C Élestin

C C Kokonendji
email: celestin.kokonendji@univ-pau.fr

Tristan Senga

Clarice G B Dem Étrio

Appropriate kernel regression on a count explanatory variable and applications

Keywords: Count data, cross-validation, discrete kernel estimator, discrete triangular distribution, nonparametric regression AMS Subject Classification: Primary : 62G08; Secondary : 62G99

We propose an appropriate nonparametric regression on a single count regressor using a recent discrete kernel approach. We adapt the Nadaraya-Watson estimator to this discrete kernel for smoothing the regression function on count data. Some properties are studied; in particular, the bandwidth selection is investigated through the cross-validation method. The proposed regression, in addition to being simple, easy to implement and effective, outperforms the competing usual regressions for small and moderate sample sizes. Using simulations and two examples from real life, the importance and the performance of discrete kernels are pointed out and compared with the optimal continuous kernel.

Introduction

Let (x i , y i ), i = 1, . . . , n, be a sample of observations in N × R, where N denotes the non-negative integers set and R is the real line. Without any specification of distributions and in the absence of an evident relation between these observations, the most natural nonparametric regression model of y i on x i is

y i = m(x i ) + e i , (1)
where y i denotes the observation of a real random variable Y i , x i is the observation of the count explanatory variable (or regressor) X i , e i is the disturbance term from the real random variable i satisfying commonly E( i ) = 0 and var( i ) = σ 2 , and m : N → R is the unknown regression function on count data that we will call regression count function for brevity. That function can be represented as m(x i ) = E(Y i |X i = x i ) for a random effect. We are interested in estimating or (discrete) smoothing this regression count function m by taking into account especially its counting structure.

Setting apart the 'frequency (or naif) estimator', there is some literature on nonparametric smoothing of discrete variables or functions dating back to the pioneering work of [START_REF] Aitchison | Multivariate binary discrimination by the kernel method[END_REF]; however, the corresponding discrete kernels are essentially for categorial data or finite discrete distributions. [START_REF] Bierens | Non-linear regression with discrete explanatory variable with an application to the earnings function[END_REF] proposed regression models which take the form of a polynomial of a linear function of the regressors with discrete explanatory variables. A non-appropriate but always used way of smoothing the regression count function m in ( 1) is simply to consider the count regressor as continuous and then apply one of the numerous techniques of nonparametric estimation of the regression continuous function (see, for example, [START_REF] Chen | Beta kernel smoothers for regression curves[END_REF][START_REF] Collomb | Estimation non-paramétrique de la régression: revue bibliographique[END_REF][START_REF] Th | Kernel estimation of regression functions[END_REF][START_REF] Michels | Asymmetric kernel functions in non-parametric regression analysis and prediction[END_REF]. Regrettably, the particular structure of counting of the regressor is not taken into account. Recently, in the particular situation of the so-called nonparametric binomial regression where the response variable Y i follows the binomial distribution B{N i , m(x i )} at each covariate x i , Okumura and Naito (2004, 2006a) needed to transform the discrete variable x before using the well-known symmetric kernel estimator of [START_REF] Nadaraya | On estimating regression[END_REF] and [START_REF] Watson | Smooth regression analysis[END_REF] for regression function on continuous data. An extension of this method for multinomial data is also given by Okumura and Naito (2006b).

In this paper we use the discrete analog of the continuous kernel estimator, introduced by Kokonendji et al. (2007b) and Senga [START_REF] Kiessé | Approche non-paramétrique par noyaux associés discrets des données de dénombrement[END_REF], for estimating the regression count function m in (1) without transforming the count (or discrete) explanatory variable. That is done by adapting the (continuous) estimator of Nadaraya-Watson to the discrete case, which is one of the oldest and simplest weighted estimators for nonparametric regression function. We attempt to illustrate the necessity of this procedure and its capacity for producing better explanations of real data by means of simulations and also two examples.

For the first example, a sales data set (Table 1), we analyse the number of a new product sold per day during the 25 first days. The 160 observations (x i , y i ), i = 1, . . . , 25, represent the day x i of the sales and the corresponding mean numbers y i ∈ {y Ai , y Bi , . . . , y Hi } of sales. The number of sale centres for each state (A, B, . . . , H) is not available except for the state H which is just one. Note that x 1 = 0 represents the first day of the product in the market and its sale results only from a previous advertising campaign; x 2 , x 3 , . . . represent the other days when the sales of the product are results also from informal advertisings of the customers. Because of the novelty of the product, we note that at the beginning there is a fast increase in the mean number of sales up to x 5 in general. After that, the mean number of sales starts to decrease, which can be a result of a limited number of customers or the lack of success of the product. We also observe that after the date x 14 there are some missing data from some centres. The aim of this type of study is to have a policy for the advertising campaign which in general is very expensive.

(Tables 1,2 and Figure 1 about here)

The second example (Table 2) concerns the study of average daily fat (kg/day) yields from milk of a single cow for each of the 35 first weeks (McCulloch, 2001, p. 40-45). The quantity of fat in the milk increases during the first fourteen weeks and decreases after. The fitted curves for two generalized linear models (see also [START_REF] Mccullagh | Generalized Linear Models[END_REF]) are presented in Figure 1. In fact, the first represents a normal model for the log-transformed response variable Y i with an identity link and the second represents a normal model for the response variable Y i with a logarithmic link. Both have the same linear predictor β 0 + β 1 x i + β 2 log x i , where x i denotes the week. We can see that neither model fits well to the data although there is an improvement for the log link model. In particular, they do not detect the plateau associated with observations x = 19, 20, . . . , 27. We will compare these results with those obtained by our new nonparametric regression.

The rest of the paper is arranged as follows. Section 2 reviews the recent discrete kernel methods to make the paper self-contained as possible. In Section 3, we define the associated discrete kernel estimator for regression count function from the Nadaraya-Watson estimator and give some properties. In particular, we establish a result on the pointwise squared error and the bandwidth selection is made by adapting the least-squares cross-validation method. The new procedure is illustrated through simulations and the two motivating examples in Section 4. Furthermore, some comparisons are pointed out within discrete kernels and also with the optimal continuous kernel (see [START_REF] Epanechnikov | Nonparametric estimates of a multivariate probability density[END_REF]. Section 5 concludes the paper and also suggests some extensions and future research topics.

Discrete kernel methods

We briefly recall the more recent discrete kernel methods for estimating (or smoothing) a probability mass function (pmf) on N. Such methods are basic for regression function on count data. See, for example, [START_REF] Izenman | Recent developments in nonparametric density estimation[END_REF] for continuous cases.

Let X 1 , . . . , X n be independent observations from a count distribution with an unknown pmf f (x) := Pr(X i = x), the discrete kernel estimator of f is expressed as

f n (x) = 1 n n i=1 K x,h (X i ) = 1 n n i=1 K (X i ; x, h) , x ∈ N. (2)
Given x ∈ N and h > 0, the associated discrete kernel

K x,h (•) = K(•; x, h) of f n (x) ≡ f n (x; h) connecting to a discrete random variable K x,h on its support ℵ x (not depending on h) is such that ℵ x ∩ N = ∅, ∪ x ℵ x ⊇ N, var(K x,h
) < ∞ and we also have:

lim h→0 E(K x,h ) = x, (3)
lim h→0 var(K x,h ) = 0. (4)
That definition unifies the notion of associated kernel that might be either continuous or discrete (Senga [START_REF] Kiessé | Approche non-paramétrique par noyaux associés discrets des données de dénombrement[END_REF]. Furthemore, the first two conditions on the supports N of the unknown pmf and ℵ x can be changed by ∅ = ℵ x ⊆ N. Then, the conditions (3) and ( 4) are important because they allow to obtain the pointwise convergence of the discrete kernel estimator (2). In fact, the basic relation in (3) reflects one of the main differences from the continuous symmetric case where the kernel functions are centred in 0 by using the ratio (x -X i )/h, and such that K x,h (•) = (1/h)K{(x -•)/h}. This condition (3) clearly points out that the discrete kernel estimator f n of f defined by ( 2) is a kind of variable kernel estimate by giving a general form K. It also allows for more flexibility to construct different (associated) discrete kernels from any discrete ditribution K. So, we shall distinguish two families of discrete kernels satisfying E(K x,h ) = x + h or E(K x,h ) = x. It is implicitly used in asymmetric continuous cases by [START_REF] Chen | Beta kernel estimators for density functions[END_REF]Chen ( , 2000a) and then by [START_REF] Scaillet | Density estimation using inverse and reciprocal inverse Gaussian kernels[END_REF]. It should be noted that all (associated) kernels satisfying (3) share the property that the shape of the kernels changes according to the value of the target x. This associated discrete kernel (or varying kernel shape) changes the amount of smoothing applied to the asymmetric kernel since its variance var(K x,h ) may or may not depend on the target x as we move away from the boundary. According to the behaviour of the variance with respect to the expectation at each target, the (associated) discrete kernels can be underdispersed (i.e. var(K

x,h ) < E(K x,h )), equidispersed (i.e. var(K x,h ) = E(K x,h )) or overdispersed (i.e. var(K x,h ) > E(K x,h )).
The last condition (4) insures an asymptotic behaviour equivalent to the frequency estimator for the discrete kernel estimator f n of f defined by [START_REF] Bierens | Non-linear regression with discrete explanatory variable with an application to the earnings function[END_REF]. Thus, we deduce several properties of the discrete kernel estimator f n of the unknown count distribution f as follows. Up to the normalizing constant C = x∈N f n (x), we assume that x → f n (x) is a pmf. Then, we have:

