0 ∇ θ f -∇ θ f ∞ sup x:J(θ ′ 0 x)=1 | f(θ ′ 0 x; θ 0 ) -f (θ ′ 0 x; θ 0 )| d F(X,Y ) (x, y).
Now condition 4.14 in Assumption 13 shows that this is o P ( θθ 0 n -1/2 ).

We have just shown that

M n θ, f = M n (θ, f ) + o P θ -θ 0 √ n + o P θ -θ 0 2 ,
on a set of probability tending to one. Furthermore, using (4.10) we deduce θθ 0 = O P n -1/2 from Theorem 1 in Sherman (1994), and since, from (4.11), on

O P n -1/2 -neighborhoods of θ 0 , M n (θ, f θ ) = 1 2 (θ -θ 0 ) ′ V (θ -θ 0 ) + 1 √ n (θ -θ 0 ) ′ W W LS n + o P n -1 ,
we can apply Theorem 2 of Sherman to conclude on the asymptotic law.

Proof for φ SD . Proceed as for φ M C , the only difference is in the fact that Ĝ does not appear in the terms where T does not appear at the numerator.

Simulation study

In this section, we tried to compare the behavior of our estimator with the estimator proposed by Burke and Lu (2005) who used the average derivative technique. We considered three configurations.

Config 1 Config 2 Config 3 ε ∼ N (0, 2) ε ∼ N (0, 1) ε ∼ N (0, 1/16) X ∼ U[-2; 2] ⊗ U[-2; 2] X ∼ U[0; 1] ⊗ U[0; 1] X ∼ B(0.6) ⊗ U[-1; 1] f (θ ′ x; θ) = 1/2(θ ′ x) 2 + 1 f (θ ′ x; θ) = 2e (0.5θ ′ x) 0.5+θ ′ x f (θ ′ x; θ) = 1 + 0.1(θ ′ x) 2 -0.2(θ ′ x -1) θ 0 = (1, 1) ′ θ 0 = (1, 2) ′ θ 0 = (1, 2) ′ C ∼ U[0, λ 1 ] C ∼ E(λ 2 ) C ∼ E(λ 3 )
The first configuration is used by [START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF] in their simulation study.

Observe that, in this model, (2.2) does not hold (this condition (2.2) is also needed in Burke and Lu's approach), but it only introduces some asymptotic bias in the estimation.

In the second configuration, there is no such problem since C is exponential. In the third configuration, we see that X does not have a Lebesgue density, but θ ′ X does. In this situation, it is expected that the average derivative techniques does not behave well since it requires that X has a density.

In each configuration, we simulated 1000 samples of different size n. For each sample, we computed θW LS , θSD , and θAD which denotes the average derivative estimate computed from the technique of [START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF]. We then evaluated θ-θ Globally, the performance of the different estimates shrinks when the proportion of censored responses increases. Performances of θW LS and θSD are globally similar. In all tested configurations, θW LS and θSD seem to perform better than θAD . As expected, in the situation where X does not have a density, θAD does not converge.

6 Appendix

Some results on Kaplan-Meier integrals

In this section, we recall some facts on the behavior of Kaplan-Meier integrals. First part of this section is devoted to the i.i.d representation of Kaplan-Meier integrals derived by [START_REF] Stute | The central limit theorem under random censorship[END_REF][START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF], first in the univariate case, then in presence of covariates. For this, define, for any function φ,

U i (φ) = φ(x, y)ψ(y, T i , δ i )dF (X,Y ) (x, y),
where ψ has been defined in Theorem 4.1. It can be easily shown that E [U i (φ)] = 0.

The following Theorem has been derived by [START_REF] Stute | Distributional convergence under random censorship when covariables are present[END_REF].

Theorem 6.1 Let φ be a function satisfying

|φ (x, y)| C 1/2 (y) dF (X,Y ) (x, y) < ∞. Then φ (x, y) d F(X,Y ) (x, y) = 1 n n i=1 δ i φ (X i , T i ) 1 -G (T i -) + 1 n n i=1 U i (φ) + o P n -1/2 .
In view of the expression (2.5) of the jumps of Kaplan-Meier estimate, this Theorem shows that, asymptotically, these jumps can be replaced by the "ideal" jumps, say

W * i = n -1 δ i [1-G (T i -)] -1
, plus some perturbation that only appears in the study of the variance (since its expectation is zero). The following lemma gives some additional precision on the difference between the jumps W in and the "ideal" jumps W * i .

Lemma 6.2 Recall that Ĝ is the Kaplan-Meier estimator for the distribution of C, W in =

n -1 δ i [1-Ĝ(T i -)] -1 and W * i = δ i [1-G(T i -)] -1
, and denote by T (n) the largest observation.

sup

t≤T (n) 1 -Ĝ(t-) 1 -G(t-) = O P (1)
and sup

t≤T (n) 1 -G(t-) 1 -Ĝ(t-) = O P (1) ; (6.18)
ii) For all 0 ≤ η ≤ 1 and ε > 0,

|W in -W * i | ≤ W * i {C (T i )} η[1/2+ε] × O P n -η/2 , (6.19)
where the O P n -η/2 factor does not depend on i.

Proof.

i) The first part of (6.18) follows from Theorem 3.2.4 in Fleming and Harrington (1991). The second part follows for instance as a consequence of Theorem 2.2 in Zhou (1991).

ii) Fix η > 0 arbitrarily. Since

τ H a C -1-2η (y)dC(y) < ∞,
for some a > 0, apply Theorem 1 in [START_REF] Gill | Large Sample Behaviour of the Product-Limit Estimator on the Whole Line[END_REF] to see that sup

y≤T (n) [C (y)] -1/2-η |Z(y)| = O P (1), (6.20)
where Z = √ n{ Ĝ -G}{1 -G} -1 is the Kaplan-Meier process. Next, the proof can be completed by using the definitions of W in , W * i , property (6.18), and elementary algebra. Proof for M W LS n . Write

M W LS n (θ, f ) -M (θ) = 2 (y -f (θ ′ 0 x; θ 0 )) {f (θ ′ 0 x; θ 0 ) -f (θ ′ x; θ)} ×d( F(X,Y ) -F (X,Y ) )(x, y) + {f (θ ′ 0 x; θ 0 ) -f (θ ′ x; θ)} 2 ×d( F(X,Y ) -F (X,Y ) )(x, y) + (y -f (θ ′ 0 x; θ 0 )) 2 d( F(X,Y ) -F (X,Y ) )(x, y). (6.21)
The last term does not depend on θ. Let

