Hypothesis Testing in Generalized Linear Models with Functional Coefficient Autoregressive Processes

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:𝑦_𝑡=𝑔𝑥^𝑇_𝑡𝛽+𝜀_𝑡,𝑡=1,2,…,𝑛, (1.1) where𝛽 is 𝑑-dimensional unknown parameter,{𝜀_𝑡,𝑡=1,2,…,𝑛} are functional coefficient autoregressive processes given by𝜀₁=𝜂₁,𝜀_𝑡=𝑓_𝑡(𝜃)𝜀_𝑡−1+𝜂_𝑡,𝑡=2,3,…,𝑛, (1.2) where {𝜂_𝑡,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 𝑓_𝑡(𝜃) is a real valued function defined on a compact set Θ which contains the true value 𝜃₀as an inner point and is a subset of 𝑅¹. The values of 𝜃₀and 𝜎² are unknown. 𝑔(⋅) is a known continuous differentiable function.

Model (1.1) includes many special cases, such as an ordinary regression model (when𝑓_𝑡(𝜃)≡0,𝑔(𝜏)=𝜏; see [1–7]), an ordinary generalized regression model (when 𝑓_𝑡(𝜃)≡0; see [8–13]), a linear regression model with constant coefficient autoregressive processes (when 𝑓_𝑡(𝜃)=𝜃, 𝑔(𝜏)=𝜏; see [14–16]), time-dependent and function coefficient autoregressive processes (when 𝑔(𝜏)=0; see [17]), constant coefficient autoregressive processes (when 𝑓_𝑡(𝜃)=𝜃, 𝑔(𝜏)=0; see [18–20]), time-dependent or time-varying autoregressive processes (when 𝑓_𝑡(𝜃)=𝑎_𝑡,𝑔(𝜏)=0; see [21–23]), and a linear regression model with functional coefficient autoregressive processes (when𝑔(𝜏)=𝜏; see [24]). Many authors have discussed some special cases of models (1.1) and (1.2) (see [1–24]). 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 as𝑦_𝑡=𝑔𝑥^𝑇_𝑡𝛽₀+𝑒_𝑡𝑒,𝑡=1,2,…,𝑛,₁=𝜂₁,𝑒_𝑡=𝑓_𝑡𝜃₀𝑒_𝑡−1+𝜂_𝑡,𝑡=2,3,…,𝑛, (2.1)(2.2) where 𝑔^′(𝜏)=(𝑑𝑔(𝜏)/𝑑𝜏)≠0,𝑓^′_𝑡(𝜃)=(𝑑𝑓_𝑡(𝜃)/𝑑𝜃)≠0. Define∏⁻¹_𝑖=0𝑓_𝑡−𝑖(𝜃₀)=1, and by (2.2), we have 𝑒_𝑡=^𝑡−1_𝑗=0^𝑗−1_𝑖=0𝑓_𝑡−𝑖𝜃₀𝜂_𝑡−𝑗. (2.3) Thus 𝑒_𝑡 is measurable with respect to the 𝜎−field 𝐻 generated by𝜂₁,𝜂₂,…,𝜂_𝑡, and𝐸𝑒_𝑡=0,Var𝑒_𝑡=𝜎²₀^𝑡−1_𝑗=0^𝑗−1_𝑖=0𝑓²_𝑡−𝑖𝜃₀. (2.4)

