hypothesis testing in generalized linear models with...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 862398, 19 pagesdoi:10.1155/2012/862398

Research ArticleHypothesis Testing in GeneralizedLinear Models with Functional CoefficientAutoregressive Processes

Lei Song,1 Hongchang Hu,1 and Xiaosheng Cheng2

1 School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China2 Department of Mathematics, Huizhou University, Huizhou 516007, China

Correspondence should be addressed to Hongchang Hu, [email protected]

Received 28 January 2012; Accepted 25 March 2012

Academic Editor: Ming Li

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

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

1. 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 coefficientautoregressive 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 errorswith zero mean and finite variance σ2, θ is a one-dimensional unknown parameter, andft(θ) is a real valued function defined on a compact set Θwhich contains the true value θ0 as

2 Mathematical Problems in Engineering

an inner point and is a subset of R1. The values of θ0 and σ2 are unknown. g(·) is a knowncontinuous differentiable function.

Model (1.1) includes many special cases, such as an ordinary regression model(when ft(θ) ≡ 0, g(τ) = τ ; see [1–7]), an ordinary generalized regression model (whenft(θ) ≡ 0; see [8–13]), a linear regression model with constant coefficient autoregressiveprocesses (when ft(θ) = θ, g(τ) = τ ; see [14–16]), time-dependent and function coefficientautoregressive processes (when g(τ) = 0; see [17]), constant coefficient autoregressive pro-cesses (when ft(θ) = θ, g(τ) = 0; see [18–20]), time-dependent or time-varying autoregres-sive processes (when ft(θ) = at, g(τ) = 0; see [21–23]), and a linear regression model withfunctional coefficient autoregressive processes (when g(τ) = τ ; see [24]). Many authors havediscussed some special cases of models (1.1) and (1.2) (see [1–24]). However, few peopleinvestigate the model (1.1) with (1.2). This paper studies the model (1.1) with (1.2). Theorganization 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 ofthe main results are presented in Section 4, with the conclusions and some open problems inSection 5.

2. The Quasi-Maximum Likelihood Estimate

Write the “true” model as

yt = g(xTt β0)+ et, t = 1, 2, . . . , n, (2.1)

e1 = η1, et = ft(θ0)et−1 + ηt, t = 2, 3, . . . , n, (2.2)

where g ′(τ) = (dg(τ)/dτ)/= 0, f ′t(θ) = (dft(θ)/dθ)/= 0. Define

∏−1i=0ft−i(θ0) = 1, and by (2.2),

we have

et =t−1∑j=0

(j−1∏i=0

ft−i(θ0)

)ηt−j . (2.3)

Thus et is measurable with respect to the σ−fieldH generated by η1, η2, . . . , ηt, and

Eet = 0, Var(et) = σ20

t−1∑j=0

(j−1∏i=0

f2t−i(θ0)

). (2.4)

Assume at first that the ηt are i.i.d. N(0, σ2), we get the log-likelihood of y2, . . . , yn

conditional on y1 given by

Φn = lnLn = − (n − 1) lnσ2

2−∑n

t=2(εt − ft(θ)εt−1

)22σ2

− (n − 1) ln 2π2

. (2.5)

Mathematical Problems in Engineering 3

At this stage we drop the normality assumption, but still maximize (2.5) to obtain QMLestimators, denoted by σ2

n, βn, θn. The estimating equations for unknown parameters in (2.5)may be written as

∂Φn

∂σ2= −n − 1

2σ2+

12σ4

n∑t=2

(εt − ft(θ)εt−1

)2, (2.6)

∂Φn

∂θ=

1σ2

n∑t=2

f ′t(θ)(εt − ft(θ)εt−1

)εt−1,

∂Φn

∂βd×1=

1σ2

n∑t=2


) ·(g ′(xTt β)xt − ft(θ)g ′

(xTt−1β)xt−1).

(2.7)

Thus, σ2n, βn, θn satisfy the following estimation equations

σ2n =

1n − 1

n∑t=2

(εt − ft

(θn)εt−1)2, (2.8)

n∑t=2

(εt − ft

(θn)εt−1)f ′t

(θn)εt−1 = 0, (2.9)

n∑t=2

(εt − ft

(θn)εt−1)(

g ′(xTt βn)xt − ft

(θn)g ′(xTt−1βn

)xt−1)= 0, (2.10)

where

εt = yt − g(xTt βn). (2.11)

Remark 2.1. If g(xTt β) = xT

t β, then the above equations become the same as Hu’s (see [24]).If ft(θ) = θ, g(xT

t β) = xTt β, 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 = (βT , θ)T . Define

Sn

(ϕ)= σ2 ∂Φn

∂ϕ= σ2

(∂Φn

∂β,∂Φn

∂θ

), Fn

(ϕ)= −σ2 ∂2Φn

∂ϕ∂ϕT. (2.12)

By (2.7), we have

Fn

(ϕ)=

⎛⎜⎜⎝

Xn

(ϕ,ω)