χ (x, y) = {y -f (θ ′ 0 x; θ 0 )}∇ θ f (x; θ 0 ) .
Using the derivability Assumption 7 and Theorem 6.1, the first term in the right-hand side of (6.21) is

2 (θ 0 -θ) ′ χ(x, y)d F(X,Y ) -F (X,Y ) (x, y) +2 (θ 0 -θ) ′ {y -f (θ ′ 0 x; θ 0 )}∇ 2 θ f (x; θ) × d( F(X,Y ) -F (X,Y ) ) (x, y) (θ 0 -θ) = 2 (θ 0 -θ) ′ 1 n n i=1 δ i χ(X i , T i ) 1 -G(T i -) -E δχ(X, T ) 1 -G(T -) +2 (θ 0 -θ) ′ 1 n n i=1 U i (χ) + o P θ -θ 0 2 , (6.22)
where the o P -rate comes from the boundedness of ∇ 2 θ f and consistency of Kaplan-Meier integrals. Furthermore, the empirical sums in (6.22) weakly converge to centered Gaussian variables at rate O P (n -1/2 ). For the second term in (6.21), rewrite it as

(θ -θ 0 ) ′ ∇ θ f θx; θ ∇ θ f θx; θ ′ d( F(X,Y ) -F (X,Y ) )(x, y) (θ -θ 0 ) .
From the boundedness of ∇ θ f , deduce that this is o P θθ 0 2 . We thus obtained (4.10). To obtain (4.11), use Theorem 6.1.

Properties of f

In this section, we derive some properties of f defined by (2.8), and show that this estimate satisfies Assumptions 6 and 13. Our approach consists of comparing f to the ideal estimator f * defined as

f * (θ ′ x; θ) = n i=1 Y * i K θ ′ X i -θ ′ x h n i=1 K θ ′ X i -θ ′ x h , (6.23)
that is the estimator based on the true (uncomputable) Y * i . Indeed, f * is a regular kernel estimator based on uncensored variables, and can be studied by traditional nonparametric kernel techniques.

Assumptions on the random variables X ′ θ.

Assumption 14 For all θ ∈ Θ, θ ′ X has a density which is continuously derivable, with uniformly bounded derivative.

Assumptions on the kernel function.

Assumption 15

• K is symmetric, positive, twice continuously differentiable function with K ′′ satisfying a Lipschitz condition.

• K(s)ds = 1.

• K has a compact support, say [-1; 1].

Assumptions on the bandwidth.

Assumption 16

• nh 8 → 0.

• nh 5 [log(n)] -1 = O(1).
The first Lemma we propose allows us to obtain uniform convergence rates for the ideal estimator f * as an immediate corollary.

Lemma 6.3 Let K be a kernel satisfying Assumption 15. Let K denote either K or its derivative. Let Z be a random variable with 4th order moment, with m

(x) = E[Z|X = x]
twice continuously differentiable, with derivatives of order 0, 1 and 2 uniformly bounded.

Consider, for d = 0, 1, and any vectors x and x ′ in X ,

g n (θ, x, x ′ , d) = 1 nh 1+d n i=1 K θ ′ X i -θ ′ x h K θ ′ X i -θ ′ x ′ h d Z i .
We have, for d = 0, 1,

sup θ,x,x ′ |g n (θ, x, x ′ , d) -E[g n (θ, x, x ′ , d)]| = O P (n -1/2 h -[d+1]/2 [log n]
f * -f = O P (n -1/2 h -1/2 [log n] 1/2 + h 2 ), ∇ θ f * -∇ θ f = O P (n -1/2 h -3/2 [log n] 1/2 + h 2 ).
Proof. For the bias terms (6.25) and (6.26), this can be done by a classical change of variables, a Taylor expansion, and the fact that uK(u)du = 0 and u 2 K(u)du < ∞.

For (6.24), first consider

g Mn n (θ, x, x ′ , d) = 1 nh 1+d n i=1 K θ ′ X i -θ ′ x h K θ ′ X i -θ ′ x ′ h d Z i 1 Z i ≤Mn .
We then follow the methodology of [START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF]. From Pakes and Pollard (1989), the family of functions indexed by (θ, x, x ′ , h) (which has a constant envelope function),

(X, Z) → K θ ′ X -θ ′ x h K θ ′ X -θ ′ x ′ h d 1
2 G = n -1/2 h -[d+1]/2 . Take M n = n 1/2 h 1/2 . It follows from Talagrand's inequality that sup θ,x,x ′ |g Mn n (θ, x, x ′ , d) -E[g Mn n (θ, x, x ′ , d)]| = O P (n -1/2 h -[d+1]/2 [log n] 1/2 ).
It remains to consider g Mn n -g n . This difference is bounded by Cn -1 h -[1+d] n i=1 |Z i |1 Z i ≥Mn for some constant C. This is a sum of positive quantities, thus we only have to show that its expectation is o P (n -1/2 h -[d+1]/2 [log n] 1/2 ). For this, apply Hölder inequality to bound this expectation by h

-[1+d] E[Z 4 ] 1/4 P(Z ≥ M n ) 3/4 . Moreover, P(Z ≥ M n ) ≤ E[Z 4 ]/M 4 n
from Tschebychev inequality, and the result follows. Proposition 6.5 below ensures that the difference between f and f * , in view of uniform consistency, is asymptotically negligible. Hence Assumption 6 can be deduced from the uniform consistency of f * . Proposition 6.5 Under Assumptions 10, 14, and Kernel Assumptions 15 and 16, we Proof

have f -f * ∞ + ∇ θ f -∇ θ f * ∞ = o P (
. Let fθ ′ X (u) = n -1 h -1 n i=1 K((θ ′ X i -u)/h). We have f (u; θ) -f * (u; θ) = 1 h n i=1 [W in -W * i ]T i K θ ′ X i -θ ′ x h fθ ′ X (u) . ( 6
| fθ ′ X (θ ′ x) -f θ ′ X (θ ′ x)| = o P (1).
Using this result on the set {f θ ′ X (θ ′ x) > c > 0}, and Lemma 6.2 ii) with η sufficiently small, we obtain the bound

| f(θ ′ x; θ)-f (θ ′ x; θ)| ≤ O P (n -η/2 h -1 )× n i=1 W * i T i C η(1/2+ε) (T i -)K θ ′ X i -θ ′ x h , (6.28)
where the O P -rate does not depend on θ nor x. Recalling the definition of W * i , consider the family of functions indexed by θ and x,