Assume at first that the 𝜂_𝑡 are i.i.d. 𝑁(0,𝜎²), we get the log-likelihood of 𝑦₂,…,𝑦_𝑛 conditional on 𝑦₁ given byΦ_𝑛=ln𝐿_𝑛()=−𝑛−1ln𝜎²2−∑^𝑛_𝑡=2𝜀_𝑡−𝑓_𝑡(𝜃)𝜀_𝑡−1²2𝜎²−(𝑛−1)ln2𝜋2. (2.5) At this stage we drop the normality assumption, but still maximize (2.5) to obtain QML estimators, denoted by𝜎²_𝑛,𝛽_𝑛,𝜃_𝑛. The estimating equations for unknown parameters in (2.5) may be written as𝜕Φ_𝑛𝜕𝜎²=−𝑛−12𝜎²+12𝜎^4𝑛_𝑡=2𝜀_𝑡−𝑓_𝑡(𝜃)𝜀_𝑡−1²,𝜕Φ_𝑛=1𝜕𝜃𝜎^2𝑛_𝑡=2𝑓^′_𝑡(𝜃)𝜀_𝑡−𝑓_𝑡(𝜃)𝜀_𝑡−1𝜀_𝑡−1,𝜕Φ_𝑛𝜕𝛽_𝑑×1=1𝜎^2𝑛_𝑡=2𝜀_𝑡−𝑓_𝑡(𝜃)𝜀_𝑡−1⋅𝑔′𝑥^𝑇_𝑡𝛽𝑥_𝑡−𝑓_𝑡(𝜃)𝑔′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1. (2.6)(2.7) Thus, 𝜎²_𝑛,𝛽_𝑛,𝜃_𝑛 satisfy the following estimation equations 𝜎²_𝑛=1𝑛−1^𝑛𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²,_𝑛𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑓^′_𝑡𝜃_𝑛𝜀̂_𝑡−1=0,_𝑛𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽_𝑛𝑥_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽_𝑛𝑥_𝑡−1=0, (2.8)(2.9)(2.10) where𝜀_𝑡̂=𝑦_𝑡−𝑔𝑥^𝑇_𝑡𝛽_𝑛. (2.11)

Remark 2.1. If 𝑔(𝑥^𝑇_𝑡𝛽)=𝑥^𝑇_𝑡𝛽, then the above equations become the same as Hu’s (see [24]). If 𝑓_𝑡(𝜃)=𝜃,𝑔(𝑥^𝑇_𝑡𝛽)=𝑥^𝑇_𝑡𝛽, 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 (𝑑+1)×1−vector 𝜑=(𝛽^𝑇,𝜃)^𝑇. Define𝑆_𝑛𝜑=𝜎²𝜕Φ_𝑛𝜕𝜑=𝜎²𝜕Φ_𝑛,𝜕𝛽𝜕Φ_𝑛𝜕𝜃,𝐹_𝑛𝜑=−𝜎²𝜕²Φ_𝑛𝜕𝜑𝜕𝜑^𝑇. (2.12) By (2.7), we have 𝐹_𝑛𝜑=𝑋_𝑛𝑈∗𝜑,𝜔_𝑛𝑡=2𝑓^′_𝑡²(𝜃)+𝑓_𝑡(𝜃)𝑓_𝑡^′′(𝜃)𝜀²_𝑡−1−𝑓_𝑡^′′(𝜃)𝜀_𝑡𝜀_𝑡−1, (2.13) where the * indicates that the elements are filled in by symmetry, 𝑋_𝑛𝜑,𝜔=−𝜎²𝜕²Φ_𝑛𝜕𝛽𝜕𝛽^𝑇,𝑈=^𝑛_𝑡=2𝑓^′_𝑡(𝜃)𝜀_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽𝑥_𝑡+𝑓^′_𝑡(𝜃)𝜀_𝑡𝑔′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1−2𝑓_𝑡(𝜃)𝑓^′_𝑡(𝜃)𝜀_𝑡−1𝑔′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1,𝜕²Φ_𝑛𝜕𝛽𝜕𝛽^𝑇1=−𝜎^2𝑛_𝑡=2𝑔^′𝑥^𝑇_𝑡𝛽𝑥_𝑡−𝑓_𝑡(𝜃)𝑔^′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1𝑔^′𝑥^𝑇_𝑡𝛽𝑥_𝑡−𝑓_𝑡(𝜃)𝑔′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1^𝑇+1𝜎^2𝑛_𝑡=2𝜀_𝑡−𝑓_𝑡(𝜃)𝜀_𝑡−1𝑔^′′𝑥^𝑇_𝑡𝛽𝑥_𝑡𝑥^𝑇_𝑡−𝑓_𝑡(𝜃)𝑔^′′𝑥^𝑇_𝑡−1𝛽𝑥_𝑡−1𝑥^𝑇_𝑡−1. (2.14) Because {𝑒_𝑡−1} and {𝜂_𝑡} are mutually independent, we have 𝐷_𝑛=𝐸𝐹_𝑛𝜑₀=𝑋_𝑛𝜑₀00^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝐸𝑒²_𝑡−1=𝑋_𝑛𝜑₀00Δ𝜃₀,𝜎₀, (2.15) where 𝑋_𝑛𝜑₀=^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1^𝑇,Δ𝜃₀,𝜎₀=^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝐸𝑒²_𝑡−1=𝜎²₀^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀^𝑡−2_𝑗=0^𝑗−1_𝑖=0𝑓²_𝑡−𝑖(𝜃)=𝑂(𝑛). (2.16) By (2.8) (2.7) and𝐸𝜂_𝑡=0, we have 𝜎²₀𝐸𝜕Φ_𝑛𝜕𝛽_𝛽=𝛽0=∑^𝑛_𝑡=2𝐸𝜂_𝑡𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1𝜎=0,₂₀𝐸𝜕Φ_𝑛𝜕𝜃_𝜃=𝜃0=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝐸𝜂_𝑡𝑒_𝑡−1=0. (2.17)