E{ f n (x)} = y∈ℵx∩N K x,h (y)f (y) = y∈ℵx f (y) Pr(K x,h = y) = E{f (K x,h )}. (5)
This leads to approximate the pointwise bias using the discrete Taylor expansion (see, for example, Schumaker, 1981, p. 343) as bias{

f n (x)} = E{f (K x,h )} -f (x) = f {E(K x,h )} -f (x) + 1 2 var(K x,h )f (2) (x) + o(h), (6)
where f (2) is the finite difference of second order

f (2) (x) =      {f (x + 2) -2f (x) + f (x -2)}/4 if x ∈ N \ {0, 1} {f (3) -3f (1) + 2f (0)}/4 if x = 1 {f (2) -2f (1) + f (0)}/2 if x = 0 (7)
which is recursively obtained through the finite difference of order k ∈ N \ {0}:

f (k) (x) = {f (k-1) (x)} (1) and f (1) (x) = {f (x + 1) -f (x -1)}/2 if x ∈ N \ {0} f (1) -f (0) if x = 0. (8)
Here, it is not necessary to suppose certain regularity or differentiability on f because it is a pmf and the finite difference substitute the derivation of the continuous case. The pointwise variance can be expressed as

var f n (x) = 1 n y∈ℵx f (y) {Pr(K x,h = y)} 2 - 1 n z∈ℵx f (z) Pr(K x,h = z) 2 = 1 n f (x) {1 -f (x)} {Pr(K x,h = x)} 2 + R n (x; h), (9)
with

R n (x; h) = 1 n   y∈ℵx\{x} f (y) {Pr(K x,h = y)} 2 + {f (x) Pr(K x,h = x)} 2   - 1 n z∈ℵx f (z) Pr(K x,h = z) 2 .
Under the condition (4) of the associated discrete kernel, we can verify that

R n (x; h) = o(1/n).
Remark 1. For nonparametric estimator in (2), a relative efficiency between two (associated) discrete kernels K 1

x,h and

K 2 x,h with E(K 1 x,h ) = E(K 2 x,h
) can be measured via [START_REF] Chen | Gamma kernel estimators for density functions[END_REF] in terms of the difference between their variances var(K 1

x,h )var(K 2

x,h ) for discrete kernels.

Since the mean integrated squared error

M ISE(n, h, K, f ) = x∈N var{ f n (x)} + x∈N bias 2 { f n (x)}
of the estimator f n of f defined in equation ( 2) is the common measure of accuracy for an estimator, we can establish the following result of convergence (see Senga [START_REF] Kiessé | Approche non-paramétrique par noyaux associés discrets des données de dénombrement[END_REF]:

Theorem 2.1 Let f be a pmf on N with lim x→∞ f (x) = 0.
Then, the nonparametric estimator in ( 2) with an associated discrete kernel satisfies

M ISE ≤ C 1 n x∈N {Pr(K x,h = x)} 2 + x∈N f {E(K x,h )} -f (x) + 1 2 var(K x,h )f (2) (x) 2 , with C 1 = f max {1 -f min } ≤ 1. Furthermore, for n → ∞ and h = h(n) → 0, we have M ISE(n, h, K, f ) → 0 if 1 n x∈N {Pr(K x,h = x)} 2 → 0 and x∈N f {E(K x,h )} -f (x) + 1 2 var(K x,h )f (2) (x) 2 → 0.
Remark 2. The M ISE of the 'frequency estimator' (with the Dirac kernel type) given by

(1/n) x∈N f (x){1 -f (x)} = (1/n) 1 -x∈N f 2 (x)
may be considered as the reference for the convergence of discrete kernel estimators.

Presented below are two examples of usual and competitive families of discrete kernels taken from Senga [START_REF] Kiessé | Approche non-paramétrique par noyaux associés discrets des données de dénombrement[END_REF]. Figure 2 gives a quick glance at the set and we summarize the main properties in Table 3. Note here that, in general, the choice of (discrete) kernel function is not asymptotically very important, such as frequency estimator; but in small and moderate samples, the kernel structure may play a more crucial role in approximating the sample distribution, especially for count random variables.

Example 2.1 (Binomial).

Consider the binomial distribution B(N, p), N ∈ N, p ∈ [0, 1]. The binomial kernel B x,h follows the binomial distribution B{x + 1, (x + h)/(x + 1)} =: B x,h with h ∈ (0, 1] and ℵ x = {0, 1, . . . , x + 1}. From Remark 1, it is better in the class of the so-called standard asymmetric discrete kernels K x,h , such as Poisson and negative binomial, having exactly E(K x,h ) = x + h and ∪ x ℵ x = N but they do not satisfy (4). This is because it is underdispersed:

var(B x,h ) = (x + h)(1 -h)/(x + 1) < x + h. Hence, from (2), the corresponding binomial kernel estimator of f is f B n (x) = 1 n n i=1 (x + 1)! X i !(x + 1 -X i )! x + h x + 1 X i 1 -h x + 1 x+1-X i , x ∈ N with X i ≤ x + 1.
Its pointwise variance can be deduced from [START_REF] Th | Kernel estimation of regression functions[END_REF] as

var{ f B n (x)} = (1 -h) 2 n f (x){1 -f (x)} x + h x + 1 2x + R B n (x; h), with R B n (x; h) 0 when n → ∞ and h = h(n) → 0. For the pointwise bias, a direct calculus gives bias{ f B n (x)} = f (x) (1 -h) x + h x + 1 x -1 + y∈ℵx\{x} f (y)B x,h (y)
which does not tend to 0 when n → ∞ and h = h(n) → 0. Thus, it follows that f B n does not converge in the sense of M ISE. However, Senga Kiessé (2008) showed that the estimator f B n of f can be better (in the sense of M ISE) than the frequency estimator for some finite sample sizes. This estimator and all others of the class are not subject to boundary bias. Furthemore, the target x is not the mean of the corresponding asymmetric kernel, but rather its mode. For the choice of the optimal bandwidth we use the well-known procedure of cross-validation.

(Figure 2 and Table 3 about here) Example 2.2 (Discrete triangular). The discrete triangular distributions, introduced by Kokonendji et al. (2007b), are useful to construct a family of symmetric discrete kernel estimators for a pmf. For given (a, x, h) ∈ N × N × (0, ∞), the associated discrete triangular kernel T a;x,h is defined through the pmf of its corresponding random variable T a;x,h on ℵ a;x = {x, x ± 1, . . . , x ± a} as

Pr(T a;x,h = y) = (a + 1) h -|y -x| h P (a, h) , y ∈ ℵ a;x ,
where P (a, h) = (2a + 1)(a + 1) h -2 a k=0 k h is the normalizing constant. The three parameters are such that a denotes the arm and is fixed, x = E(T a;x,h ) is the center and represents the target, and h is the order which corresponds to the bandwidth. The particular case T 0;x,h provides the Dirac random variable at x. From (2), the class of discrete triangular kernel estimators is given, for fixed a = 0, as

f Ta n (x) = 1 n n i=1 (a + 1) h -|X i -x| h (2a + 1)(a + 1) h -2 a k=0 k h , x ∈ N.
Its pointwise variance can be written from [START_REF] Th | Kernel estimation of regression functions[END_REF] as

var{ f Ta n (x)} = 1 n f (x) {1 -f (x)} (a + 1) h P (a, h) 2 + R Ta n (x; h),
with lim h→0 (a + 1) h /P (a, h) = 1 and R Ta n (x; h) → 0 when n → ∞ and h = h(n) → 0. While its pointwise bias can be directly obtained by using [START_REF] Chen | Gamma kernel estimators for density functions[END_REF] as follows:

bias{

f Ta n (x)} = 1 2 V (a, h)f (2) (x) + o(h),
where f (2) is as given in [START_REF] Chen | Beta kernel smoothers for regression curves[END_REF] and

V (a, h) = var(T a;x,h ) = {a(2a + 1)(a + 1) h+1 /3 -2 a k=0 k h+2 }/P (a, h) (10)
tends to 0 when h → 0. The condition (4) holds for T a;x,h and we can therefore apply Theorem 2.1 to get the convergence of f Ta n in the sense of M ISE. From Remark 1, T a 1 ;x,h is more efficient than T a 2 ;x,h when a 1 < a 2 . However, for fixed a = 0, these discrete triangular kernel estimators induce a boundary bias on the left of N because the set ∪ x ℵ a;x = {-a, . . . , -1} ∪ N contains strictly the support N of the unknown pmf f . An original solution is proposed by Kokonendji, Senga [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF] to solve this situation while preserving the structure of the local symmetry of the associated discrete kernel around every target. The bandwidth selection is here made essentially by cross-validation method.

Nadaraya-Watson discrete estimator

Following the (continuous) weighted estimator of [START_REF] Nadaraya | On estimating regression[END_REF] and [START_REF] Watson | Smooth regression analysis[END_REF], the discrete analog for m in ( 1) is here defined by

m n (x) = n i=1 ω x (X i )Y i , x ∈ N, where ω x (X i ) = K x,h (X i ) n i=1 K x,h (X i ) = ω x,h (X i ) (11)
represents the weight such that n i=1 ω x,h (X i ) = 1 with the convention 0/0 = 0, and K x,h (•) denotes an associated discrete kernel as given in [START_REF] Bierens | Non-linear regression with discrete explanatory variable with an application to the earnings function[END_REF]. The bandwidth h ≡ h(n, K) determines the (discrete) smoothness of the estimate. For an appropriate discrete kernel, very small bandwidths almost reproduce the data while extremely large bandwidths yield a constant estimate for the regression count function.

