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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 862398, 19 pages
http://dx.doi.org/10.1155/2012/862398
Research Article

Hypothesis Testing in Generalized Linear Models with Functional Coefficient Autoregressive Processes

1School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
2Department of Mathematics, Huizhou University, Huizhou 516007, China

Received 28 January 2012; Accepted 25 March 2012

Academic Editor: Ming Li

Copyright © 2012 Lei Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper studies the hypothesis testing in generalized linear models with functional coefficient autoregressive (FCA) processes. The quasi-maximum likelihood (QML) estimators are given, which extend those estimators of Hu (2010) and Maller (2003). Asymptotic chi-squares distributions of pseudo likelihood ratio (LR) statistics are investigated.

1. Introduction

Consider the following generalized linear model: where is -dimensional unknown parameter, are functional coefficient autoregressive processes given by where are independent and identically distributed random variable errors with zero mean and finite variance , is a one-dimensional unknown parameter, and is a real valued function defined on a compact set which contains the true value as an inner point and is a subset of . The values of and are unknown. is a known continuous differentiable function.

Model (1.1) includes many special cases, such as an ordinary regression model (when; see [17]), an ordinary generalized regression model (when ; see [813]), a linear regression model with constant coefficient autoregressive processes (when , ; see [1416]), time-dependent and function coefficient autoregressive processes (when ; see [17]), constant coefficient autoregressive processes (when , ; see [1820]), time-dependent or time-varying autoregressive processes (when ; see [2123]), and a linear regression model with functional coefficient autoregressive processes (when; see [24]). Many authors have discussed some special cases of models (1.1) and (1.2) (see [124]). However, few people investigate the model (1.1) with (1.2). This paper studies the model (1.1) with (1.2). The organization of this paper is as follows. In Section 2, some estimators are given by the quasi-maximum likelihood method. In Section 3, the main results are investigated. The proofs of the main results are presented in Section 4, with the conclusions and some open problems in Section 5.

2. The Quasi-Maximum Likelihood Estimate

Write the “true” model as where . Define, and by (2.2), we have Thus is measurable with respect to the field generated by, and

Assume at first that the are i.i.d. , we get the log-likelihood of conditional on given by At this stage we drop the normality assumption, but still maximize (2.5) to obtain QML estimators, denoted by. The estimating equations for unknown parameters in (2.5) may be written as Thus, satisfy the following estimation equations where

Remark 2.1. If , then the above equations become the same as Hu’s (see [24]). If ,, then the above equations become the same as Maller’s (see [15]). Thus we extend those QML estimators of Hu [24] and Maller [15].
For ease of exposition, we will introduce the following notations, which will be used later in the paper. Let vector . Define By (2.7), we have where the * indicates that the elements are filled in by symmetry, Because and are mutually independent, we have where By (2.8) (2.7) and, we have

3. Statement of Main Results

In the section pseudo likelihood ratio (LR) statistics for various hypothesis tests of interest are derived. We consider the following hypothesis: When the parameter space is restricted by a hypothesis , letbe the corresponding QML estimators of , and let be minus twice the log-likelihood, evaluated at the fitted parameters. Also let be the “deviance” statistic for testing against. From (2.5) and (2.8), and similarly

In order to obtain our results, we give some sufficient conditions as follows.(A1) is positive definite for sufficiently large and where and denotes the maximum in absolute value of the eigenvalues of a symmetric matrix.(A2) There is a constant such that(A3)andexist and are bounded, andis twice continuously differentiable, , .

Theorem 3.1. Assume (2.1), (2.2) and (A1)–(A3).(1)Suppose and is a continuous function, holds. Then (2) Suppose , holds. Then (3) Suppose , holds. Then

4. Proof of Theorem

To prove Theorem 3.1, we first introduce the following lemmas.

Lemma 4.1. Suppose that (A1)–(A3) hold. Then, for all , where

Proof. Similar to proof of Lemma 4.1 in Hu [24], here we omit.

Lemma 4.2. Suppose that (A1)–(A3) hold. Then , and where are on the line ofand.