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:𝐻₁∶𝑔(⋅),𝑓(⋅)arecontinuousfunctions,and𝑓^′(⋅)≠0,𝜎²₀>0. (3.1) When the parameter space is restricted by a hypothesis 𝐻_0𝑗, 𝑗=1,2,…,let𝛽_𝑗𝑛,𝜃_𝑗𝑛,𝜎²_𝑗𝑛be the corresponding QML estimators of 𝛽,𝜃,𝜎², and let𝐿_𝑗𝑛=−2Φ_𝑛𝛽_𝑗𝑛,𝜃_𝑗𝑛,𝜎²_𝑗𝑛 (3.2) be minus twice the log-likelihood, evaluated at the fitted parameters. Also let 𝐿_𝑛=−2Φ_𝑛𝛽_𝑛,𝜃_𝑛,𝜎²_𝑛,𝑑_𝑗𝑛=𝐿_𝑗𝑛−𝐿_𝑛 (3.3) be the “deviance” statistic for testing 𝐻_0𝑗against𝐻₁. From (2.5) and (2.8),𝐿_𝑛=(𝑛−1)ln𝜎²_𝑛+(𝑛−1)(1+ln2𝜋) (3.4) and similarly𝐿_𝑗𝑛=(𝑛−1)ln𝜎²_𝑗𝑛+(𝑛−1)(1+ln2𝜋). (3.5)

In order to obtain our results, we give some sufficient conditions as follows.(A1)𝑋_𝑛=∑^𝑛_𝑡=2𝑥_𝑡𝑥^𝑇_𝑡 is positive definite for sufficiently large 𝑛 andlim_𝑛→∞max_{1≤𝑡≤𝑛}𝑥^𝑇_𝑡𝑋_𝑛⁻¹𝑥_𝑡=𝑂𝑛^−𝛼1,∀𝛼∈2,1,lim_𝑛→∞sup|𝜆|_max𝑋_𝑛^−1/2𝑍_𝑛𝑋_𝑛^−𝑇/2<1, (3.6) whereZ_𝑛=(1/2)∑^𝑛_𝑡=2(𝑥_𝑡𝑥^𝑇_𝑡−1+𝑥_𝑡−1𝑥^𝑇_𝑡) and |𝜆|_max(⋅)denotes the maximum in absolute value of the eigenvalues of a symmetric matrix.(A2) There is a constant 𝛼>0such that^𝑡_𝑗=1^𝑗−1_𝑖=0𝑓²_𝑡−𝑖(𝜃)≤𝛼,max_{1≤𝑗≤𝑛}^𝑛_𝑡=𝑗+1^{𝑡−𝑗−1}_𝑖=0𝑓_𝑡−𝑖𝜃₀≤𝛾. (3.7)(A3)𝑓^′_𝑡(𝜃)=𝑑𝑓_𝑡(𝜃)/𝑑𝜃≠0and𝑓_𝑡^′′(𝜃)=𝑑𝑓^′_𝑡(𝜃)/𝑑𝜃exist and are bounded, and𝑔(⋅)is twice continuously differentiable, 0<𝑚≤max_𝑢|𝑔′(𝑢)|≤𝑀<∞, 0<𝑚≤max_𝑢|𝑔^′′(𝑢)|≤𝑀<∞.

Theorem 3.1. Assume (2.1), (2.2) and (A1)–(A3).(1)Suppose 𝐻₀₁∶𝑓_𝑡(𝜃)=𝜃 and 𝑔(𝑢) is a continuous function,𝜎²₀>0 holds. Then 𝑑^𝐷_1𝑛−→𝜒²₁,𝑛⟶∞. (3.8)(2) Suppose 𝐻₀₂∶𝑓_𝑡(𝜃)=𝜃,𝑔(𝑢)=𝑢,𝜎²₀>0 holds. Then 𝑑^𝐷_2𝑛−→𝜒²₁,𝑛⟶∞. (3.9)(3) Suppose 𝐻₀₃∶𝑓_𝑡(𝜃)=𝜃,𝑔(𝑢)=𝑒^𝑢/(1+𝑒^𝑢),𝜎²₀>0 holds. Then 𝑑^𝐷_3𝑛−→𝜒²₁,𝑛⟶∞. (3.10)

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 𝐴>0, sup_{𝜑∈𝑁𝑛(𝐴)}𝐷_𝑛^−1/2𝐹_𝑛𝜑𝐷_𝑛^−𝑇/2−Φ−→_𝑛^𝑃0,𝑛→∞, (4.1) where Φ_𝑛=diagI_𝑑,∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1Δ_𝑛𝜃₀,𝜎₀,𝑁_𝑛(𝐴)=𝜑∈𝑅^𝑑+1∶𝜑−𝜑₀^𝑇𝐷_𝑛𝜑−𝜑₀≤𝐴². (4.2)(4.3)

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

Lemma 4.2. Suppose that (A1)–(A3) hold. Then 𝜑_𝑛→𝜑₀, 𝜎²_𝑛→𝜎²₀ and X_𝑛𝛽^∗,𝛽^∗∗,𝜃_𝑛⟶X_𝑛𝜑₀, (4.4) where 𝛽^∗,𝛽^∗∗ are on the line of𝛽₀and𝛽_𝑛.