For investigating the bias and variance of the discrete kernel estimator m n (x) ≡ m n (x; h) of m, it is common to write m n (x) as the ratio:

m n (x) = N n (x; h)/D n (x; h), with D n (x; h) = 1 n n i=1 K x,h (X i ) = f n (x) and N n (x; h) = 1 n n i=1 Y i K x,h (X i ).
The expressions of the expectation and the variance of D n (x; h) are given in ( 6) and ( 9), respectively, where f is assumed to be the unknown pmf of any X i . Then, from m(x) = E(Y i |X i = x), we may write successively:

E{N n (x; h)} = E{Y 1 K x,h (X 1 )} = z∈ℵx m(z)f (z) Pr(K x,h = z) = E{(mf )(K x,h )} = (mf ){E(K x,h )} + var(K x,h ) 2 (mf ) (2) (x) + o(h),
with (mf ) (2) = m (2) f + 2m (1) f (1) + mf (2) . By using the conditional expectations of Y i and of Y 2 i on X i , we calculate the variance of N n (x; h) as follows:

var{N n (x; h)} = 1 n y∈ℵx E(Y 2 1 |X 1 = y)f (y){Pr(K x,h = y)} 2 - 1 n z∈ℵx E(Y 1 |X 1 = z)f (z) Pr(K x,h = z) 2 = 1 n E(Y 2 1 |X 1 = x) -f (x)E 2 (Y 1 |X 1 = x) f (x) {Pr(K x,h = x)} 2 + r n (x; h), (12)
with

r n (x; h) = 1 n y∈ℵx\{x} E(Y 2 1 |X 1 = y)f (y){Pr(K x,h = y)} 2 + 1 n {E(Y 1 |X 1 = x)f (x) Pr(K x,h = x)} 2 - 1 n    z∈ℵx\{x} E(Y 1 |X 1 = z)f (z) Pr(K x,h = z)    2 .

Mean squared error

We here formulate a result on the mean squared error

M SE(x) = var { m n (x)} + bias 2 { m n (x)} , x ∈ N,
of the estimator m n of the regression count function m connecting to the pmf f of the regressor.

Proposition 3.1 For x ∈ N, let m(x) = E(Y |X = x) and f (x) = Pr(X = x) defined on N → R.
As n → ∞ and h = h(n) → 0, for all x such that f (x) = 0, the nonparametric regression estimator m n (x) of m(x) with an associated discrete kernel possesses the following bias and variance:

bias{ m n (x)} = m (2) (x) + 2m (1) (x) f (1) f (x) var(K x,h ) 2 + O(1/n) 2 + o(h) and var{ m n (x)} = E(Y 2 1 |X 1 = x) -f (x)E 2 (Y 1 |X 1 = x) nf (x) {Pr(K x,h = x)} 2 + o 1 n .
Then, we have

M SE(x) = m (2) (x) + 2m (1) (x) f (1) f (x) 2 var 2 (K x,h ) 4 + E(Y 2 1 |X 1 = x) -f (x)E 2 (Y 1 |X 1 = x) nf (x) {Pr(K x,h = x)} 2 + o h 2 + 1 n .
Proof. From Bosq and Lecoutre (1987; p. 119-121) for continuous case with here g = mf , we can write

m n (x) = m(x) + N n (x; h) -g(x) f (x) - g(x){D n (x; h) -f (x)} f 2 (x) - {N n (x; h) -g(x)}{D n (x; h) -f (x)} f 2 (x) + N n (x; h) f 3 (x) {D n (x; h) -f (x)} 2 {1 + o(1)} p.s. ( 13
) with D n (x; h) = n -1 n i=1 K x,h (X i ) and N n (x; h) = n -1 n i=1 Y i K x,h (X i ).
Under the condition (3) of the associated discrete kernel, we get

E{D n (x; h)} -f (x) = var(K x,h ) 2 f (2) (x) + o(h)
and

E{N n (x; h)} -(mf )(x) = var(K x,h ) 2 (mf ) (2) (x) + o(h),
with (mf ) (2) = m (2) f + 2m (1) f (1) + mf (2) . Furthermore, we can show that

E[N n (x; h){D n (x; h) -f (x)} 2 ] = O(1/n) 2 + O(1/n) + E{D n (x; h) -f (x)} 2 E{N n (x; h)}.
Thus, we obtain

E{ m n (x)} -m(x) = (mf ) (2) (x) f (x) - mf (2) (x) f (x) var(K x,h ) 2 + E{D n (x; h) -f (x)} 2 E{N n (x; h)} f 3 (x) - f (x)E[{N n (x; h) -g(x)}{D n (x; h) -f (x)}] f 3 (x) + O(1/n) 2 + o(h).
Then, in the previous expression we express

E{D n (x; h) -f (x)} 2 E{N n (x; h)} -f (x)E[{N n (x; h) -g(x)}{D n (x; h) -f (x)}] = O(1/n) 2 + o(h),
which finally leads to bias{ m n (x)} = m (2) (x) + 2m (1)

(x) f (1) f (x) var(K x,h ) 2 +O(1/n) 2 +o(h).
(14) For the variance of m n , from [START_REF] Mangasarian | Best summation formulae and discrete splines[END_REF], we obtain:

var{ m n (x)} = var{N n (x; h)} f 2 (x) + g 2 (x) f 4 (x) var{D n (x; h)} -2 g(x) f 3 (x) cov{N n (x; h), D n (x; h)} + o 1 n .
By using the variance of N n in ( 12), var{D n (x;

h)} = O(1/n) and cov{N n (x; h), D n (x; h)} = O(1/n), we get var{ m n (x)} = E(Y 2 1 |X 1 = x) -f (x)E 2 (Y 1 |X 1 = x) nf (x) {Pr(K x,h = x)} 2 +o 1 n .
(15) The expressions of the expectation ( 14) and the variance (15) of m n (x) allow to deduce the mean squared error at x ∈ N.

We can apply the previous result to binomial and associated discrete triangular kernels as in Examples 2.1 and 2.2.

Hence, the global error can be investigated through the M ISE. For a practical comparison of the amount of nonparametric regression obtained by some discrete or continuous kernels (with a given bandwidth h), we will use the well-known coefficient of determination, R 2 , which quantifies the proportion of variation of the response variables y i explained by the non-intercept regressor x i :

R 2 = n i=1 ( y i -y) 2 n i=1 (y i -y) 2 , ( 16
)
with y i = m n (x i ; h), y = n -1 (y 1 + . . . + y n ) and

1 n n i=1 (y i -y) 2 = 1 n n i=1 ( y i -y) 2 + 1 n n i=1 (y i -y i ) 2 .

Bandwidth selection

In this context of discrete kernel regression on count data, the bandwidth selection is also obtained by the well-known least-squares cross-validation method (see, for example, Hardle and Marron, 1985). In fact, for a given associated discrete kernel, the optimal bandwidth is h cv = arg min h>0 CV (h) with

CV (h) = 1 n n i=1 {Y i -m -i (X i ; h)} 2 M (X i ), (17)
where

m -i (X i ; h) = n j =i Y j K X i ,h (X j )/ n j =i K X i ,h (X j )
is the leave-one-out kernel estimator of m n (X i ; h) and 0 ≤ M (•) ≤ 1 is a weight to avoid difficulties caused by dividing by zero or by the slow convergence rate caused by boundary effects.

Note that, for practical use of the weight M (•) in ( 17), we can consider M (X i ) = ω X i ,h (X i ) ≡ M (X i ; h) defined by [START_REF] Izenman | Recent developments in nonparametric density estimation[END_REF] and depending on h.

Illustrations with discussion

In this section we present a simulation study and we apply the discrete kernel estimator m to both data sets of Tables 1 and2. In order to analyse and then compare the effects of discrete and continuous kernels, we complete Example 2.1 with two other discrete kernels, Poisson and negative binomial, and we add the optimal continuous kernel of [START_REF] Epanechnikov | Nonparametric estimates of a multivariate probability density[END_REF]. Indeed, the Poisson and negative binomial kernels are standard asymmetric discrete kernels on the same support ℵ x = N with h > 0 such that

P x,h (z) = (x + h) z e -(x+h) z! , z ∈ N,
and

N B x,h (z) = (x + z)! x!z! x + h 2x + 1 + h z x + 1 2x + 1 + h x+1 , z ∈ N,
respectively. As for the Epanechnikov kernel, it is defined by

K E (z) = 3 4 (1 -z 2 ), z ∈ [-1, 1].
For the following illustrations, and given (discrete or continuous) kernel, we first determine a bandwidth using generally the cross-validation procedure with the weight M ≡ 1. Then, we calculate the corresponding (weigthed) coefficient of determination R 2 . Finally, we give some plots using linear interpolation between the regression points. We do not examine here the model diagnostics in terms of the residual study. However, we will use sometimes the term of better fit in the sense of M ISE if the bandwidth is selected by the cross-validation method.

Simulation study

The regression count function considered is

m(x) = 2 x x! , x ∈ N.
Table 4 contains the optimal average integrated squared errors and their standard errors for the estimators based on 1000 replications. For each simulation, the optimal discrete smoothing bandwidths are given by the cross-validation method. The optimal integrated squared errors are determined by using the optimal bandwidths. The results in Table 4 show that the associated discrete triangular kernels with small arms perform much better than the binomial and the Epanechnikov ones, even if the sample size is so large. We do not recommend the use of Poisson and negative binomial kernels because they are neither underdispersed nor satisfied (4). Finally, we note that the simulated average integrated squared errors of the best discrete and continuous kernel estimators are closed as the sample size n increased.