{(T, δ, X) → δT [1 -G(T -)] -1 C η(1/2+ε) (T -)K((θ ′ X -θ ′ x)/h)}.
This family is Euclidean (see Lemma 22 in Nolan and Pollard, 1987) for an enveloppe

δT [1 -G(T -)] -1 C η(1/2+ε) (T -) which is, for η = 1/2,
| n i=1 W * i T i C η(1/2+ε) (T i -)K θ ′ X i -θ ′ x h | = O P (h) + O P (n -1/2 ).
Finally, back to (6.28), this shows that f -

f ∞ = O P (n -1/4 ) = o P (1). Similarly, ∇ θ f -∇ θ f ∞ = O P (n -1/4 h -1
W * i J(θ ′ 0 X i )(T i -f (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ))
Θn =O P (n -1 ), (6.29)

h 2 n i=1 W * i J(θ ′ 0 X i )( f (θ ′ 0 X i ; θ 0 ) -f * (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ))
Θn =O P (n -1 ), (6.30)

n i=1 W * i J(θ ′ 0 X i )( f (θ ′ 0 X i ; θ 0 ) -f * (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ)) Θn =o P (n -1/2 ), (6.31) n i=1
W * i (f (θ ′ 0 X i ; θ 0 )f * (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ))

Θn =o P (n -1/2
W * i J(θ ′ 0 X i )(f (θ ′ 0 X i ; θ 0 ) -f * (θ ′ 0 X i ; θ 0 ))(∇ θ f (X i ; θ) -∇ θ f * (X i ; θ)) = o P (n -1/2 ),
which can be done following the lines of Lemma C2 in Delecroix & al. (2008). Similarly, Proposition (6.7) allows to replace f by f * .

Proof of Proposition 6.7. We only prove (6.30) and (6.32) since the others are similar.

We first prove (6.30). This can be done by studying separately the different terms arising by differentiation with respect to θ in the definition of f . We will only study the term coming from the differentiation of the numerator (since the other is similar), that is

1 nh 2 i,j δ i J(θ ′ 0 X i )( f(θ ′ 0 X i ; θ 0 ) -f * (θ ′ 0 X i ; θ 0 ) 1 -G(T i -) K ′ θ ′ X i -θ ′ X j h f -1 θ ′ X (θ ′ X i )[W * j -W jn ]T j .
By bounding |K ′ | by K ′ ∞ and using the convergence rate of f * , it is easily seen that the terms for i = j can be removed from this double sum, arising an o P (n -1/2 ) term uniform in θ. Applying (6.27), we then get that the above quantity is, up to an o P (n -1/2 )term,

1 h 3 i =j,k W * i J(θ ′ 0 X i )K θ ′ 0 X i -θ ′ 0 X k h [W * k -W kn ]T k ×K ′ θ ′ X i -θ ′ X j h [W * j -W jn ]T j fθ ′ X (θ ′ X i ) -1 fθ ′ 0 X (θ ′ 0 X i ) -1 .
Again, using Lemma 6.2 ii) with η = 1/2, and bounding K ′ by K ′ ∞ allows us to remove the terms for j = k and i = k. For the rest of this triple sum, apply Lemma 6.2 ii) with η = 1 and bound K ′ by K ′ ∞ . If follows that the left-hand side of (6.30) is bounded, uniformly in θ, by

O P (n -1 h -2 ) n i =k,j =k,i =j W * j W * k W * i C 1/2+ε (T j -)C 1/2+ε (T k -)|T j ||T k |K θ ′ 0 X i -θ ′ 0 X k h .
The last sum as finite expectation (and does not depend on θ) from Assumption 10.

For (6.32), again, we will consider only the part of ∇ θ f coming from the differentiation of the numerator, this means that we are trying to bound

1 h 2 i,j W * i J(θ ′ 0 X i )[f (θ ′ 0 X i ; θ 0 )-f * (θ ′ 0 X i ; θ 0 )]K ′ θ ′ X i -θ ′ X j h T j [W * j -W jn ] fθ ′ X (θ ′ X i ) -1 . (6.33)
First, let S τ be the double sum deduced from (6.33) by introducing 1 T j ≤τ for some τ < τ H . From [START_REF] Gill | Large Sample Behaviour of the Product-Limit Estimator on the Whole Line[END_REF], sup t≤τ | Ĝ(t)-G(t)||1-G(t)| -1 = O P (n -1/2 ), and consequently, sup j |W * j -W jn |1 T j ≤τ = O P (n -1/2 ). Now, using the uniform convergence rate of f * and bounding K ′ by K ′ ∞ shows that S τ = o P (n -1/2 ) for any τ < τ H . To obtain a bound for (6.33), we then have to make τ tend to τ H . For this, we use the same Cramer-Slutsky argument as [START_REF] Stute | The central limit theorem under random censorship[END_REF] in his proof of the Central Limit Theorem under censoring. Using Lemma 6.2 ii) with η = 1, observe that

|S τ H -S τ | ≤ O P (n -1/2 h -2 ) f * -f ∞ 1 h n i=1 K θ ′ 0 X i -θ ′ 0 X j h 1 T j ≥τ W * j C 1/2+ε (T j -)W * i .

  , Delecroix et Hristache, 1999, Xia et Li, 1999, Xia, Tong, et Li, 1999, Delecroix, Hristache et Patilea, 2006), average derivative based estimation (Powell, Stock et Stoker, 1989, Härdle et Stoker, 1989, Hristache et al., 2001a, 2001b), and iterative methods (Weisberg et Welsh, 1994, Chiou et Müller, 1998, Bonneu et Gba, 1998, Xia et Härdle, 2002).