Proof. Similar to proof of Theorem 3.1 in Hu [24], we easily prove that𝜑_𝑛→𝜑₀, and 𝜎²_𝑛→𝜎²₀. Since (4.4) is easily proved, here we omit the proof (4.4).

Proof of Theorem 3.1. Note that 𝑆_𝑛(𝜑_𝑛)=0and𝐹_𝑛(𝜑_𝑛) are nonsingular. By Taylor’s expansion, we have 0=𝑆_𝑛𝜑_𝑛=𝑆_𝑛𝜑₀−𝐹_𝑛𝜑_𝑛𝜑_𝑛−𝜑₀, (4.5) where 𝜑_𝑛=𝑎𝜑_𝑛+(1−𝑎)𝜑₀ for some 0≤𝑎≤1. Since 𝜑_𝑛∈𝑁_𝑛(𝐴), also 𝜑_𝑛∈𝑁_𝑛(𝐴). By (4.1), we have 𝐹_𝑛𝜑_𝑛=𝐷_𝑛^1/2Φ_𝑛+𝐴_𝑛𝐷_𝑛^𝑇/2. (4.6) Thus 𝐴_𝑛 is a symmetric matrix with𝐴−→_𝑛^𝑃0. By (4.5) and (4.6), we have 𝐷_𝑛^𝑇/2𝜑_𝑛−𝜑₀=𝐷_𝑛^𝑇/2𝐹_𝑛⁻¹𝜑_𝑛𝑆_𝑛𝜑₀=Φ_𝑛+𝐴_𝑛⁻¹𝐷_𝑛^−1/2𝑆_𝑛𝜑₀. (4.7) Let𝑆_𝑛(𝜑),𝐹_𝑛(𝜑)denote𝑆_𝑛^(𝛽)(𝜑),𝑆_𝑛^(𝜃)(𝜑),and𝐹_𝑛^(𝛽)(𝜑),𝐹_𝑛^(𝜃)(𝜑), respectively. By (4.7), we have Φ_𝑛𝐷_𝑛^𝑇/2𝛽_𝑛−𝛽₀,𝜃_𝑛−𝜃₀=𝐷_𝑛^−1/2𝑆_𝑛^(𝛽)𝜑₀,𝑆_𝑛^(𝜃)𝜑₀+𝑜_𝑃(1). (4.8) Note that Φ_𝑛𝐷_𝑛^𝑇/2=𝑋_𝑛^𝑇/2𝜑₀00∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1Δ_𝑛𝜃₀,𝜎₀,𝐷_𝑛^−1/2=𝑋_𝑛^−1/2𝜑₀001Δ_𝑛𝜃₀,𝜎₀. (4.9) By (2.15), (4.2) and (4.8), we get 𝑋_𝑛^𝑇/2𝜑₀𝛽_𝑛−𝛽₀=𝑋_𝑛^−1/2𝜑₀𝑆_𝑛^(𝛽)𝜑₀+𝑜_𝑃(1)=𝑋_𝑛^−1/2𝜑₀^𝑛_𝑡=2𝜂_𝑡𝑔^′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔^′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1+𝑜_𝑃(1),_𝑛𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1𝜃_𝑛−𝜃₀=𝑆_𝑛^(𝜃)𝜑₀+𝑜_𝑃Δ_𝑛𝜃₀,𝜎₀=^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1+𝑜_𝑃Δ_𝑛𝜃₀,𝜎₀. (4.10)(4.11) Note that 𝜀_𝑡=𝑦_𝑡−𝑔𝑥^𝑇_𝑡𝛽=𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡𝛽₀−𝛽+𝑒_𝑡 (4.12) By (2.1), (2.11) and (4.12), we have 𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1=𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛+𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1. (4.13) By (4.13) and (2.10), we have 𝜀_𝑡̂^𝑛_𝑡=2−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²=^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1×𝛽₀−𝛽_𝑛+𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1=^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛+^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1=^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1. (4.14) By (4.13), we have 𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛=𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1−𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1. (4.15) By (4.15), we have 𝑔′𝑥^𝑛_𝑡=2^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛²=^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²+^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²−2^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1=^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²−^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1². (4.16) By (4.14) and (4.16), we have 