(Table 4 about here)

Average daily fat

Table 5 and Figure 3 present the corresponding results for the nonparametric regressions for the data in Table 2. The associated discrete triangular kernel with a = 2, h cv = 0.1 and R 2 = 99.140% represents the most interesting results (in both senses of MISE and R 2 ) among all these nonparametric regressions. Then, we have the binomial (R 2 = 97.179%) and the Epanechnikov (R 2 = 96.967%). These three kernels point out the plateau associated with observations x i = 19, 20, . . . , 27. Note that the Poisson and negative binomial kernels do not detect this behaviour, similarly to the generalized linear models discussed in the Introduction; they underestimate or overestimate most of the y-values.

For these data with both single regressor and endogenous observations, we note that the nonparametric regression by the discrete triangular kernel with a = 1 gives a better fit in both senses of M ISE and R 2 , but not in the sense of better representation (or regression curve) because it almost reproduces the data. So, we improve the (discrete) smoothing by taking other values of the arm a which is a free parameter and depends on the user. However, for the associated discrete triangular kernel with a = 2, we also obtained a smoother regression curve (or representation) by changing the value of the discrete smoothing parameter (h = 0.5 with R 2 = 97.321%). For a = 4, the associated discrete triangular kernel provides R 2 ∈ {96.445, 93.101, 90.126, 88.843} according to the values of the bandwidths h ∈ {0.1, 0.3, 0.7, 1}, respectively. This shows that the obtained regression curves are smoother while the R 2 does not change too much. (We omit here to present their corresponding tables and figures.) The main question here is to find an optimal arm when we use discrete triangular kernels for getting a smoother regression curve with a better R 2 associated to the optimal bandwidth, which is an open problem.

(Table 5 and Figure 3 about here)

In general, if the optimal bandwidth h cv gives an optimal nonparametric regression in the sense of R 2 (and therefore M ISE). Use of other values of the discrete smoothing parameter h may also give a good fit in the sense of R 2 and a smoother curve or representation, providing broader choice for h according to the purpose of the user.

Sales data

Table 6 and Figure 4 show the results for the nonparametric regressions for the data in Table 1, obtained by discrete kernels (triangular with a ∈ {1, 2}, binomial, Poisson and negative binomial) and the Epanechnikov kernel.

(Table 6 and Figure 4 about here) Among discrete kernels, the discrete triangular with a ∈ {1, 2} and binomial kernels give the best results for the nonparametric regressions in both senses of M ISE and R 2 which are around 95%, 92% and 80%, respectively. Again the Poisson and negative binomial kernel regressions either underestimate (x < 10) or overestimate (x > 10) the y-values. Omitting the six strange values (with * ), the R 2 values increase for all the kernels but the order of performance remains the same as the previous results.

The R 2 value for the (continuous) Epanechnikov kernel is smaller than the R 2 value for the discrete triangular. So, the associated discrete triangular kernels are more appropriate for this data set.

Concluding remarks

In this work, we have introduced an appropriate and efficient nonparametric regression on a single count regressor [START_REF] Aitchison | Multivariate binary discrimination by the kernel method[END_REF]. The discrete kernels used are easy to implement and directly applicable to the count variable without any transformation. According to the discrete kernels, in particular binomial and associated discrete triangular with small arms, simulation results and both data sets demonstrate the usefulness of the proposed method which is the best one or competitive with respect to the optimal continuous kernel regression. Since there is not for instance an optimal associated discrete kernel, one of the reasons of these good fits comes from the small variance and also the finite support of these interesting discrete kernels which are binomial and discrete triangular with small arms. Another reason is suggested by the very good approximation between these discrete kernels and the Gaussian kernel, which has the advantage of being one of the best in the (symmetric) continuous kernel nonparametric regression. Some evident and practical extensions of the model (1) are for several regressors. The first one is to consider that all the regressors are counts. The multivariate (associated) discrete kernel can be a product of some univariate ones from the same or different discrete distributions. The second and more useful extension is to investigate a nonparametric regression which admits a mixture of discrete and continuous explanatory variables using also the method of appropriate kernels. For the part of the discrete (or discretized) regressors, we must distinguish count regressors from categorial ones. At the same time, we can improve by using semiparametric regression, where the model of the endogenous variable y i on the p × 1 vector of regressors x i = (x 1i , . . . , x pi ) ′ contains a parametric function g(x i ; β) and a nonparametric factor m(x i ) such that y i = g(x i ; β)m(x i ) + e i for all i = 1, 2, . . . , n. 2) by some discrete kernels and the continuous Epanechnikov kernel

Figure 1 :

 1 Figure 1: Two generalized linear models from data of Table2, with the best R 2 = 0.76790[START_REF] Mcculloch | An Introduction to Generalized Linear Mixed Models, 46a Reunião Anual da RBRAS -9o SEAGRO[END_REF]

Figure 2 :

 2 Figure 2: Behaviour of some discrete kernels for count distributions of Senga Kiessé (2008) at the target y = x = 5 with the same bandwidth h = 0.1

Figure 3 :

 3 Figure 3: Nonparametric regressions of average daily fat (Table2) by some discrete kernels and the continuous Epanechnikov kernel

Table 1 :

 1 Sales data (withdenotes a missing observation and * can be considered as a strange value)

	x i	y Ai y Bi	y Ci y Di	y Ei	y F i	y Gi y Hi
	0	7.2	9.7	5.0	9.0 12.0	7.0	9.2	7
	1 16.4 15.9 * 5.0 16.1 16.2 15.4 16.2	16
	2 21.4 18.9 22.2 19.7 17.2 20.7 19.9	23
	3 22.0 19.7 24.2 20.5 18.1 22.4 21.1	20
	4 20.4 19.2 23.0 19.8 17.6 21.7 20.5	23
	5 18.2 18.1 20.4 18.4 16.8 19.8 * 8.8	24
	6 16.1 16.6 17.7 16.6 15.7 17.6 * 6.5	12
	7 14.2 15.0 15.2 14.8 14.5 15.3 * 4.0	13
	8 12.6 13.5 13.0 13.0 13.3 13.3 * 1.8	9
	9 11.1 12.0 11.1 11.5 12.1 11.5	9.9	9
	10	9.9 10.6	9.5 10.0 10.9	9.9	8.5	8
	11	8.7	9.4	8.2	8.8	9.8	8.6	7.5	10
	12	7.8	8.3	7.1	7.7	8.7	7.4	6.8	8
	13	6.9	7.3	6.2	6.7	7.8	6.5	6.3	7
	14	-	6.4	5.4	5.9	6.9	5.6	5.9	2
	15	5.4	5.6	4.7	-	6.1	4.9	5.6 * 12
	16	4.8	-	4.1	4.6	5.4	-	-	3
	17	4.3	-	3.6	-	-	3.8	-	5
	18	-	3.8	-	-	-	3.3	-	4
	19	-	-	-	3.2	3.7	-	4.3	2
	20	-	2.9	-	2.8	3.2	-	-	2
	21	-	-	-	-	2.9	2.3	3.5	5
	22	-	-	-	-	2.5	-	3.2	5
	23	-	-	2.1	1.8	2.2	-	-	2
	24	-	-	1.9	-	-	-	2.5	-

Table 2

 2

	, with the best R 2 =

Table 2 :

 2 Average daily fat (kg/day) yields from milk of a single cow for each of the 35 first weeks[START_REF] Mcculloch | An Introduction to Generalized Linear Mixed Models, 46a Reunião Anual da RBRAS -9o SEAGRO[END_REF]

	x i	1	2	3	4	5	6	7	8	9	10	11	12
	y i 0.31 0.39 0.50 0.58 0.59 0.64 0.68 0.66 0.67 0.70 0.72 0.68
	x i	13	14	15	16	17	18	19	20	21	22	23	24
	y i 0.65 0.64 0.57 0.48 0.46 0.45 0.31 0.33 0.36 0.30 0.26 0.34
	x i	25	26	27	28	29	30	31	32	33	34	35
	y i 0.29 0.31 0.29 0.20 0.15 0.18 0.11 0.07 0.06 0.01 0.01

Table 3 :

 3 Summary of properties of some discrete kernel estimators (Senga[START_REF] Kiessé | Approche non-paramétrique par noyaux associés discrets des données de dénombrement[END_REF] Type of discrete E(K x,h ) var(K x,h ) lim

	h→0	var(K x,h ) Convergence	Cross-	Excess Symmetry Remarks

Table 4 :

 4 Simulated optimal average integrated squared errors and their standard errors (in parentheses) for discrete and Epanechnikov kernel estimators. The results multiplied by 10 3 are given 20 11.77 (8.2) 24.51 (19.9) 43.87 (49.6) 55.06 (107.2) 65.98 (51.2) 82.73 (122.5) 50 7.01 (8.4) 17.00 (14.1) 27.03 (26.5) 32.02 (31.6) 69.10 (36.1) 87.01 (43.6) 80 4.47 (4.5) 14.13 (8.7) 18.18 (13.6) 23.30 (19.8) 69.42 (27.6) 88.45 (42.9) 100 3.67 (2.9) 12.68 (7.9) 17.03 (11.4) 21.93 (16.5) 70.51 (25.7) 92.12 (37.3)

	n	Triang.1	Triang.2	Binomial	Epanech.	Poisson	Neg. bin.