( 1993 )

 1993 proposed to use an estimator based on the Kaplan-Meier estimator of F . Recall the definition of Kaplan-Meier estimator,

  2 and the proof of Theorem 4.1). However, Assumption 10 could be replaced by Assumption 9 if we were to use an estimator (not necessarly kernel estimator) which would not require Assumption 10 to satisfy the proper convergence assumptions. Note that this kind of assumption is classical in studying regression models with censored responses. Although it is not mentioned in[START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF], a similar assumption is implicitly needed to obtain equation (2.29) of[START_REF] Lai | Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression[END_REF]. In their proof of Lemma A.7 page 199 of[START_REF] Burke | Censored multiple regression by the method of average derivatives[END_REF], the authors refer to equation(2.29) page 275 of[START_REF] Lai | Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression[END_REF]: this only holds under the condition C3 of[START_REF] Lai | Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression[END_REF] which basically controls the tail behavior of the distributions.

  y) , and let W W LS = E U W LS 2 and W SD = E U SD 2 . Let M n and M ∞ denote respectively either M W LS n and M, or M SD n and M * . We have, under Assumptions 1 to 10,

6. 2

 2 Proof of Theorem 4.1 In this section, we show that the criterion M W LS n and M SD n satisfy the conditions (4.10) and (4.11). The same properties can be also shown for the synthetic data estimators of Leurgans (1987) and Lai & al. (1995). More precisions can be found in Delecroix & al. (2008). For the sake of simplicity, we only prove it for M W LS n since the proof for M SD n uses similar arguments.

  Z≤Mn , satisfies the uniform entropy condition of Proposition 1 in[START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF] (condition (ii) in their Proposition 1). The other assumptions in their Proposition 1 hold with β = σ = CM, for some constant C not depending on M. We then can apply Talagrand's inequality (see[START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF][START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF] Talagrand, 1994), with σ

1 ). Corollary 6 . 6

 166 Under the Assumptions of Proposition 6.5, f satisfies Assumption 6 and condition (4.13) in Assumption 13.

  under Assumption 1 and 2. Hence, if Y * were available, the same regressions techniques as in the uncensored case could be applied to Y * . Of course, Y * can not be computed, since it depends on the unknown function G. But Y *

	can be easily estimated (which is not the case for Y ) by replacing G by its Kaplan-Meier
	estimate. For i = 1, ..., n we obtain

  ). Now, to prove the corollary, we have to show the uniform consistency of f * and ∇ θ f * , which can be done applying Theorem 2 in[START_REF] Einmahl | Uniform in bandwidth consistency of kerneltype function estimators[END_REF]. Let • Θn denote the supremum of the absolute value over Θ n . Under the Assumptions of Proposition 6.5, we have

	The following Proposition allows us to obtain that f satisfies conditions (4.14) and
	(4.15) of Assumption 13.
	Proposition 6.7 h 2 n
	i=1

  ). Under the assumptions of Proposition 6.5, f satisfies conditions(4.14) and (4.15) of Assumption 13.Proof of Corollary 6.8. To prove (4.15), according to Proposition 6.7, it remains

		(6.32)
	Corollary 6.8 to show that
		n
	sup
	θ∈Θn	i=1

The last part does not depend on θ and its expectation tends to zero as τ → τ H , while the rest is O P (n -1/2 ), using the convergence rate of f * and the Assumptions 16. Then the Cramer-Slutsky argument of Stute allows us to conclude.

The only condition that still needs to be checked is that u → f(u; θ 0 ) ∈ F , where F is defined in Assumption 12. This can be done if we specify this class of functions. If

, which can be done by using the same method as in Proposition 6.7 to replace f by f * .

Trimming

In the following proposition, we show that the trimming J n (θ ′ n x) can be replaced by J(θ ′ 0 x) modulo arbitrary small terms. Proposition 6.9 Let, for any function φ,

We have

Proof. For any δ > 0, we have, with probability tending to one,

where

As in Delecroix, Hristache, Patilea (2006) page 737-738, we have

Note that P(n 1/2 Z n ≥ δ) ≤ P(Z n ≥ δ), which tends to zero as δ tends to zero.


C Chabert

E Touron

A Saturnin

G Krivtchik

G Martin

Ph Miranda

F Courtin

J L Girotto

G Mathonnière

S Gabriel

R Eschback

A Allou

R Boccaccio

F X Giffard

A Bonin

P Casoli

F Laugier

B Carlier

G Senentz

PROSPECTIVE INVENTORY OF RADIOACTIVE MATERIALS AND WASTE PRODUCED BY THE FRENCH NUCLEAR FLEET ACCORDING TO VARIOUS OPTIONS

In accordance with the French Act of 28 June 2006 on the sustainable management of radioactive materials and waste, this paper summarises the technical characterisation of prospective scenarios using different fuel cycle options: open cycle, recycling of plutonium and uranium in PWRs (current option for the French nuclear power fleet), multiple recycling of plutonium in SFRs, and multiple recycling of plutonium in PWRs. This information has been submitted by the CEA to the Ministry of Energy within the scope of Article 51 of the Ministerial Order dated 23 February on the French National Radioactive Materials and Waste Management Plan (PNGMDR) [Ref. 1].

Rather than imagining the large-scale replacement of PWRs by fast reactors within a short period of time (the assumption of some past studies), it was decided to study a scenario involving a more progressive deployment on the basis of existing materials and facilities. This solution appears to be better suited to the dynamics of technical progress in the field, while providing greater flexibility to adapt to societal changes. The path for this scenario is marked by successive milestones (stages), with each corresponding to an increased deployment of fast reactors with their own increasingly ambitious objectives. Phase A corresponds to the current state of the French nuclear reactor fleet wherein plutonium and uranium are recycled in mixed-oxide (MOX) and enriched reprocessed uranium (ERU) fuels in pressurised water reactors (PWR). Phase B consists in recycling spent MOX fuel from PWRs in a limited number of SFRs. The objective of this phase is to stabilise the quantities of spent MOX fuels from light water reactors. Phase C is designed to be able to stabilise the plutonium inventory by deploying a symbiotic fleet comprising UOX-PWRs, MOX-PWRs and SFRs. The objective of phase D is to deploy a fleet of reactors that no longer burns natural uranium. There are two possible options for a nuclear fleet that can generally be considered as self-sufficient, i.e. D1, a homogeneous fleet with 100% SFRs, and D2, a mixed fleet comprising breeder SFRs producing plutonium and PWRs fuelled with 100% MOX to burn this plutonium. However, SFRs may not become economically competitive in the next few decades if uranium resources remain readily available, and MOX spent fuels may start to pile up at the back-end of the fuel cycle unless alternative plutonium management solutions in PWRs are found. In this study, advanced fuel batches, called CORAIL and MIX, are applied to enable multiple recycling in standard PWRs. The CORAIL concept involves placing both MOX fuel rods with depleted uranium support structures and UOX fuel rods in the same fuel assembly. A configuration based on 84 MOX fuel rods and 181 UOX fuel rods was studied in the early 2000s, which is why it has been chosen for the first scenario in this paper. As the enrichment of UOX rods is maximised at 5%, the plutonium content is adapted to make up for its loss of fissile quality with each recycling phase. The MIX concept is based on a fuel assembly containing only MOX fuel rods with enriched uranium support structures. The purpose in this case is to limit the plutonium content in the fuel to a level similar to that of current MOX fuels and to meet any additional needs in fissile nuclei by providing enriched uranium in the support. The greater the isotopic degradation of plutonium, the greater the enrichment will be, which will be increased with each recycling phase to make sure 235 U is stabilised at around 3% to 4%. Three plutonium contents have been considered in our study: 8%, 9.54% and 12%. This paper assesses the material flows and inventories for these various options on the basis of two approaches: 1) static (or in equilibrium) by supposing that each option is artificially maintained long enough, and 2) dynamic (for transition scenarios). The total disposal surface area required for each of the different options is also discussed on the basis of the Andra disposal concept. The assessment of the resulting waste volumes depending on the option is given in another paper, also presented at this conference [Ref. 2].