𝜀_𝑡̂^𝑛_𝑡=2−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²=^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²−^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛². (4.17) By (4.15), we have 𝜀_𝑡̂^𝑛_𝑡=2−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²=^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²+^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛²+2^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛. (4.18) Thus, by (4.17) and (4.18), we have 𝑔′𝑥^𝑛_𝑡=2^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛²+^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛=0. (4.19) Since 𝜂_𝑡=𝑒_𝑡−𝑓_𝑡(𝜃₀)𝑒_𝑡−1, we have ^𝑛_𝑡=2𝑒_𝑡−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²=^𝑛_𝑡=2𝜂_𝑡+𝑓_𝑡𝜃₀𝑒_𝑡−1−𝑓_𝑡𝜃_𝑛𝑒_𝑡−1²=^𝑛_𝑡=1𝜂²_𝑡+^𝑛_𝑡=2𝑓_𝑡𝜃₀−𝑓_𝑡𝜃_𝑛²𝑒²_𝑡−1+2𝑓_𝑡𝜃₀−𝑓_𝑡𝜃_𝑛𝜂_𝑡𝑒_𝑡−1. (4.20) Thus, by (4.17), (4.20) and mean value theorem, we have (𝑛−1)𝜎²_𝑛=^𝑛_𝑡=2𝜀_𝑡̂−𝑓_𝑡𝜃_𝑛𝜀̂_𝑡−1²=^𝑛_𝑡=1𝜂²_𝑡+^𝑛_𝑡=2𝑓_𝑡𝜃₀−𝑓_𝑡𝜃_𝑛²𝑒²_𝑡−1+2𝑓_𝑡𝜃₀−𝑓_𝑡𝜃_𝑛𝜂_𝑡𝑒_𝑡−1−^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛²=^𝑛_𝑡=1𝜂²_𝑡+𝜃₀−𝜃_𝑛^2𝑛_𝑡=2𝑓^′_𝑡²𝜃𝑒²_𝑡−1+2𝜃₀−𝜃_𝑛^𝑛_𝑡=2𝑓^′_𝑡𝜃𝑒_𝑡−1𝜂_𝑡−^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃_𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽_𝑛², (4.21) where 𝜃=𝑎𝜃₀+(1−𝑎)𝜃_𝑛 for some 0≤𝑎≤1.
It is easy to know that𝛽_𝑛−𝛽₀^𝑇𝑋_𝑛𝜑₀𝛽_𝑛−𝛽₀=^𝑛_𝑡=2𝜂_𝑡𝑋_𝑛^−1/2𝜑₀𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑝(1). (4.22) By Lemma 4.2 and (4.22), we have (𝑛−1)𝜎²_𝑛=^𝑛_𝑡=1𝜂²_𝑡+𝜃₀−𝜃_𝑛^2𝑛_𝑡=2𝑓^′_𝑡²𝜃𝑒²_𝑡−1+2𝜃₀−𝜃_𝑛^𝑛_𝑡=2𝑓^′_𝑡𝜃𝑒_𝑡−1𝜂_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋_𝑛^−1/2𝛽^∗,𝛽^∗∗,𝜃_𝑛𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑃(1)=^𝑛_𝑡=1𝜂²_𝑡+𝜃₀−𝜃_𝑛^2𝑛_𝑡=2𝑓^′_𝑡²𝜃𝑒²_𝑡−1+2𝜃₀−𝜃_𝑛^𝑛_𝑡=2𝑓^′_𝑡𝜃𝑒_𝑡−1𝜂_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋_𝑛^−1/2𝜑₀𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑃(1). (4.23) Hence, by (4.11), we have 𝜃_𝑛−𝜃₀=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃Δ_𝑛𝜃₀,𝜎₀∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1. (4.24) By (4.24), we have 𝜃₀−𝜃_𝑛^2𝑛_𝑡=2𝑓^′_𝑡²𝜃𝑒²_𝑡−1+2𝜃₀−𝜃_𝑛^𝑛_𝑡=2𝑓^′_𝑡𝜃𝑒_𝑡−1𝜂_𝑡=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1^2𝑛𝑡=2𝑓^′_𝑡²𝜃𝑒²_𝑡−1∑+2^𝑛𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1^𝑛𝑡=2𝑓^′_𝑡𝜃𝑒_𝑡−1𝜂_𝑡+𝑜_𝑃(1)=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1^2𝑛𝑡=2𝑓^′_𝑡𝜃₀+𝑜(1)²𝑒²_𝑡−1∑+2^𝑛𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃1∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1⋅^𝑛_𝑡=2𝑓_𝑡^′′𝜃₀+𝑜(1)𝑒_𝑡−1𝜂_𝑡+𝑜_𝑃(1)=∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1²∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1−2∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1²∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃(1)∑=−^𝑛𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1²∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1+𝑜_𝑃(1). (4.25) By