U

∗n∑t=2

((f ′t2(θ) + ft(θ)f ′′

t (θ))ε2t−1 − f ′′

t (θ)εtεt−1)

⎞⎟⎟⎠, (2.13)


where the ∗ indicates that the elements are filled in by symmetry,

Xn

(ϕ,ω)= −σ2

(∂2Φn

∂β∂βT

),

U =n∑t=2

(f ′t(θ)εt−1g

′(xTt β)xt + f ′

t(θ)εtg′(xTt−1β)xt−1 − 2ft(θ)f ′

t(θ)εt−1g′(xTt−1β)xt−1),

∂2Φn

∂β∂βT= − 1

σ2

n∑t=2

(g ′(xTt β)xt − ft(θ)g ′

(xTt−1β)xt−1)(

g ′(xTt β)xt − ft(θ)g ′

(xTt−1β)xt−1)T

+1σ2

n∑t=2


)(g ′′(xTt β)xtx

Tt − ft(θ)g ′′

(xTt−1β)xt−1xT

t−1).

(2.14)

Because {et−1} and {ηt} are mutually independent, we have

Dn = E(Fn

(ϕ0))

=

⎛⎜⎜⎝

Xn

(ϕ0)

0

0n∑t=2

f ′t2(θ0)Ee2t−1

⎞⎟⎟⎠ =

(Xn

(ϕ0)

0

0 Δ(θ0, σ0)

), (2.15)

where

Xn

(ϕ0)=

n∑t=2

(g ′(xTt β0)xt − ft(θ0)g ′

(xTt−1β0

)xt−1)(

g ′(xTt β0)xt − ft(θ0)g ′

(xTt−1β0

)xt−1)T

,

Δ(θ0, σ0) =n∑t=2

f ′t2(θ0)Ee2t−1 = σ2

0

n∑t=2

f ′t2(θ0)

t−2∑j=0

(j−1∏i=0

f2t−i(θ)

)= O(n).

(2.16)

By (2.8) (2.7) and Eηt = 0, we have

σ20E

(∂Φn

∂β

∣∣∣∣β=β0

)=

n∑t=2

Eηt(g ′(xT

t β0)xt − ft(θ0)g ′(xT

t−1β0)xt−1)= 0,

σ20E

(∂Φn

∂θ

∣∣∣∣θ=θ0

)=

n∑t=2

f ′t(θ0)E

(ηtet−1

)= 0.

(2.17)

3. Statement of Main Results

In the section pseudo likelihood ratio (LR) statistics for various hypothesis tests of interestare derived. We consider the following hypothesis:

H1 : g(·), f(·) are continuous functions, and f′(·)

/= 0, σ20 > 0. (3.1)


When the parameter space is restricted by a hypothesisH0j , j = 1, 2, . . . , let βjn, θjn, σ2jn be the

corresponding QML estimators of β, θ, σ2, and let

Ljn = −2Φn

(βjn, θjn, σ

2jn

)(3.2)

be minus twice the log-likelihood, evaluated at the fitted parameters. Also let

Ln = −2Φn

(βn, θn, σ

2n

),

djn = Ljn − Ln

(3.3)

be the “deviance” statistic for testing H0j against H1. From (2.5) and (2.8),

Ln = (n − 1) ln σ2n + (n − 1)(1 + ln 2π) (3.4)

and similarly

Ljn = (n − 1) ln σ2jn + (n − 1)(1 + ln 2π). (3.5)

In order to obtain our results, we give some sufficient conditions as follows.

(A1) Xn =∑n

t=2 xtxTt is positive definite for sufficiently large n and

limn→∞

max1≤t≤n

xTt X

−1n xt = O

(n−α), ∀α ∈

(12, 1], lim

n→∞sup |λ|max

(X−1/2

n ZnX−T/2n

)< 1, (3.6)

where Zn = (1/2)∑n

t=2(xtxTt−1 + xt−1xT

t ) and |λ|max(·) denotes the maximum in absolute valueof the eigenvalues of a symmetric matrix.

(A2) There is a constant α > 0 such that

t∑j=1

(j−1∏i=0

f2t−i(θ)

)≤ α, max

1≤j≤n

∣∣∣∣∣∣n∑

t=j+1

(t−j−1∏i=0

ft−i(θ0)

)∣∣∣∣∣∣≤ γ. (3.7)

(A3) f ′t(θ) = dft(θ)/dθ /= 0 and f ′′

t (θ) = df ′t(θ)/dθ exist and are bounded, and g(·) is

twice continuously differentiable, 0 < m ≤ maxu|g ′(u)| ≤ M < ∞, 0 < m ≤maxu|g ′′(u)| ≤ M < ∞.

Theorem 3.1. Assume (2.1), (2.2) and (A1)–(A3).