I. TECHNICAL PERFORMANCE OF THE FUEL CYCLE OPTIONS -STATIC APPROACH

All the cases studied in this paper are based on the assumption of a nuclear reactor fleet generating about 430 TWe/y. Table I summarises the main characteristics of the options involving the progressive deployment of SFRs up to the fuel cycle closure option (D2 and D1). The characteristics in the table confirm that the gradual transition from phase A through to phase D would lead to improvements -with each phase -in the sustainable management of materials, particularly in the consumption of natural uranium (additional savings of 5% between phase A and phase B, and around 43% between B and C, before reaching 100% in phase D). Spent PWR-MOX and ERU in storage can be recycled and thus optimised, which simultaneously stabilises the inventories of all the spent fuels regardless of their type. This confirms the relevance of such a progressive approach, which also makes it possible to improve the industrial maturity of SFRs and the related fuel cycle with each phase, even though its integration in the second half of this century remains relatively small.

The information used to compare the technical characteristics of the scenarios using the MIX and CORAIL concepts is given in Table II

II. FUEL CYCLE TRANSITION SCENARIOS

Two types of transitional scenarios have been chosen and compared. In all cases, the initial conditions applied are those that currently exist in France, i.e. once-through recycling of Pu in PWRs (phase A). o Scenarios involving the progressive deployment of SFRs -Scenario ABCD1 involves the successive deployment of all phases, starting with phase A (once-through recycling of Pu in PWRs), then phase B (stabilisation of spent PWR MOX by recycling in a few SFRs), followed by phase C (stabilisation of the Pu inventory using a symbiotic PWR-SFR fleet), and finally phase D1 (100% SFR fleet). -Scenario ABCD2 ends with a hybrid fleet comprising SFRs and 100% MOX PWRs -Scenario ABD2 is an alternative to scenario ABCD2 where deployment of D2 is accelerated (this scenario could apply in the case where the price of natural uranium rises rapidly or there is a shortage of natural uranium). o Scenarios involving the multiple recycling of Pu in PWRs -Scenarios implementing the MIX concept for the multiple recycling of Pu until its total inventory has stabilised. Three alternatives have been defined according to the acceptable Pu concentration in the fuel: 8%, 9.54%1 and 12%. We also investigated a MIX scenario based on the recycling of reprocessed uranium through the management of ERU PWRs.

-The scenario implementing the CORAIL concept for the multiple recycling of Pu until its total inventory has stabilised.

II.A. Scenarios involving the progressive deployment of SFRs

The transition from phase A to phase B corresponds to the implementation of spent PWR MOX fuel treatment and to the deployment of the first SFR units required to use the plutonium resulting from this treatment. The start-up of commercial SFRs is expected occur 25 years after the industrial commissioning of the Astrid-600 MW reactor (assumed in this study to be in 2039), i.e. in the mid-2060s. This timescale takes into account the need for sufficient feedback from the operation of Astrid and for realistic lead times for technical and regulatory actions. The chosen reference option consists of directly including Astrid in the objective of stabilising the spent MOX fuels in phase B. In this case, phase B assumes the deployment of 2 SFRs in addition to the demonstrator. These are paired, enabling additional savings to be made on the cost of these 2 reactors. Phase C starts with the implementation of spent SFR fuel processing from the first commercial units deployed in phase B and the deployment of additional SFRs to recycle the plutonium that has been recovered. It was decided in the study to start the phase by 2090, a date which corresponds to a period of deployment of new reactors and coincides with the renewal of the fuel cycle facilities, the lifetime of which is assumed to be 50 years in this study. The transition from C to D will occur during a period of renewal of the fleet, i.e. between 2150 and 2180. An accelerated transition from B to D has also been characterised, enabling the assessment of issues which could be encountered in the event of early tension on uranium prices. The scenario in this case envisages the transition to D by 2090. An example of the changing composition of the fleet over time is given in Figure 1 for transitional scenario ABCD1. All the SFR deployment scenarios stabilise the storage of spent fuels, unlike the option of continuing with the current situation (Scenario A) for which the spent fuel inventories continue to increase, albeit much less quickly than in the context of an open cycle (Fig. 4). It should therefore be noted that in 2200 for example, the extension of the current situation would lead to accumulating around 57,000 t of spent fuel, comprising: 45,000 t of UOX and ERU plus 12,000 t of MOX. For the open cycle, it would lead within the same timescale to accumulating around 220,000 t of spent UOX fuel.

Figure 4: Variation in the interim storage of spent fuel for the transitional scenarios (kt)

II.B. Scenarios involving the multiple recycling of Pu in PWRs

The scenarios implementing the MIX and CORAIL concepts have been established using the same assumptions as those for the SFR deployment scenarios with regard to the description of the current PWR fleet and its renewal. It should be remembered that the CORAIL concept used here is the one studied in the early 2000s and that the concept is currently being optimised by Framatome.

It has not been possible to assess the optimised version for this study.