200 2.43 (1.4) 10.20 (4.0) 12.60 (5.6) 14.81 (15.0) 76.48 (21.2) 102.17 (23.4) 500 1.86 (0.7) 8.46 (2.0) 9.92 (2.5) 9.96 (10.0) 83.45 (17.3) 116.13 (22.4) 1000 1.60 (0.4) 8.09 (1.4) 9.19 (1.5) 8.27 (7.9) 89.69 (13.0) 126.78 (20.1)

Table 5 :

 5 R-squared (in %) of nonparametric regressions of average daily fat (Table2) by some discrete kernels and the continuous Epanechnikov kernel Kernel Triang.2 (a) Triang.2 (b) Binomial Epanech. Poisson Neg. bin.Figure 4: Nonparametric regressions of sales data (Table1) by some discrete kernels and the continuous Epanechnikov kernel

	h cv	0.1	0.5 *	0.101	4.0	0.151	0.224
	R 2	99.140	97.321	97.179	96.967	67.057	49.435

Table 6 :

 6 R-squared (in %) of nonparametric regressions of sales data (Table1) by some discrete kernels and the continuous Epanechnikov kernel

	Data	Kernel Triang.1 Triang.2 Epanech. Binomial Poisson Neg. bin.
	Complete	h cv	0.558	0.132	2.427	0.064	0.206	0.327
		R 2	95.155	92.502	82.877	80.086 56.065	40.094
	Incomplete	h cv	0.170	0.052	2.121	0.078	0.209	0.327
	(without * )	R 2	94.075	92.922	93.915	89.151 63.159	45.230

Acknowledgment

This work was partially supported by grants from FAPESP and CNPq.


Belkacem Abdous
email: belkacem.abdous@msp.ulaval.ca

Célestin C Kokonendji
email: celestin.kokonendji@univ-fcomte.fr

Tristan Senga Kiessé
email: tristan.senga-kiesse@onf.fr

On semiparametric regression for count explanatory variables

Keywords: Asymptotic bias and variance, Discrete associated kernel, Discrete finite difference, Multiplicative correction factor, Nonparametric regression, Nadaraya-Watson estimator, Logistic regression. 2000 MSC: 62G08, 62J02, 62G99

This work deals with a semiparametric estimation of a count regression function m that can be represented as a product of an unknown discrete parametric function r and an unknown discrete "smooth" function ω. We propose an estimation procedure in two steps: first, we construct an approximation r of r, then we use a discrete associated kernel method to estimate non-parametrically the multiplicative correction factor ω = m/ r. The asymptotic and small-sample properties of the proposed estimator are investigated. Its comparison with the classical Nadaraya-Watson type count regression estimator shows that an improvement in terms of bias is achieved.

Introduction

Let (X 1 , Y 1 ), (X 2 , Y 2 ), . . . , (X n , Y n ) be a sequence of i.i.d. random variables defined on N d × R and such that

Y i = m(X i ) + i , (1)
where m is an unknown regression function and the i 's are assumed to have zero mean and finite variance. Parametric and nonparametric estimation of the regression function m are classical problems. The continuous case has received much more attention than the discrete one. Semiparametric approaches have been considered by several authors as well. For a recent account of the (multiplicative or additive) combinations of parametric and nonparametric kernel estimators, see, e.g., [START_REF] Martins-Filho | A class of improved parametrically guided nonparametric regression estimators[END_REF], [START_REF] Su | Nonparametric prewhitening estimators for conditional quantiles[END_REF], [START_REF] Fan | Local quasi-likelihood with a parametric guide[END_REF]. While, nonparametric estimation of discrete functions such as probability mass or count regression functions is definitely less popular. An extension of the well known kernel technique has been investigated by [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF]Kokonendji et al. ( , 2009b)).

In this paper we focus on the semiparametric estimation of the unknown count regression function m given by (1). More precisely, we investigate the following form:

m(x) = r(x; β)ω(x)

(2) = : m ω (x; β), for x ∈ N d , where r(•; β) is a parametric function that depends on unknown parameter β = (β 1 , . . . , β p ) T , and ω(•) is a nonparametric multiplicative correction function. As shown later on, by splitting the estimation problem of the conditional mean function m(x) = E(Y |X = x) up into a parametric procedure for r and a kernel-based technique for ω, we improve the asymptotic bias of the proposed estimator.

The rest of this paper is organized as follows. Section 2 briefly recalls the nonparametric count regression estimator of m and investigates its asymptotic bias and variance expansions. In Section 3, we present our semiparametric estimation of (2) for d = 1. The proposed estimator is constructed in two steps: first, we use a generalized least squared criterion to find an estimate β of the parameter β in r(x; β); then a discrete associated kernel approach [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF](Kokonendji et al. , 2009a) ) is used to provide a nonparametric estimator of the multiplicative correction factor ω(x; β) = m(x)/r(x; β). The asymptotic bias and variance of the obtained estimator are investigated and compared to those of a nonparametric discrete associated kernel estimator. The extension to higher dimension is tackled as well. Section 4 provides two applications on real data, while proofs of all results of Section 3 are postponed to Section 5.

Nonparametric count regression estimator

In this section we summarize some asymptotic expansions of the nonparametric count regression estimator introduced by Kokonendji et al. (2009b). Indeed, suppose that we are interested in a count regression function m(x) = E(Y |X = x) in which the r.v. Y and X belong to R and N respectively; without loss of generality, we here present the univariate case with d = 1 (see Section 3.3). Then, the well-known Nadaraya-Watson kernel estimate might be extended to the case of count regression as follows

m n (x) = n i=1 Y i K x,h (X i ) n j=1 K x,h (X j ) , x ∈ N, (3)
where h = h(n) > 0 is an arbitrary sequence of smoothing parameters that fulfills lim n→∞ h(n) = 0, while K x,h (•) is a suitably chosen discrete associated kernel function (e.g. [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF]. The kernel function K x,h (•) is itself a probability mass function (p.m.f.) with support S x that does not depend on h and such that x ∈ S x . Furthermore, we also impose the following two conditions:

lim h→0 E(K x,h ) = x, (4)
lim h→0 V ar(K x,h ) = 0, (5)
where K x,h is the discrete random variable whose p.m.f. is K x,h (•). See also [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] for a review of discrete associated kernels and asymptotic behaviour for discrete smoothing using (4) and (5).

To deal with derivatives of any count function g : N → R, we will use finite differences instead of the usual differentiation on R. For k ∈ N \ {0}, we will put

g (k) (x) = {g (k-1) (x)} (1) and g (1) (x) =      {g(x + 1) -g(x -1)}/2 if x ∈ N \ {0} g(1) -g(0) if x = 0.
(6) For instance, it follows that the finite difference of second order is given by

g (2) (x) =              {g(x + 2) -2g(x) + g(x -2)}/4 if x ∈ N \ {0, 1} {g(3) -3g(1) + 2g(0)}/4 if x = 1 {g(2) -2g(1) + g(0)}/2 if x = 0. ( 7
)
Note that the finite differences g (k) are always defined no matter the smoothness of g.

The following theorem presents the asymptotic bias and variance of the nonparametric estimator m n (x) in (3).

Theorem 1 Let f be a p.m.f. of a discrete r.v. X defined on N. Assume that f (x) = Pr(X = x) > 0 for a given x ∈ N. Furthermore, suppose that the bandwidth h = h(n) > 0 satisfies lim n→∞ h = 0 and that the discrete kernel K x,h (•) fulfills assumptions (4)-( 5). Then, the bias and variance of m n (x) admit the following expansions

Bias{ m n (x)} = m (2) (x) + 2m (1) (x) f (1) f (x) V ar(K x,h ) 2 +O 1 n +o(h), (8)
V ar{ m n (x)} = V ar(Y |X = x) nf (x) {Pr(K x,h = x)} 2 + o 1 n , (9)
where f (1) , m (1) and m (2) are finite differences as defined in ( 6) and ( 7).

The proof is given in Section 5.

Semiparametric count regression estimator

Following Kokonendji et al. (2009a) for p.m.f., we here adapt the semiparametric approach of count regression function m in (1) using expression in (2) with a parametric start r(•; β) and a nonparametric correction function ω(•).

In fact, given a sample (X i , Y i ), i = 1, . . . , n, it uses any parametric estimation procedure to provide an estimation of β involved in r(x; β), and it serves the nonparametric discrete associated kernel procedure to estimate the correction function ω(x) = m(x)/r(x; β) by the following Nadaraya-Watson type estimator

ω n (x; β) = n i=1 Y i K x,h (X i ) r(X i ; β) n j=1 K x,h (X j ) , x ∈ N, (10)
where β = ( β 1 , . . . , β p ) T is an estimate of β constructed in the previous step (by generalized least squared method for example). Upon combining these two estimation steps, we end up with the following semiparametric estimator m n of m

m n (x) = r(x; β) ω n (x; β) = r(x; β) n i=1 Y i K x,h (X i ) r(X i ; β) n j=1 K x,h (X j ) . (11)
Next, let us examine the asymptotic bias and variance of the proposed estimator (11) and compare them to those of the traditional estimator (3). These asymptotics will be tackled under two settings: known and unknown parametric functionr r(x; β). Besides, we will investigate the multivariate case as well.

Known parametric start

Suppose that based on a goodness-of-fit test or any a priori knowledge about m we decide to fix the parametric start in (2) and put r 0 (x) = r(x; β 0 ). Then, by letting m(x) = r 0 (x)ω(x) and by using the estimation procedure described above, it is easy to see that a semiparametric estimate of m is given by

m n (x) = r 0 (x) ω n (x) = n i=1 Y i K x,h (X i ) n j=1 K x,h (X j ) × r 0 (x) r 0 (X i ) , x ∈ N. ( 12
)
The following theorem exhibits the asymptotic expansions of m n (x)'s bias and variance

Theorem 2 Let x be a given point in N satisfying f (x) = Pr(X = x) > 0.