Proof. Similar to proof of Theorem 3.1 in Hu [24], we easily prove that, and . Since (4.4) is easily proved, here we omit the proof (4.4).

Proof of Theorem 3.1. Note that and are nonsingular. By Taylor’s expansion, we have where for some . Since , also . By (4.1), we have Thus is a symmetric matrix with. By (4.5) and (4.6), we have Letdenoteand, respectively. By (4.7), we have Note that By (2.15), (4.2) and (4.8), we get Note that By (2.1), (2.11) and (4.12), we have By (4.13) and (2.10), we have By (4.13), we have By (4.15), we have By (4.14) and (4.16), we have By (4.15), we have Thus, by (4.17) and (4.18), we have Since , we have Thus, by (4.17), (4.20) and mean value theorem, we have where for some .
It is easy to know that By Lemma 4.2 and (4.22), we have Hence, by (4.11), we have By (4.24), we have By Lemma 4.2, we have Now, we prove (3.8). By (4.12), we have Note that From (4.28), we have By (2.8) and (2.10), we have From (4.30), we obtain that By (4.29), (4.31) and Lemma 4.2, we have By (3.3)–(3.5), we have Under the , and by (4.26), (4.32) and (4.33), we have It is easily proven that Thus, by (4.33)–(4.35), we finish the proof of (3.8).
Next we prove (3.9). Under, , and , we have Hence By (2.8), (2.10), we have From (4.38), we obtain, Thus, by (4.37), (4.39) and Lemma 4.2, we have By (3.3)–(3.5), we have Under the, by (4.26), (4.40), and (4.41), we obtain Thus, by (4.35), (4.42), (3.9) holds.
Finally, we prove (3.10). Under, we have Thus By (2.8) and (2.10), we have From (4.45), we obtain By (4.44), (4.46) and Lemma 4.2, we have By (3.3)–(3.5), we know that Under the , by (4.26), (4.47) and (4.48), we have Thus, (3.10) follows from (4.48), (4.49), and (4.35). Therefore, we complete the proof of Theorem 3.1.

5. Conclusions and Open Problems

In the paper, we consider the generalized linear mode with FCA processes, which includes many special cases, such as an ordinary regression model, an ordinary generalized regression model, a linear regression model with constant coefficient autoregressive processes, time-dependent and function coefficient autoregressive processes, constant coefficient autoregressive processes, time-dependent or time-varying autoregressive processes, and a linear regression model with functional coefficient autoregressive processes. And then we obtain the QML estimators for some unknown parameters in the generalized linear mode model and extend some estimators. At last, we use pseudo LR method to investigate three hypothesis tests of interest and obtain the asymptotic chi-squares distributions of statistics.

However, several lines of future work remain open.

(1) It is well known that a conventional time series can be regarded as the solution to a differential equation of integer order with the excitation of white noise in mathematics, and a fractal time series can be regarded as the solution to a differential equation of fractional order with a white noise in the domain of stochastic processes (see [25]). In the paper, is a conventional nonlinear time series. We may investigate some hypothesis tests by pseudo LR method when theis a fractal time series (the idea is given by an anonymous reviewer). In particular, we assume that where is strictly decreasing sequence of nonnegative numbers, is a constant sequence, and is the Riemann-Liouville integral operator of order given by where is the Gamma function, and is a piecewise continuous on and integrable on any finite subinterval of (See [25, 26]). Fractal time series may have a heavy-tailed probability distribution function and has been applied various fields of sciences and technologies (see [25, 2732]). Thus it is very significant to investigate various regression models with fractal time series errors, including regression model (1.1) with (5.1).

(2) We maybe investigate the others hypothesis tests, for example::; :; :; : and is a continuous function,;:, ;:, .

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments which have led to this much improved version of the paper. The paper was supported by Scientific Research Item of Department of Education, Hubei (no. D20112503), Scientific Research Item of Ministry of Education, China (no. 209078), and Natural Science Foundation of China (no. 11071022, 11101174).

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