In addition, although this point needs to be consolidated by additional studies, it has been considered that the industrial deployment of these concepts would be theoretically possible in 2045, a timescale that seems at this stage in the studies to be reasonable for commissioning new fuel cycle plants and qualification of these new fuel products. These transitional scenarios have been defined so as to reveal and test the capacity of these concepts for multi-recycling and using the existing spent fuel inventories. To do this, the following principles are applied: recycling of all the spent MOX fuel stocks as quickly as possible, then stabilisation of the plutonium inventory and all the spent fuel inventories. Once all the spent MOX fuel has been recycled, the spent MIX and CORAIL fuel is reprocessed.

On a preliminary basis, a joint plutonium and reprocessed uranium multi-recycling scenario was established in order to measure its feasibility (MIX scenario including management of ERU PWRs).

An example of a timescale for the deployment of reactors for the MIX scenario including management of ERU EPRs is given in Figure 5.

The number of MIX PWRs deployed to achieve stabilisation of the Pu inventory depends on the plutonium content chosen in the concept. Thus the higher the content, the more the plutonium consumption in the MIX reactor increases and the fewer reactors with MIX fuel need to be introduced into the fleet to compensate for the Pu production in the reactors loaded with UOX fuel. The rate of incorporating MIX in the fleet thus varies between 29% and 37% according to the Pu content being considered (between 12% and 8%). It should be noted that the assumption is made in these studies that the MIX or CORAIL reactors are started up with cores completely loaded with these fuels (no simulated start-up cores). A large proportion of the Pu is therefore mobilised to supply these reactors, and the management of MOX PWRs is then stopped.

Figure 5: Variation in the composition of the fleet for the 9.54% MIX scenario with ERU management Figure 6 shows the cumulative consumption of natural uranium for the different scenarios; it can be seen that self-sufficiency with respect to natural uranium can only be achieved by opting for the closed fuel cycle, i.e. deployment of SFRs. The consumption of natural uranium in the 12% MIX scenario (6700 tonnes/year) is lower compared with the 8% MIX scenario (7000 tonnes/ year). Degradation in the fissile quality of Pu caused by its successive recycling is compensated by its enrichment in 235 U using an enriched support structure. When the Pu content in the MIX fuel increases, the need for 235 U enrichment to maintain the required cycle length therefore decreases. Compared with scenario A, the consumption of natural uranium in the MIX scenarios without ERU management is always higher by about 10 to 15% depending on the Pu content. The sharp decrease in the neutron flux in the thermal range because of the plutonium isotopes in the MIX fuel deteriorates the efficiency of 235 U which, in turn, is less likely to fission. The 235 U content found in a MIX fuel at unloading is about 2% compared with 0.8% for an irradiated standard UOX fuel. This reprocessed uranium with excellent fissile properties therefore needs to be recycled in order to efficiently manage the uranium resources. Any remaining uranium should thus be recycled in MIX and UOX fuels through ERU PWR management strategies in order to reduce the consumption of natural uranium compared with scenario A wherein the current situation is continued. Additional savings in natural uranium of about 9% can be made for a total of 25 to 30% compared with an open fuel cycle. MIX 9,54% MIX 9,54% avec URE CORAIL is highly concentrated in spent fuels (representing about 55%).

Figure 7 shows the evolving inventory of minor actinides for the MIX and CORAIL scenarios before making a comparison with the minor actinide inventories for scenarios A and ABCD1. The increased production of minor actinides due to the implementation of ERU management strategies in a mixed fleet is mostly due to neptunium. The 236 U contained in ERU fuel leads to a strong increase in the production of 237 Np under irradiation. The composition of the minor actinide inventory in Pu multiple recycling scenarios in PWRs is similar to that of scenarios involving the deployment of SFRs, i.e. 27.5% Np, 70% Am and 2.5% Cm. Nevertheless, MIX and CORAIL fleets will generate more minor actinides than SFR fleets, as shown in Figure 7. This difference can be explained by the fact that the fissile quality of plutonium deteriorates with each recycling in the MIX and CORAIL configurations, and by the behaviour of plutonium isotopes with even numbers of protons which are neutron absorbers in the thermal spectrum. From an isotopic perspective, an increase in the production of 243 Am has been detected in these concepts which is closely related to the high proportion of 242 Pu produced. Curium-244 is produced by neutron reactions with 243 Am, therefore resulting in an increase in the level of curium. Minor actinides are mainly found in vitrified waste packages.

III. DISPOSAL SURFACE AREAS REQUIRED FOR THE DIFFERENT FUEL CYCLE OPTIONS

The waste disposal surface area required for the different options has been assessed in terms of the thermo-hydro mechanical (THM) and thermal design criteria (90°C on waste package surfaces) relying on data provided by Andra. This study also includes cases wherein the spent fuel is placed directly in disposal. However, the study on the possible direct disposal of non-reprocessed spent fuel is not sufficiently advanced for our current assessment. This is particularly true with respect to assessing the underground disposal surface area requirements for large volumes of spent fuel with high decay heat releases. This restricts the results of our study with respect to scenarios involving the direct disposal of spent fuel.

Nevertheless, the disposal surface area required for high-level waste (vitrified waste packages) has been fully assessed, and the results for all fuel cycle options (once-through, twice-through recycling and multiple recycling in PWRs, and multiple recycling in SFRs) are reported in this paper. This study does not pre-empt the location of the disposal site for the waste considered. However, we did assume that the deep geological repository was located in clayey rock similar to the layer of argillaceous rock studied in the Meuse/ Haute-Marne region for the Cigéo project. This assumption allowed us to base our concepts on the knowledge collected from the project. Sensitivity analyses were also conducted:

-Duration of interim storage: a period of 80 years was chosen as the reference. A 5% reduction in the surface area required is possible if this storage period is extended to 100 years. -Depth of the clayey rock: a reference depth of -550 m was chosen. A 5% reduction in the surface area required is possible if this depth is increased by 30 m. -Parameters of the clayey rock (thermal conductivity, Young's modulus and porosity): Applying the more favourable conditions of the Cigeo region leads to a reduction in the centre-to-centre distance between disposal cells by about 50%, i.e. a reduction in the surface area of the same order of magnitude.

The surface area required for HLW (vitrified packages) associated with phases A, B, C, D1, D2, MIX and CORAIL for a period of 60 years has been estimated for an interim storage period of 80 years (see Table III).

The smallest surface area for vitrified waste packages corresponds to phase A as the packages contain low levels of minor actinides (contribution of 241 Am) and are therefore less problematic in terms of thermal releases. The surface area estimated for phase B is about 50% higher than that estimated for phase A. These preliminary results characterise the disposal conditions for vitrified waste to which spent fuel must be added (phase A in the case where there is no transition to the subsequent phases).