Assume that the regression function satisfies m(x) = r 0 (x)ω(x) with r 0 (x) = r(x; β 0 ) being a fixed start. Then, provided that

h = h(n) → 0 for n → ∞, the estimator m n (x) verifies Bias{ m n (x)} = r 0 (x)ω (2) (x) + 2r 0 (x)ω (1) (x) f (1) f (x) V ar(K x,h ) 2 +O 1 n + o(h), (13)
V ar m n (x) = V ar(Y |X = x) nf (x) {Pr(K x,h = x)} 2 + o 1 n , (14)
where f (1) , ω (1) and ω (2) are finite differences as defined in ( 6) and (7).

Proof. See Section 5.

A comparison of the asymptotic expansions of the semiparametric estimator m n in (12) and the nonparametric discrete Nadaraya-Watson type estimator m n in (3) shows that the leading terms in the variance expressions are identical, while the bias expressions are different. More precisely, since under the assumption m(x) = r 0 (x)ω(x) one has

m (1) = (r 0 ω) (1) = r 0 ω (1) + r (1) 0 ω m (2) = (r 0 ω) (2) = r 0 ω (2) + 2r
(1) 0 ω (1) + r

(2) 0 ω, it follows that the difference between the leading terms in Bias{ m n (x)} and Bias{ m n (x)} is given by:

r (2) 0 (x)ω(x) + 2r (1) 0 (x)ω (1) (x) + 2r (1) 0 (x)ω(x) f (1) f (x) V ar(K x,h ) 2 .
Unfortunately, the sign of this quantity can be either positive or negative.

Unknown parametric start

Next, let us consider the semiparametric count regression estimator m n in (11) of m in ( 2) with an estimate β of β (still constructed by generalized least squared method for example). In the situation where the parametric function r(x; β) is misspecified, the estimator β of β converges in probability to a certain value β 0 such that r(x; β 0 ) is the best approximant to m(x) with respect to the Kullback-Leibler distance

x∈N m(x) log m(x) r(x; β) =: d {m(•), r(•; β)} of r(x; β)
from the true function m(x); see, for example, [START_REF] White | Maximum likelihood estimation of misspecified models[END_REF].

A Taylor expansion of the ratio r(x; β)/r(X i ; β) around β 0 provides

r(x; β) r(X i ; β) = exp{log r(x; β) -log r(X i ; β)} . = r 0 (x) r 0 (X i ) 1 -{u 0 (X i ) -u 0 (x)} T ( β -β 0 ) + 1 2 ( β -β 0 ) T M (x, X i )( β -β 0 ) ,
where .

= denotes an asymptotic equivalent, and the matrix M (x,

X i ) is M (x, X i ) = v 0 (x) -v 0 (X i ) + {u 0 (x) -u 0 (X i )} {u 0 (X i ) -u 0 (x)} T
such that u 0 (x) and v 0 (x) are the gradient and the Hessian matrix with respect to β, respectively, evaluated in β 0 . Using the above expansion, the estimator m n in (11) can be approximated as

m n (x) . = m 0 n (x) + A n (x) + 1 2 B n (x), (15)
where m 0 n is the estimator in ( 12) with fixed start and the two others terms are

A n (x) = n i=1 [{u 0 (X i ) -u 0 (x)} T ( β -β 0 )]Y i K x,h (X i ) n j=1 K x,h (X j ) × r(x; β 0 ) r(X i ; β 0 ) , B n (x) = n i=1 {( β -β 0 ) T M (x, X i )( β -β 0 )}Y i K x,h (X i ) n j=1 K x,h (X j ) × r(x; β 0 ) r(X i ; β 0 ) .
We then formulate the following result.

Theorem 3 Let x be a given point in N satisfying f (x) = Pr(X = x) > 0.

Assume that the parametric function r 0 (x) = r(x; β 0 ) is the best approximant of the regression function m(x) under the Kullback-Leibler criterion, with ω(x) = m(x)/r 0 (x) being a multiplicative correction factor. Then, provided that n → ∞ and h = h(n) → 0, the estimator m n (x) in ( 11) satisfies

Bias{ m n (x)} = r 0 (x)ω (2) (x) + 2r 0 (x)ω (1) (x) f (1) f (x) V ar(K x,h ) 2 +O 1 n + o(h), V ar m n (x) = V ar(Y |X = x) nf (x) {Pr(K x,h = x)} 2 + o 1 n ,
where f (1) , ω (1) and ω (2) are finite differences as defined in ( 6) and ( 7).

Proof. See Section 5.

Looking at the previous theorem, the asymptotic bias and variance are the same as in Theorem 2 for the case of the fixed start r 0 . Hence, the proposed estimator m n in (11) of m can once again be shown to be better or not than the nonparametric one in (3). See, for example, Kokonendji et al. (2009a) for some illustrations in the case of count data distributions.

Multivariate case

The multidimensional estimators (3) and ( 11) are easily obtained by using a multiplicative d-univariate kernel

K x,h (X i ) = d k=1 K [k] x k ,h k (X ik ), with x = (x 1 , . . . , x d ) T , h = (h 1 , . . . , h d ) T , X i = (X i1 , . . . , X id ) T and K [k]
x k ,h k (•) is a univariate discrete associated kernel satisfying (4) and ( 5) for all k = 1, . . . , d. We now consider the d-dimensional discrete functions m(•), r(•; β) and ω(•) in (2) with their corresponding partial finite differences in the sense of ( 6) and (7), for example, g

(1)

k (x) = ∂g(x)/∂x k and g (2) kl (x) = ∂ 2 g(x)/(∂x k ∂x l )
and so on.

The asymptotic bias and variance of the d-dimensional discrete Nadaraya-Watson estimator replacing ( 8) and ( 9), respectively, are given by

Bias{ m n (x)} = 1 2 d k=1    m (2) kk (x) + 2m (1) k (x)   f (1) k f   (x)    V ar(K [k] x k ,h k ) +O 1 n + o h 2 1 + h 2 2 + . . . + h 2 d h 1 h 2 . . . h d , V ar{ m n (x)} = V ar(Y |X = x) nf (x) d k=1 {Pr(K [k] x k ,h k = x k )} 2 + o 1 n .
The corresponding expression for the bias of the new estimator given in Theorem 3 is

Bias{ m n (x)} = 1 2 d k=1    r 0 (x)ω (2) kk (x) + 2r 0 (x)ω (1) k (x)   f (1) k f   (x)    V ar(K [k] x k ,h k ) +O 1 n + o h 2 1 + h 2 2 + . . . + h 2 d h 1 h 2 . . . h d ,
while the variance is the same as that of the multidimensional discrete Nadaraya-Watson estimator above. Similarly to that for univariate case, the change of bias is clearly point out.

Applications

In this section two practical datasets of demography and economy are used for applying the nonparametric and semiparametric discrete regression approaches in comparison with classical parametric regression models. The first example concerns mortality rates [START_REF] Copas | Non-parametric graduation using kernel methods[END_REF]. For each age x ∈ N, a population of size s x is exposed to risk, associated to a number of deaths d x . The crude death rates m 0 (x) is calculated as

m 0 (x) = d x s x , x ∈ N.
The frequency mortality rate is generally increasing with respect to the age, having a small numbers of deaths at younger ages. The graduated values obtained by applying a logistic model does not fit well to data (see Figure 1). Indeed, the parametric model produces oversmoothing and, in particular, does not succeed to describe the behaviour on the ages x ∈ {42, 44, 45, 46}. The second example is a sales dataset with multiple y i at a given x i (Kokonendji et al., 2009b). We analyse the amount of daily sales of a new product during the 24 first days. The 151 observations (x i , y i ), i = 1, . . . , 24, represent the day x i and the corresponding mean of sales numbers y i ∈ {y Ai , y Bi , . . . , y Hi }. The number of sale centres for each state (A, B, . . ., H) is not available except for the state H where this number is equal to one. We apply generalized linear models (GLM) in comparison with nonparametric and semiparametric models for fitting the sales data (Figure 2). The given examples indicate that the use of continuous semiparametric model may provide fitted values at any point of the real line numbers, while the predictor is an ordinal variable. This motivates to recommend a discrete model that focuses on ordinal covariates and has the same nature. Hence, the nonparametric correction in the two examples is available only for discrete predictors.

For the discrete semiparametric estimator, the logistic model and GLM are used as start functions. The measure of error used is the root mean squared error RMSE defined as

RM SE = n j=1 (y j -y j ) 2 n ,
where y j is the adjustement of the j-th observation y j and n is the number of observations.

Parametric models

The GLM represents a normal model for the response variable Y i with a logarithmic link. It has a linear predictor based on a combination of explanatory variables such as

y i = θ 1 + θ 2 x i + θ 2 log x i + e i , x i ∈ N.
The nonlinear model corresponds to a logistic one for the situation of population growth towards a limited value. It is given by

y i = θ L 1 1 + exp - x i -θ L 2 θ L 3 + e i , x i ∈ N.
The fixed effect parameter θ L 1 is the asymptote towards which the population grows. The parameter θ L 2 is the midpoint and corresponds to the time at which y i = θ L 1 /2. The parameter θ L 3 is the scale and represents the distance on the time axis between the midpoint and the point where the response is θ L 1 /(1 + e -1 ).