Lemma 4.2, we have (𝑛−1)𝜎²_𝑛=^𝑛_𝑡=1𝜂²_𝑡−∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1²∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1−^𝑛_𝑡=2𝜂_𝑡𝑋_𝑛^−1/2𝛽^∗,𝛽^∗∗,𝜃_𝑛𝑔^′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔^′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑃(1)=^𝑛_𝑡=1𝜂²_𝑡−∑^𝑛_𝑡=2𝑓^′_𝑡𝜃₀𝜂_𝑡𝑒_𝑡−1²∑^𝑛_𝑡=2𝑓^′_𝑡²𝜃₀𝑒²_𝑡−1−^𝑛_𝑡=2𝜂_𝑡𝑋_𝑛^−1/2𝜑₀𝑔^′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝑓_𝑡𝜃₀𝑔^′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑃(1). (4.26) Now, we prove (3.8). By (4.12), we have 𝜀_𝑡̂(1)=𝑦_𝑡−𝑔𝑥^𝑇_𝑡𝛽_1𝑛=𝑔^′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥^𝑇_𝑡𝛽₀−𝛽_1𝑛+𝑒_𝑡. (4.27) Note that 𝜀_𝑡−𝑓_𝑡𝜃₀𝜀_𝑡−1=𝑔′𝑥^𝑇_𝑡𝛽^∗𝑥^𝑇_𝑡−𝑓_𝑡𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗𝑥^𝑇_𝑡−1𝛽₀−𝛽+𝜂_𝑡. (4.28) From (4.28), we have 𝜀_𝑡̂(1)−𝜃_1𝑛𝜀̂_𝑡−1(1)=𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥^𝑇_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥^𝑇_𝑡−1𝛽₀−𝛽_1𝑛+𝜂_𝑡. (4.29) By (2.8) and (2.10), we have 0=^𝑛_𝑡=2𝜀_𝑡̂(1)−𝜃_1𝑛𝜀̂_𝑡−1(1)𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1=^𝑛_𝑡=2𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥^𝑇_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥^𝑇_𝑡−1𝛽₀−𝛽_1𝑛𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1+^𝑛_𝑡=2𝜂_𝑡𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1=𝛽₀−𝛽_1𝑛^𝑇𝑋_1𝑛𝛽^∗_1𝑛,𝛽^∗∗_1𝑛,𝜃_1𝑛+^𝑛_𝑡=2𝜂_𝑡𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1. (4.30) From (4.30), we obtain that 𝛽_1𝑛−𝛽₀=𝑋⁻¹_1𝑛𝛽^∗_1𝑛,𝛽^∗∗_1𝑛,𝜃_1𝑛^𝑛_𝑡=2𝜂_𝑡𝑔′𝑥^𝑇_𝑡𝛽_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽_1𝑛𝑥_𝑡−1. (4.31) By (4.29), (4.31) and Lemma 4.2, we have (𝑛−1)𝜎²_1𝑛=^𝑛_𝑡=2𝜀_𝑡̂(1)−𝜃_1𝑛𝜀̂_𝑡−1(1)²=^𝑛_𝑡=1𝜂²_𝑡+𝛽₀−𝛽_1𝑛^𝑇𝑋_1𝑛𝛽^∗_1𝑛,𝛽^∗∗_1𝑛,𝜃_1𝑛𝛽₀−𝛽_1𝑛+2𝛽₀−𝛽_1𝑛^𝑇𝑛_𝑡=2𝜂_𝑡𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋^−1/2_1𝑛𝛽^∗_1𝑛,𝛽^∗∗_1𝑛,𝜃_1𝑛𝑔′𝑥^𝑇_𝑡𝛽^∗_1𝑛𝑥_𝑡−𝜃_1𝑛𝑔′𝑥^𝑇_𝑡−1𝛽^∗∗_1𝑛𝑥_𝑡−1²+𝑜_𝑝(1)=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋^−1/2_1𝑛𝜑₀𝑔′𝑥^𝑇_𝑡𝛽₀𝑥_𝑡−𝜃₀𝑔′𝑥^𝑇_𝑡−1𝛽₀𝑥_𝑡−1²+𝑜_𝑝. (4.32) By (3.3)–(3.5), we have 𝑑_1𝑛=𝐿_1𝑛−𝐿_𝑛=()𝜎𝑛−1ln^21𝑛𝜎²_𝑛=()𝜎𝑛−1^21𝑛𝜎²_𝑛−1+𝑜_𝑃(1). (4.33) Under the 𝐻₀₁, and by (4.26), (4.32) and (4.33), we have (𝑛−1)𝜎²_1𝑛−𝜎²_𝑛𝜎²_𝑛=∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1²𝜎²_𝑛∑^𝑛_𝑡=2𝑒²_𝑡−1+𝑜_𝑃(1)=∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1²𝜎²₀∑^𝑛_𝑡=2𝑒²_𝑡−1+𝑜_𝑃(1). (4.34) It is easily proven that ∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1𝜎₀∑^𝑛_𝑡=2𝑒²_𝑡−1⟶𝑁(0,1). (4.35) Thus, by (4.33)–(4.35), we finish the proof of (3.8).