Option D1 requires a significantly smaller disposal area compared with the other phases involving multiple recycling. This is due to the lower flow of waste packages (about 40% less) thanks to the lower production of minor actinides in SFRs. Phases C and D2 require similar disposal surface areas owing to the importance of two input parameters, i.e. 1) annual flow of waste packages which is determined by the alpha particle limit on vitrified waste due to the increased levels of some minor actinide radioisotopes ( 241 Am, 243 Am and 244 Cm), and 2) the thermal power which is mainly due to 241 Am. These results are due to the higher quantity of minor actinides contained in the waste, which is due to the larger quantity of treated MOX fuel compared with other cases. These conclusions remain valid for the MIX and CORAIL phases which require disposal surface areas similar to those of phases C and D2; both these phases have a large proportion of PWR MOX fuel.

Though it has not been assessed in this study, a large quantity of spent fuel emitting high thermal releases in phases 0, A and B will also require disposal and thus a large disposal surface area. At this stage of the study, we have only indicated the thermal power released over 80 years by the quantity of spent fuel produced in one year. It will be important to clarify the required disposal surface areas in subsequent studies carried out in collaboration with Andra (Figures 8, 9 and 10).

These studies have shown that the thermal power of 241 Am in practically all high-level waste categories for disposal (including 238 Pu to a certain extent for spent fuel) is the predominant factor for estimating the required surface areas, considering the recent changes in the design criteria for clayey-type rock configurations and the thermo-hydro-mechanical criterion. We need to further investigate how spent fuel in storage should be processed so as to limit the production of 241 Am through the radioactive decay of 241 Pu during interim storage in order to optimise the surface area required for disposal.

IV. CONCLUSION

The main results of these studies can be summarised as follows:

Progressive deployment of SFRs with the multiple recycling of uranium and plutonium:

The characterisations provided herein confirm that transitioning from once-through recycling (current solution) to a closed fuel cycle improves the key parameters of a sustainable fuel cycle when the uranium market prices justify investing in an SFR fleet (natural uranium consumption, optimisation of spent MOX and ERU fuel, and production of final waste).

Multiple recycling of uranium and plutonium in PWRs:

The preliminary assessments of the MIX and CORAIL concepts show they are capable of recycling spent MOX and ERU fuel and of stabilising the spent fuel and plutonium inventories.

Compared with the once-through recycling of uranium and plutonium, these concepts reduce the need for natural uranium when reprocessed uranium can be recycled, which in turn generates a saving over 25% in natural resources compared with an open fuel cycle. The production of minor actinides is increased by about 30% compared with oncethrough recycling. Among the different concepts discussed, CORAIL (MOX and UOX rods in the same fuel assembly) and MIX (MOX fuel on enriched uranium support structures) currently appear to be the most accessible considering the status of French knowledge in the field of oncethrough recycling in PWRs. Their deployment on a fleet-scale could be envisaged in 20 to 30 years' time. At this stage, the preliminary studies have not revealed any unacceptable issues with reactor operation, but these results remain to be consolidated by additional engineering studies. For deployment on a fleet scale, these concepts require reactors capable of handling MIX and CORAIL fuels, with the number of these reactors varying depending on the concept and fuel assembly configuration chosen, e.g. 30% of the fleet for a MIX management strategy. They also require new fuel cycle plants, particularly facilities capable of managing the increased plutonium levels.

In the case of a transitional scenario between the current once-through recycling solution and the full deployment of SFRs, the option of being able to implement multiple recycling in PWRs would give us time to improve the processes and industrial maturity of the fuel cycle plants needed for SFRs, considering that equilibrium in the annual plutonium and MOX fuel flows occurs somewhere between the flows needed for once-through recycling and those expected to be required for an SFR fleet. Multiple recycling in PWRs deteriorates the fissile quality of plutonium which is currently stored as a resource for the deployment of SFRs (e.g. about 45% 239 Pu + 241 Pu for the MIX concept at equilibrium, compared with about 50% for spent MOX fuel and a little more than 60% for spent UOX fuel). It would be possible to stabilise or even reduce the plutonium inventory with a multiple recycling solution.

It should be pointed out that the technical performance (i.e. savings in natural uranium resources and reduced waste production) of multiple recycling in PWRs is much lower than that achieved by fast reactors, which remain the most relevant choice for a closed fuel cycle. How complex the PWR fuel cycle becomes by implementing multiple recycling instead of oncethrough recycling must also be assessed in terms of the refined flows of recycled materials, the detailed analysis of the industrial feasibility, and the technical progress required for its implementation in the reactors and fuel cycle plants.

Waste disposal surface areas:

The waste disposal surface areas have mainly been assessed on the basis of the thermo-hydromechanical criterion in collaboration with Andra. At this stage of the study, only the disposal surface area required for vitrified waste packages has been assessed for the different options. These results highlight the relevance of using a closed fuel cycle to reduce the surface area required for waste disposal, it being understood that there would no longer be any spent fuel in disposal, contrary to phases A and B. These assessments are still ongoing in collaboration with Andra.

Fig. 1 :Figure 2

 12 Fig. 1: Changing composition of the fleet for transitional scenario ABCD1

Figure 2 :

 2 Figure 2: Cumulative consumption of natural uranium (kt) for the SFR deployment scenarios and comparison with the scenarios involving oncethrough recycling and the open cycleThese scenarios achieve the objective of stabilising the plutonium inventory at a value close to 1200 tonnes (Figure3). Conversely, the overall plutonium inventory continues to increase linearly in the case of the open cycle scenario, or at a slower rate during once-through recycling of Pu in PWRs (Scenario A continuing the current situation). Accelerated transition scenario ABD2 stabilises the Pu inventory from 2120 on.

