Hypothesis Testing in Generalized Linear Models with Functional Coefficient Autoregressive Processes

Lei Song,Xiaosheng Cheng,Hongchang Hu

doi:10.1155/2012/862398

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.

Highlights

Consider the following generalized linear model: yt g xtT β εt, t 1, 2, . . . , n, 1.1 where β is d-dimensional unknown parameter, {εt, t 1, 2, . . . , n} are functional coefficient autoregressive processes given by ε1 η1, εt ft θ εt−1 ηt, t 2, 3, . . . , n, 1.2 where {ηt, t 1, 2, . . . , n} are independent and identically distributed random variable errors with zero mean and finite variance σ2, θ is a one-dimensional unknown parameter, and ft θ is a real valued function defined on a compact set Θ which contains the true value θ0 as Mathematical Problems in Engineering an inner point and is a subset of R1
Many authors have discussed some special cases of models 1.1 and 1.2 see 1–24
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

Summary

Introduction

Consider the following generalized linear model: yt g xtT β εt, t 1, 2, . . . , n, 1.1 where β is d-dimensional unknown parameter, {εt, t 1, 2, . . . , n} are functional coefficient autoregressive processes given by ε1 η1, εt ft θ εt−1 ηt, t 2, 3, . . . , n, 1.2 where {ηt, t 1, 2, . . . , n} are independent and identically distributed random variable errors with zero mean and finite variance σ2, θ is a one-dimensional unknown parameter, and ft θ is a real valued function defined on a compact set Θ which contains the true value θ0 as Mathematical Problems in Engineering an inner point and is a subset of R1. Consider the following generalized linear model: yt g xtT β εt, t 1, 2, . N, 1.1 where β is d-dimensional unknown parameter, {εt, t 1, 2, . N} are independent and identically distributed random variable errors with zero mean and finite variance σ2, θ is a one-dimensional unknown parameter, and ft θ is a real valued function defined on a compact set Θ which contains the true value θ0 as Mathematical Problems in Engineering an inner point and is a subset of R1. Many authors have discussed some special cases of models 1.1 and 1.2 see 1–24. This paper studies the model 1.1 with 1.2.

The Quasi-Maximum Likelihood Estimate

Statement of Main Results

Proof of Theorem

Conclusions and Open Problems