Next we prove (3.9). UnderH₀₂∶𝑓_𝑡(𝜃)=𝜃, 𝑔(𝑢)=𝑢, and 𝑦_𝑡=𝑥^𝑇_𝑡𝛽₀+𝑒_𝑡, we have𝜀_𝑡̂(2)=𝑦_𝑡−𝑥^𝑇_𝑡𝛽_2𝑛=𝑥^𝑇_𝑡𝛽₀−𝑥^𝑇_𝑡𝛽_2𝑛+𝑒_𝑡=𝑥^𝑇_𝑡𝛽₀−𝛽_2𝑛+𝑒_𝑡. (4.36) Hence 𝜀_𝑡̂(2)−𝜃_2𝑛𝜀̂_𝑡−1(2)=𝑥^𝑇_𝑡𝛽₀−𝛽_2𝑛+𝑒_𝑡−𝜃_2𝑛𝑥^𝑇_𝑡−1𝛽₀−𝛽_2𝑛+𝑒_𝑡−1=𝑥^𝑇_𝑡−𝜃_2𝑛𝑥^𝑇_𝑡−1𝛽₀−𝛽_2𝑛+𝜂_𝑡. (4.37) By (2.8), (2.10), we have 0=^𝑛_𝑡=2𝜀_𝑡̂(2)−𝜃_2𝑛𝜀̂_𝑡−1(2)𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1=^𝑛_𝑡=2𝑥^𝑇_𝑡−𝜃_2𝑛𝑥^𝑇_𝑡−1𝛽₀−𝛽_2𝑛𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1+^𝑛_𝑡=2𝜂_𝑡𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1. (4.38) From (4.38), we obtain, 𝛽_2𝑛−𝛽₀=𝑋⁻¹_2𝑛𝜃_2𝑛^𝑛_𝑡=2𝜂_𝑡𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1. (4.39) Thus, by (4.37), (4.39) and Lemma 4.2, we have (𝑛−1)𝜎²_2𝑛=^𝑛_𝑡=2𝜀_𝑡̂(2)−𝜃_2𝑛𝜀̂_𝑡−1(2)²=^𝑛_𝑡=1𝜂²_𝑡+𝛽₀−𝛽_2𝑛^𝑇𝑋_2𝑛𝜃_2𝑛𝛽₀−𝛽_2𝑛+2𝛽₀−𝛽_2𝑛^𝑇𝑛_𝑡=2𝜂_𝑡𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1^𝑇𝑋⁻¹_2𝑛𝜃_2𝑛^𝑛_𝑡=2𝜂_𝑡𝑥_𝑡−𝜃_2𝑛𝑥_𝑡−1=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋^−1/2_2𝑛𝜃₀𝑥_𝑡−𝜃₀𝑥_𝑡−1²+𝑜_𝑝(1). (4.40) By (3.3)–(3.5), we have 𝑑_2𝑛=𝐿_2𝑛−𝐿_𝑛=()𝜎𝑛−1ln^22𝑛𝜎²_𝑛=()𝜎𝑛−1^22𝑛𝜎²_𝑛−1+𝑜_𝑃(1). (4.41) Under the𝐻₀₂, by (4.26), (4.40), and (4.41), we obtain (𝑛−1)𝜎²_2𝑛−𝜎²_𝑛𝜎²_𝑛=∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1²𝜎²_𝑛∑^𝑛_𝑡=2𝑒²_𝑡−1+𝑜_𝑃(1)=∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1²𝜎²₀∑^𝑛_𝑡=2𝑒²_𝑡−1+𝑜_𝑃(1). (4.42) Thus, by (4.35), (4.42), (3.9) holds.