Figure 3 :

 3 Figure 3: Variation in the overall Pu inventory (tons) for the SFR deployment scenarios

Figure 6 :

 6 Figure 6: Consumption of natural uranium (kt) in the MIX and CORAIL scenarios compared with scenarios A, ABD2 and ABCD1 The main objective of these scenarios is to 'absorb' the spent MOX fuel inventories as quickly as possible so we can test and validate the capacity of the CORAIL and MIX concepts to optimise existing fuels. This objective can be reached by deploying the MIX scenarios for 15 years or the CORAIL scenario for 25 years on the condition that PWR MOX fuels are reprocessed at a much faster rate. The industrial feasibility of achieving such reprocessing rates during this period of time remains to be demonstrated. The additional 10 years for the CORAIL concept are due to the tradeoff between maintaining a reasonable inventory of separated Pu and maintaining a nominal reprocessing capacity that reflects the equilibrium to be reached in the future. The total plutonium inventories for the different scenarios become relatively stable from 2060 once the fleet has reached equilibrium. Achieving stability is the second most important objective of these Pu multiple recycling scenarios in a thermal fleet once all of the spent MOX fuel has been recycled. It is reached about 15 years after having deployed the first MIX or CORAIL management strategies in PWRs (2045). It can be seen that the Pu inventory can be stabilised with inventories ranging between 640 and 680 tonnes in 2100, for example, in contrast with the regular increase in the Pu inventory for scenario A (continuation of the current situation) which generates about 1000 tonnes at the same period in time. This Pu inventory

Figure 7 :

 7 Figure 7: Minor actinide inventory (tons) for the MIX and CORAIL scenarios compared with scenarios A and ABCD1

Figure 8 :Figure 10 :

 810 Figure 8: Disposal surface areas required for HLW (m 2 /TWh)

TABLE I :

 I Progressive deployment of SFRs -Main characteristics for each phase

		A	B	C	D2	D1	Open cycle (0)
	Fraction of SFRs in	0%	4.5%	31%	73%	100%	0
	the fleet
	Unat consumption	6000	5700	3400	0	0	7500
	(t/year)
	Pu inventory	 7.2	 6.8	Stabilised	Stabilised	Stabilised	 9.9
	(t/year)
	Minor actinide	 3.3	 3.1	 4.4	 3.4	 2.3	 2.7
	inventory (t/year)
	Increase in the spent	180	130	0	0	0	950
	fuel inventory (t/y)

TABLE II :

 II Multiple recycling of Pu in PWRs -Main characteristics for each phase

		A	CORAIL	MIX 9.54%	MIX 9.54%	D1	Open cycle
				with ERU	without ERU		(0)
				management	management
	Fraction of MIX/CORAIL or SFRs in	0%	87%	35%	35%	100%	0
	the fleet
	Unat consumption (t/year)	6000	6300	5500	6900	0	7500
	Pu inventory (t/year)	 7.2	Stabilised	Stabilised	Stabilised	Stabilised	 9.9
	Minor actinide inventory (t/year)	 3.3	 4.2	 4.5	 4.2	 2.3	 2.7
	Increase in spent fuel inventory (t/y)	180	0	0	0	0	950

The Pu concentration of 9.54% corresponds to the "MOX parity" management strategy that EDF asked the French regulator to examine. The regulator accepted a Pu concentration of 9.08% in the end, which is sufficient for the management of MOX fuels in the coming years.

ACKNOWLEDGMENTS

The authors are grateful to EDF, Orano and Framatome for their support on this work through the CEA research programme on the back-end of the fuel cycle. They would also like to thank Andra for its help in assessing the impact of the different fuel cycle options on waste disposal.


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Assessing the homogeneity of an alumina/zircon powder mixture at various scales by scanning electron microscopy

Martin Giraud, S. Vaudez, Cendrine Gatumel, G. Bernard-Granger, Henri

Berthiaux

To cite this version:

Martin Giraud, S. Vaudez, Cendrine Gatumel, G. Bernard-Granger, Henri Berthiaux. Assessing the homogeneity of an alumina/zircon powder mixture at various scales by scanning electron microscopy. PACRIM 13 -The 13rd Pacific Rim Conference of Ceramic Societies, Oct 2019, Okinawa, Japan. cea-02614141

Martin GIRAUD 1,2 , Stéphane VAUDEZ 2 , Cendrine GATUMEL 1 , Guillaume BERNARD-GRANGER 2 , Henri BERTHIAUX 1 1 : Université de Toulouse, IMT Mines Albi, UMR CNRS 5302, RAPSODEE, Campus Jarlard, F-81013, Albi, France 2 : 1 CEA, DEN, MAR, DMRC, SFMA, LFC, F-30207 Bagnols-sur-Cèze Cedex, France

Powder mixing is a fundamentally important unit operation in industries handling powders. The purpose of such an operation is to reach a given homogeneity, which has to be defined at a certain scale, the observation scale, in order to be relevant. This scale is defined by the final product's requirements.

Moreover, the homogeneity of a mixture can hardly be known exactly, and its assessment is always a scientific issue due to theoretical and practical limitations.

The most obvious limitation comes from the fact that it is rarely possible to get an exhaustive analysis of the mixture, since most analysis techniques require sampling. Thus, homogeneity has generally to be assessed from a restrained number of samples using statistical considerations. Finally, the analysis technique must enable a good differentiation of the powders while fulfilling the sampling procedure's requirements, specifically in terms of size and number of the samples.

In this study, alumina (Al2O3) and zircon (ZrO2) powders have been used as simulants for an industrial process combining mixing and ball milling steps. Homogeneity of the blend among the whole process is crucial. Various binary mixtures of alumina (Al2O3) and zircon (ZrO2) powders have been prepared using two different techniques: in a Turbula® mixer and in a dry ball mill. Within the ball mill, a reduction of the particle's size takes place simultaneously with the mixing operation. This allows homogenization at a much finer scale. Therefore, the knowledge of the homogeneity of the mixture at different scales is needed in order to identify the influence of different operational parameters on the homogenization of the mixtures during both processes.

The sampling was performed by drawing several portions, around two grams, into the final blends obtained by the Turbula® mixer and the ball mill. Each sample was then shaped into pellets in order to allow Scanning Electrons Microscopy observations without disrupting the samples. SEM has been used to perform subsampling of the pellets. Indeed, the back-scattered electrons signal analysis allows a very good differentiation of alumina and zircon due to their different atomic numbers. This technique also allows assessing the homogeneity of the mixtures at various scales by varying the magnification, and thus varying the size of the sub-samples. It is then relevant for both mixing paths.

Then, a MatLab® program has been developed in order to perform image analysis by theresolding and binarizing the SEM images. The program was also designed to perform statistical treatment of the samples and sub-samples providing sampling procedure for both scales. In particular, the number and size of the samples and sub-samples needed was computed.

Finally, this study provides a whole methodology to assess homogeneity of alumina and zircon mixtures at various scales for Turbula® mixing and ball milling.

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