Discrete associated kernel

For (x, a) ∈ N × N and h > 0, the symmetric discrete triangular associated kernel K a;x,h on support S a;x = {x, x ± 1, . . . , x ± a} has discrete probability distribution given by

K a;x,h (z) = (a + 1) h -|z -x| h P (a, h) , ∀z ∈ S a;x ,
where P (a, h) = (2a + 1)(a + 1) h -2 a k=0 k h . That kernel fulfills assumptions (4) -( 5). Note that a general discrete triangular associated kernel has been proposed by [START_REF] Kokonendji | Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions[END_REF]; the corresponding R package is also available (see Senga [START_REF] Senga Kiessé | The R package for general discrete triangular distributions[END_REF].

Concerning the regression estimators with discrete triangular associated kernels, the optimal bandwidth parameter is selected by the cross-validation method; refer to Kokonendji et al. (2009) for discrete case and, [START_REF] Hardle | Optimal bandwidth selection in nonparametric regression function estimation[END_REF] for continuous one. For a given associated discrete kernel, the optimal bandwidth is h cv = arg min h>0 CV (h) with

CV (h) = 1 n n i=1 {Y i -m -i (X i ; h)} 2 , where m -i (X i ; h) = n j =i Y j K X i ,h (X j )/ n j =i K X i ,h (X j )
is the leave-one-out kernel estimator of m n (X i ; h). However, this method does not always converge and, thus, other values of the bandwidth parameter h ∈ {0.5, 3.5} are proposed to point out its influence on goodness-of-fit or discrete smoothness. Here, this procedure is applied only for discrete nonparametric estimator but some works are in progress on its development for discrete semiparametric estimator. For the arm a ∈ N, a value equals to 1 or 2 is sufficient because the error connected to the adjustment is increasing with respect to the arm a [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF]; hence, we consider the discrete triangular associated kernel a = 2.

Mortality rates

The Figure 1 and Table 1 present the results connected to death rates data. For this dataset, the cross-validation procedure does not converge, consequently we use the two values of h proposed previously. First, using semiparametric estimator with discrete triangular kernel a = 2, h = 0.5 provides the best adjustement which corresponds to the smallest RMSE; the performance of nonparametric and semiparametric approaches are closed and outperform the parametric model. Then, by changing the bandwidth value in h = 3.5 improves the amount of smoothing provided by the two discrete associated kernel estimators. Finally, the semiparametric estimator stays the best one in comparison to the nonparametric and the difference between the performance of the two estimators is more clear.

Sales data

The Figure 2 and Table 2 present the results corresponding to sales data.

For this example, the cross-validation method gives an optimal bandwidth value h cv = 0.132 for discrete nonparametric estimator; in comparison, we Fig. 2. GLM, nonparametric and semiparametric regressions using symmetric discrete triangular associated kernels on sales data 5 Proofs

Proof of Theorem 1

Let us rewrite the estimator m n (x) in (3), as follows [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF]Theorem 2.4) and, similarly, the convergence of N n (x; h) to mf (x) can de deduced. From here, following [START_REF] Bosq | Théorie de l'Estimation Fonctionnelle[END_REF]p. 119-121), we can write

m n (x) = N n (x; h) D n (x; h) , with D n (x; h) = n -1 n j=1 K x,h (X j ) and N n (x; h) = n -1 n i=1 Y i K x,h (X i ). The convergence of D n (x; h) to f (x) is established by
m n (x) = m(x) + 1 f (x) {N n (x; h) -(mf )(x)} - (mf )(x) f 2 (x) {D n (x; h) -f (x)} - 1 f 2 (x) {N n (x; h) -(mf )(x)}{D n (x; h) -f (x)} + N n (x; h) f 3 (x) {D n (x; h) -f (x)} 2 {1 + o(1)} a.s. ( 16
)
The expectation of D n (x; h) can be approximated as

E{D n (x; h)} = E{f (K x,h )} = f {E(K x,h )} + 1 2 V ar(K x,h )f (2) (x) + o(h),
where f (2) is the finite difference of second order as in (7). Similarly, for the expectation of N n (x; h), we have

E{N n (x; h)} = E{Y 1 K x,h (X 1 )} = z∈Sx m(z)f (z) Pr(K x,h = z) = E{(mf )(K x,h )} = (mf ){E(K x,h )} + 1 2 V ar(K x,h )(mf ) (2) (x) + o(h),
Thus, under the condition (4) of discrete associated kernel, we obtain

E{D n (x; h)} -f (x) = 1 2 V ar(K x,h )f (2) (x) + o(h) (17)
and

E{N n (x; h)} -(mf )(x) = 1 2 V ar(K x,h )(mf ) (2) (x) + o(h). (18)
Next, we have

E[N n (x; h){D n (x; h) -f (x)} 2 ] = E [N n (x; h) -E{N n (x; h)}][D n (x; h) -E{D n (x; h)}] 2 + 2[E{D n (x; h)} -f (x)]Cov{D n (x; h), N n (x; h)} + E[{D n (x; h) -f (x)} 2 ]E{N n (x; h)} = O(1/n) 2 + O(1/n) + E{D n (x; h) -f (x)} 2 E{N n (x; h)}. (19)
To get the last inequality, we used the fact that

Cov{D n (x; h), N n (x; h)} = 1 n [E{Y 1 K 2 x,h (X 1 )} -E{Y 1 K x,h (X 1 )}E{K x,h (X 2 )}] = O(1/n). and E [N n (x; h) -E{N n (x; h)}][D n (x; h) -E{D n (x; h)}] 2 = O(1/n) 2 .
Similar arguments enable to see that

E{D n (x; h) -f (x)} 2 E{N n (x; h)} -f (x)E[{N n (x; h) -(mf )(x)}{D n (x; h) -f (x)}] = O(1/n) + o(h). (20)
Upon plugging the expansions ( 17)-( 20) into ( 16), we end up with

E{ m n (x)} -m(x) = (mf ) (2) (x) f (x) - mf (2) (x) f (x) V ar(K x,h ) 2 + O(1/n) + o(h).
Finally, to obtain (8), simply note that (mf

) (2) = m (2) f + 2m (1) f (1) + mf (2) .
As for the variance expansion (9), it follows from ( 16) that

V ar{ m n (x)} = V ar{N n (x; h)} f 2 (x) + (mf ) 2 (x) f 4 (x) V ar{D n (x; h)} -2 (mf )(x) f 3 (x) Cov{N n (x; h), D n (x; h)} + o 1 n . (21)
The variance of D n (x; h) might be written as follows

V ar {D n (x; h)} = 1 n y∈Sx f (y) {Pr(K x,h = y)} 2 - 1 n    y∈Sx f (y) Pr(K x,h = y)    2 = 1 n f (x){Pr(K x,h = x)} 2 - 1 n f 2 (x) + R n (x; h), with R n (x; h) = 1 n y∈Sx\{x} f (y) {Pr(K x,h = y)} 2 + 1 n f 2 (x) - 1 n   f (x) + y∈Sx {f (y) -f (x)} Pr(K x,h = y)   2 . ( 22
)
This quantity becomes negligible under the hypothesis (4) and ( 5) of discrete associated kernel; i.e. for any x ∈ N, R n (x; h) → 0 when n → ∞ and h = h(n) → 0. Indeed, let y ∈ S x \ {x} we can find a constant η = η(y) > 0 such that

Pr (K x,h = y) ≤ Pr(|K x,h -x| > η) ≤ 1 η 2 E{(K x,h -x) 2 } = 1 η 2 [V ar(K x,h ) + {E(K x,h ) -x} 2 ] → 0 as h → 0,
and for y = x we deduce the asymptotic modal probability Pr(K x,h = x) → 1 when h → 0.

Similarly, the variance of N n (x; h) can be expressed

V ar{N n (x; h)} = 1 n y∈Sx E(Y 2 1 |X 1 = y)f (y){Pr(K x,h = y)} 2 - 1 n    z∈Sx E(Y 1 |X 1 = z)f (z) Pr(K x,h = z)    2 = 1 n E(Y 2 1 |X 1 = x)f (x){Pr(K x,h = x)} 2 + 1 n y∈Sx\{x} E(Y 2 1 |X 1 = y)f (y){Pr(K x,h = y)} 2 - 1 n   f (x)E(Y 1 |X 1 = x) + y∈Sx {f (y) -f (x)}E(Y 1 |X 1 = y) Pr(K x,h = y)   2 = 1 n f (x){Pr(K x,h = x)} 2 {V ar(Y |X = x) + m 2 (x)} - 1 n (mf ) 2 (x) + Q n (x; h),
where

Q n (x; h) = 1 n y∈Sx\{x} E(Y 2 |X = x)f (y) {Pr(K x,h = y)} 2 + 1 n (mf ) 2 (x) - 1 n   (mf )(x) + y∈Sx {(mf )(y) -(mf )(x)} Pr(K x,h = y)   2 .
The same arguments as those used for R n (x; h) in ( 22) might be used to show that Q n (x; h) goes to 0 as h → 0. Finally, the relationship in (21) together with the above asymptotic expansions and the fact that Cov{N n (x; h), D n (x; h)} = O(1/n) lead to the desired result (9).

