STRONG CONSISTENCY OF ESTIMATORS FOR HETEROSCEDASTIC PARTLY LINEAR REGRESSION MODEL UNDER DEPENDENT SAMPLES

In this paper we are concerned with the heteroscedastic regression model C œ B  1Ð> Ñ  / " Ÿ 3 Ÿ 8Ñ / 3 3 3 3 3 3 " 5 under correlated errors , where it is assumed that , the design points are known and nonrandom, 5 3 3 3 3 3 œ 0Ð? Ñ ÐB ß > ß ? Ñ and and are unknown functions. The interest lies in the slope parameter . 1 0 " Assuming the unobserved disturbance are negatively associated, we study the /3 issue of strong consistency for two different slope estimators: the least squares estimator and the weighted least squares estimator. Partial Linear Model, Negatively Associated Sample, Weighted


Introduction
Consider the following heteroscedastic partial linear regression model: where is an unknown parameter of interest, , are nonrandom design " 5 # % functions defined on a closed interval of the real line .M V Clearly, the model (1.1) reduces to the partial linear model when and are i.i.d. 5 5 oe / This was first introduced by Engle et al. [11], and further studied by Heckman [14], Speckman [25], Chen [7], Chen and Shiau [8,9], Hong and Cheng [15,16], Hamilton and Truong [13], and Mammen and Van de Geer [20] among others.Various estimators for " and were given by using different methods such as the kernel method, the penalized 1Ð † Ñ spline method, the piecewise constant smooth method, the smoothing splines and the trigonometric series approach.
For the more general case where we only assume that are i.i.d. in the model (1.1), / 3 asymptotic properties have also been studied by various authors.For instance, Chen, Ren, and Hu [10] and Gao, Chen and Zhao [12] established the strong consistency and the asymptotic normality, respectively, for the least squares estimator and weighted least squares estimator of , based on nonparametric estimates of and .

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0Ð † Ñ 1Ð † Ñ All the work discussed so far assumes that the errors are independent.The purpose of / 3 this paper is to study some asymptotic properties of the slope estimators when are / 3 correlated.In particular, we shall study the strong consistency of several commonly used estimators of the slope when the errors are negatively associated (NA), whose definition " / 3 will be given later in the paper.
The paper is organized as follows.In Section 2, we shall introduce the least squares estimators (LSE) and weight least squares estimators (WLSE) of the slope parameter .The " main results are given in Section 3, while their proofs will be provided in Section 4. From (C1) and (C2) , we obtain that, when is large enough, Remark 2.1:

Main Results
When the errors are i.i.d., Chen et al.
[10] showed the strong consistency for the estimators The main aim of this paper is to investigate whether Theorems 3.1 and 3.2 still hold when the random errors , are NA random variables./ " Ÿ 3 Ÿ 8 3 A finite family of random variables is said to be Definition: Ö\ ß " Ÿ " Ÿ 8× 3 negatively associated (NA) if for every pair of disjoint subsets and of , whenever and are coordinatewise increasing and such that the covariance exists.An 0 0 " # infinite family of random variables is NA if every finite subfamily is NA.The notion of negative association was first introduced by Alam and Saxena [1] and studied in detail by Joag-Dev and Proschan [17].Because of its wide applications in multivariate statistical analysis and systems reliability, the notion of NA have recently received considerable attention.For convergence results, we refer to Joag-Dev and Proschan [17] for fundamental properties, Matula [21] for the three series theorem, Shao and Su [24] for law of the iterated logarithm, Liang [18] for complete convergence, and Liang and Jing [19] for strong laws.Asymptotic properties of estimates related to NA samples have also been studied extensively.Cai and Roussas [3] studied uniformly strong consistency, convergence rates, and asymptotic distribution of Kaplan-Meier estimator of distributed function with random censored failure times, and Cai and Roussas [4] gave Berry-Esseen bounds for smooth estimates of a distribution function.Roussas [23] investigated consistency of the kernel estimate of a probability density function.Amini and Bozorgnia [2] dealt with the consistency and complete convergence of sample quantiles.
However, there have been few asymptotic results for the estimators of parametric and nonparametric component in partial linear model regression under negatively associated error's structure.In the next two theorems, we shall give some results similar to those in Theorems 3.1 and 3.2 under negatively associated error's structure.Furthermore, we wish to consider the uniformly strong consistency for the estimator of under NA random errors.1Ð † Ñ But first, we list a weaker assumption than (C3): (C3 max , Our main results are as follows: Theorem 3.

Proofs of the Main Results
In  Again, we only prove (3.8) as the proof of (3.9) is similar.It is easy to see Ð33Ñ  The proof of (3.10) is similar to that of (3.8) in Theorem 3.3 and hence omitted here.Ð33Ñ Ñ ÐB ß > ß ?Ñ points, are the response variables, are random errors,

8 3 3
Since independent random samples are a special case of NA random Remark 3.1: samples and (C3 ) is weaker than (C3), Theorems 3.3-3.4extend Theorems 3.1-3.2 to NA w sample setting and weaken the restriction for the weight function.Moreover, we eliminate the condition (C1) in Theorems 3.1-3.2.Ð33Ñ