Finally, we prove (3.10). Under𝐻₀₃, we have𝜀_𝑡̂(3)=𝑦_𝑡−𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛=𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥^𝑇_𝑡𝛽₀−𝛽_3𝑛+𝑒_𝑡. (4.43) Thus 𝜀_𝑡̂(3)−𝜃_3𝑛𝜀̂_𝑡−1(3)=𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥^𝑇_𝑡𝛽₀−𝛽_3𝑛+𝑒_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥^𝑇_𝑡−1𝛽₀−𝛽_3𝑛−𝜃_3𝑛𝑒_𝑡−1=𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥^𝑇_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥^𝑇_𝑡−1𝛽₀−𝛽_3𝑛+𝜂_𝑡. (4.44) By (2.8) and (2.10), we have 0=^𝑛_𝑡=2𝜀_𝑡̂(3)−𝜃_3𝑛𝜀̂_𝑡−1(3)𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1=^𝑛_𝑡=2𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥^𝑇_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥^𝑇_𝑡−1𝛽₀−𝛽_3𝑛×𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1+^𝑛_𝑡=2𝜂_𝑡𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1=𝛽₀−𝛽_3𝑛^𝑇𝑋_3𝑛𝛽^∗_3𝑛,𝛽^∗∗_3𝑛,𝜃_3𝑛+^𝑛_𝑡=2𝜂_𝑡𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1. (4.45) From (4.45), we obtain 𝛽_3𝑛−𝛽₀=𝑋⁻¹_3𝑛𝛽^∗_3𝑛,𝛽^∗∗_3𝑛,𝜃_3𝑛^𝑛_𝑡=2𝜂_𝑡𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1. (4.46) By (4.44), (4.46) and Lemma 4.2, we have (𝑛−1)𝜎²_3𝑛=^𝑛_𝑡=2𝜀_𝑡̂(3)−𝜃_3𝑛𝜀̂_𝑡−1(3)²=^𝑛_𝑡=1𝜂²_𝑡+𝛽₀−𝛽_3𝑛^𝑇𝑋_3𝑛𝛽^∗_3𝑛,𝛽^∗∗_3𝑛,𝜃_3𝑛𝛽₀−𝛽_3𝑛+2𝛽₀−𝛽_3𝑛^𝑇𝑛_𝑡=2𝜂_𝑡𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥_𝑡−1=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋^−1/2_3𝑛𝛽^∗_3𝑛,𝛽^∗∗_3𝑛,𝜃_3𝑛×𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛1+𝑒^𝑥𝑇𝑡𝛽^∗_3𝑛²𝑥^𝑇_𝑡−𝜃_3𝑛𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛1+𝑒^{𝑥𝑇𝑡−1}𝛽^∗∗_3𝑛²𝑥^𝑇_𝑡−1²=^𝑛_𝑡=1𝜂²_𝑡−^𝑛_𝑡=2𝜂_𝑡𝑋^−1/2_3𝑛𝜑₀𝑒^{𝑥𝑇𝑡𝛽0}1+𝑒^{𝑥𝑇𝑡𝛽02}𝑥_𝑡−𝜃₀𝑒^{𝑥𝑇𝑡−1𝛽0}1+𝑒^{𝑥𝑇𝑡−1𝛽02}𝑥_𝑡−1²+𝑜_𝑝(1). (4.47) By (3.3)–(3.5), we know that 𝑑_3𝑛=𝐿_3𝑛−𝐿_𝑛=()𝜎𝑛−1ln^23𝑛𝜎²_𝑛=()𝜎𝑛−1^23𝑛𝜎²_𝑛−1+𝑜_𝑃(1). (4.48) Under the 𝐻₀₃, by (4.26), (4.47) and (4.48), we have (𝑛−1)𝜎²_3𝑛−𝜎²_𝑛𝜎²_𝑛=∑^𝑛_𝑡=2𝜂_𝑡𝑒_𝑡−1²𝜎²₀∑^𝑛_𝑡=2𝑒²_𝑡−1+𝑜_𝑃(1). (4.49) 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, {𝜀_𝑡}is a conventional nonlinear time series. We may investigate some hypothesis tests by pseudo LR method when the{𝜀_𝑡}is a fractal time series (the idea is given by an anonymous reviewer). In particular, we assume that^𝑝_𝑖=0𝑎_𝑝−𝑖𝐷^𝑣𝑖𝜀_𝑡=𝜂_𝑡, (5.1) where 𝑣_𝑝,𝑣_𝑝−1,…,𝑣₀ is strictly decreasing sequence of nonnegative numbers, 𝑎_𝑖 is a constant sequence, and 𝐷^𝑣 is the Riemann-Liouville integral operator of order 𝑣>0 given by𝐷^𝑣ℎ(𝑡)=1Γ(𝑣)^𝑡₀(𝑡−𝑢)^𝑣−1ℎ(𝑢)𝑑𝑢, (5.2) where Γ is the Gamma function, and ℎ(t) is a piecewise continuous on(0,∞) and integrable on any finite subinterval of [0,∞)(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, 27–32]). 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:𝐻₀₄:𝑓_𝑡(𝜃)=0,𝑔(𝑢)=u,𝜎²₀>0; 𝐻₀₅:𝑓_𝑡(𝜃)=𝜃,𝑔(𝑢)=0,𝜎²₀>0; 𝐻₀₆:𝑓_𝑡(𝜃)=0,𝑔(𝑢)=𝑒^𝑢/(1+𝑒^𝑢),𝜎²₀>0; 𝐻₀₇:𝑓_𝑡(𝜃)=𝑎_𝑡 and 𝑔(𝑢) is a continuous function,𝜎²₀>0;𝐻₀₈:𝑓_𝑡(𝜃)=𝑎_𝑡,𝑔(𝑢)=𝑢, 𝜎²₀>0;𝐻₀₉:𝑓_𝑡(𝜃)=𝑎_𝑡,𝑔(𝑢)=𝑒^𝑢/(1+𝑒^𝑢), 𝜎²₀>0.

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).