Proof of Theorem 2

The proof follows the same lines as those of Theorem 1. In fact, we write m n (x) = H n (x; h)/F n (x; h) to establish an expression similar to ( 16) with

H n (x; h) = n -1 n i=1 {r 0 (x)/r 0 (X i )}Y i K x,h (X i ) and F n (x; h) = n -1 n j=1 K x,h (X j ) = D n (x; h).
In order to express the expectation of m n (x), we first calculate

E{H n (x; h)} = r 0 (x)E[{r 0 (X 1 )} -1 Y 1 K x,h (X 1 )] = r 0 (x) z∈Sx {r 0 (z)} -1 f (z)m(z) Pr(K x,h = z) = r 0 (x)E{(ωf )(K x,h )} = r 0 (x) (ωf ){E(K x,h )} + 1 2 V ar(K x,h )(ωf ) (2) (x) + o(h) .
Thus, under condition (4) of discrete associated kernel, we get

E{H n (x; h)} = (mf )(x) + 1 2 V ar(K x,h )r 0 (x)(ωf ) (2) (x) + o(h).
In addition, we also have

Cov{F n (x; h), H n (x; h)} = r 0 (x) n E {r 0 (X 1 )} -1 Y 1 K 2 x,h (X 1 ) - r 0 (x) n E {r 0 (X 1 )} -1 Y 1 K x,h (X 1 ) E {K x,h (X 2 )} = O(1/n).
Hence, it suffices to use the same arguments as in the proof of Theorem 1 to obtain the bias of m n (x).

As for the variance of m n (x), the proof being quite similar to that of m n (x) in Theorem 1, details are therefore omitted.

Proof of Theorem 3

In order to establish the results of Theorem 3, the difference β -β 0 will be expressed as an average of i.i.d. variables with mean zero plus remainder term as in (23). Indeed, let P be the generating joint distribution of (X, Y ) and P n its corresponding empirical distribution. We consider functional estimators of β of the form β = T (P n ) for which β = T (P ) realizes the best approx-imant r(x; β 0 ) to m(x) with respect to some distance measure, for instance the Kullback-Leibler distance. We then introduce the influence function of dimension p (of β)

I(X, Y ) = lim →0 [T {(1 -)P + δ (X,Y ) } -T (P )]/ ,
which has zero mean and finite covariance matrix, and where δ (X,Y ) denotes the unit point mass in (X, Y ). Hence, one has

β -β 0 = 1 n n i=1 I(X i , Y i ) + b n + n , (23)
where b/n can be considered as the bias of the estimator β and the last term

n is such that E( n ) = O(n -2 ).
In addition, we note

U i = r 0 (x) r 0 (X i ) Y i K x,h (X i ), V i = r 0 (x) r 0 (X i ) Y i K x,h (X i ){u 0 (x) -u 0 (X i )} T , W i = r 0 (x) r 0 (X i ) Y i K x,h (X i )M (x, X i ), Z i = K x,h (X i ),
and we then write U * = (1/n) n i=1 U i and the same for V * , W * and Z * . Thus, it comes that m

0 n (x) = U * /Z * , A n (x) = ( β -β 0 )V * /Z * and B n (x) = ( β -β 0 ) T W * ( β -β 0 )/Z * .
Looking first to the expectation of the estimator m n such that

E{ m 0 n (x)} . = E{ m 0 n (x)} + E{A n (x)} + 1 2 E{B n (x)}.
Let us recall that the expectation E{ m 0 n (x)} can be deduced from (13) of Theorem 2 as E{ m 0 n (x)} = bias{ m n (x)} + r 0 (x)ω(x).

For A n (x), we reformulate it as shown in Eq. ( 16). Thus, to calculate E{A n (x)}, we need E{( β -β 0 )V * }, E(Z * ) and Cov{( β -β 0 )V * , Z * }. Note that the expectation of Z * = D n (x; h) is already found in the proof of Theorem 1. From (23) and E{I(X i Y i )} = 0 for all i, we get

E{( β -β 0 )V * } = n -1 E{V 1 I(X 1 , Y 1 )} + n -1 bE(V 1 ) + O(n -2 ) = o(hn -1 ) + O(n -2 ).
Indeed, using both the discrete Taylor expansion and the condition (4) of the discrete associated kernel, we have successively

E{V 1 I(X 1 Y 1 )} = E r 0 (x) r 0 (X 1 ) Y 1 K x,h (X 1 )I(X 1 , Y 1 ){u 0 (x) -u 0 (X 1 )} T = z r 0 (x) r 0 (z) K x,h (z){u 0 (x) -u 0 (z)} T f (z) y yI(z, y)g(y|z) = r 0 (x) z Pr(K x,h = z){u 0 (x) -u 0 (z)} T f (z)q(z) r 0 (z) = r 0 (x) [u 0 (x) -u 0 {E(K x,h )}] T s{E(K x,h )} + 1 2 r 0 (x)V ar(K x,h ) [u 0 (x) -u 0 {E(K x,h )}] T s{E(K x,h )} (2) + o(h) = - 1 2 r 0 (x)V ar(K x,h ) {u (2) 0 } T (x)s(x) + 2{u (1) 0 } T (x)s (1) (x) + o(h) = a(x) + o(h), where a(x) = (-1/2)r 0 (x)V ar(K x,h ) {u (2) 0 } T (x)s(x) + 2{u (1) 0 } T (x)s (1) (x) , q(x) = E{I(X, Y )Y |X = x}, s(x) = f (x)q(x
)/r 0 (x). Hence, under condition (5) on the variance of K x,h , we have E{V 1 I(X 1 Y 1 )} and E(V 1 ) which are of order o(h). Thus, in the reformulation of A n (x) as seen in ( 16), n -1 a(x) takes the place of mf such that

E{( β -β 0 )V * } -n -1 a(x) = o(hn -1 ) + O(n -2 ).
Then, we express

Cov{( β -β 0 )V * , Z * } = Cov(V * I * , Z * ) + n -1 Cov(bV * , Z * ) + Cov( n V * , Z * ) = o(hn -1 ) + O(n -2 ),
where Cov(V * I * , Z * ) = n -1 E(Z 1 )E(V 1 I 1 ) + O(n -2 ), n -1 Cov(bV * , Z * ) = n -1 bE(Z 1 )E(V 1 ) + O(n -2 ) and E(V 1 ) = o(h). In addition, from the decomposition of A n (x) as in ( 16), we have E[( β -β 0 )V * {Z * -f (x)} 2 ] = O(n -2 + n -1 ) + o(h). Finally, we obtain E(A n ) = o(hn -1 + h) + O(n -2 + n -1 ).

In the same manner, we calculate the expectation E(B n ) = o(hn -1 + h) + O(n -2 + n -1 ). We omit here to present all the calculus and we give only the mains results. Firstly, we obtain E{( β -β 0 )W * ( β -β 0 ) T } = n -1 T r{E(W 1 )E(I 1 I T 1 )} + O(n -2 ) = o(hn -1 ) + O(n -2 ) and E(W 1 ) = o(h). Then, we get Cov{( β -β 0 )W * ( β -β 0 ) T , Z * } = O(n -2 ).

Lastly, we find

E[{( β -β 0 )W * ( β -β 0 ) T }{Z * -f (x)} 2 ] = O(n -2 + n -1 ) + o(h)
which appears in the decomposition of B n (x) as in ( 16). The previous term allow to calculate E(B n ).

Finally, the expectation of the estimator m n results in E{ m n (x)} = E{ m 0 n (x)}+ O(n -2 + n -1 ) + o(hn -1 + h) which provides the desired result asymptotically. The expression of V ar{ m 0 n (x)} is arleady found in ( 14) of Theorem 2. For the variance of A n , by establishing an expression of V ar{A n (x)} similarly to (21), it follows that we need to calculate V ar{( β -β 0 )V * } = E{( β -β 0 )(W * ) 2 ( β -β 0 ) T } + O(n -2 ) = n -1 E(W 1 )E(I 1 I T 1 )E(B 1 ) T + O(n -2 ) = O(n -2 ) + o(h 2 n -1 ).

Then, by taking into account the others terms resulting from the presentation of the variance of A n (x) as in (21), it ensues that V ar{A n (x)} = O(n -2 + n -1 ) + o(h 2 n -1 + n -1 ). Furthermore, it can be proved

Cov{ m 0 n (x), A n (x)} = E U * ( β -β 0 )V * (V * ) 2 -E{ m 0 n (x)}E{A n (x)} = o(hn -1 ) + O(n -2 )
because in the expression of E{ m 0 n A n } the more influential terms are

E U * ( β -β 0 )V * = E(U * I * V * ) + n -1 bE(U 1 )E(V 1 ) + O(n -2 ) = o hn -1 + O(n -2 )
and Cov{U * ( β-β 0 )V * , (Z * ) 2 } which can be also shown to be of order o(hn -1 )+ O(n -2 ).

For B n , without give the details here, it can proved that V ar{B n (x)} = o(h 2 n -2 )+O(n -2 ), Cov{ m 0 n (x), B n (x)} = o(hn -2 )+O(n -2 ) and also Cov{A n (x), B n (x)} = o(h 2 n -1 ) + O(n -2 ).

Finally, we obtain V ar{ m n (x)} = V ar{ m 0 n (x)} + O(n -2 ) + o(hn -1 + hn -2 + h 2 n -2 + n -1 ) and it ensues the desired asymptotic result.

Fig. 1 .

 1 Fig. 1. Logistic regression, nonparametric and semiparametric regressions using symmetric discrete triangular associated kernels on death data

From ( 15 2

 152 ), we express the variance of m n (x) asV ar{ m n (x)} . = V ar{ m 0 n (x)} + V ar{A n (x)} + 1 4 V ar{B n (x)} +Cov{ m 0 n (x), A n (x)} + Cov{ m 0 n (x), B n (x)} + 1 Cov{A n (x), B n (x)}.

Acknowledgments

We sincerely thank the anonymous referee and Associate Editor for their valuable and constructive comments.