(1) Suppose H01 : ft(θ) = θ and g(u) is a continuous function, σ20 > 0 holds. Then

d1nD−→ χ2

1, n −→ ∞. (3.8)


(2) Suppose H02 : ft(θ) = θ, g(u) = u, σ20 > 0 holds. Then

d2nD−→ χ2

1, n −→ ∞. (3.9)

(3) Suppose H03 : ft(θ) = θ, g(u) = eu/(1 + eu), σ20 > 0 holds. Then

d3nD−→ χ2

1, 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,

supϕ∈Nn(A)

∥∥∥D−1/2n Fn

(ϕ)D−T/2

n −Φn

∥∥∥ P−→ 0, n → ∞, (4.1)

where

Φn = diag

(Id,

∑nt=2 f

′t2(θ0)e2t−1

Δn(θ0, σ0)

), (4.2)

Nn(A) ={ϕ ∈ Rd+1 :

(ϕ − ϕ0

)TDn

(ϕ − ϕ0

) ≤ A2}. (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 ϕn → ϕ0, σ2n → σ2

0 and

Xn

(β∗, β∗∗, θn

)−→ Xn

(ϕ0), (4.4)

where β∗, β∗∗ are on the line of β0 and βn.

Proof. Similar to proof of Theorem 3.1 in Hu [24], we easily prove that ϕn → ϕ0, and σ2n →

σ20 . Since (4.4) is easily proved, here we omit the proof (4.4).

Proof of Theorem 3.1. Note that Sn(ϕn) = 0 and Fn(ϕn) are nonsingular. By Taylor’s expansion,we have

0 = Sn

(ϕn

)= Sn

(ϕ0) − Fn

(ϕn

)(ϕn − ϕ0

), (4.5)


where ϕn = aϕn + (1 − a)ϕ0 for some 0 ≤ a ≤ 1. Since ϕn ∈ Nn(A), also ϕn ∈ Nn(A). By (4.1),we have

Fn

(ϕn

)= D1/2

n

(Φn + An

)DT/2

n . (4.6)

Thus An is a symmetric matrix with AnP−→ 0. By (4.5) and (4.6), we have

DT/2n

(ϕn − ϕ0

)= DT/2

n F−1n

(ϕn

)Sn

(ϕ0)=(Φn + An

)−1D−1/2

n Sn

(ϕ0). (4.7)

Let Sn(ϕ), Fn(ϕ) denote S(β)n (ϕ), S(θ)

n (ϕ), and F(β)n (ϕ), F(θ)

n (ϕ), respectively. By (4.7), we have

ΦnDT/2n

(βn − β0, θn − θ0

)= D−1/2

n

(S(β)n

(ϕ0), S

(θ)n

(ϕ0))

+ oP (1). (4.8)

Note that

ΦnDT/2n =

⎛⎜⎜⎜⎝

XT/2n

(ϕ0)

0

0

(∑nt=2 f

′t2(θ0)e2t−1

)√Δn(θ0, σ0)

⎞⎟⎟⎟⎠,

D−1/2n =

⎛⎜⎝

X−1/2n

(ϕ0)

0

01√

Δn(θ0, σ0)

⎞⎟⎠.

(4.9)

By (2.15), (4.2) and (4.8), we get

XT/2n

(ϕ0)(

βn − β0)= X−1/2

n

(ϕ0)S(β)n

(ϕ0)+ oP (1)

= X−1/2n

(ϕ0) n∑

t=2

ηt(g ′(xTt β0)xt − ft(θ0)g ′

(xTt−1β0

)xt−1)+ oP (1),

(4.10)

n∑t=2

f ′t2(θ0)e2t−1

(θn − θ0

)= S

(θ)n

(ϕ0)+ oP

(√Δn(θ0, σ0)

)

=n∑t=2

f ′t(θ0)ηtet−1 + oP

(√Δn(θ0, σ0)

).

(4.11)

Note that

εt = yt − g(xTt β)= g ′(xTt β

∗)xTt

(β0 − β

)+ et (4.12)


By (2.1), (2.11) and (4.12), we have

εt − ft(θn)εt−1 =

(g ′(xTt β

∗)xTt − ft

(θn)g ′(xTt−1β

∗∗)xTt−1)(

β0 − βn)+(et − ft

(θn)et−1).

(4.13)

By (4.13) and (2.10), we have

n∑t=2

(εt − ft

(θn)εt−1)2

=n∑t=2

(εt − ft

(θn)εt−1)((

g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)

×(β0 − βn

)+(et − ft

(θn)et−1))

=n∑t=2

(εt − ft

(θn)εt−1)(

g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn)

+n∑t=2

(εt − ft

(θn)εt−1)(

et − ft(θn)et−1)

=n∑t=2

(εt − ft

(θn)εt−1)(

et − ft(θn)et−1).

(4.14)

By (4.13), we have

(g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn)=(εt − ft

(θn)εt−1)−(et − ft

(θn)et−1).

(4.15)

By (4.15), we have

n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

=n∑t=2

(εt − ft

(θn)εt−1)2

+n∑t=2

(et − ft

(θn)et−1)2

− 2n∑t=2

(εt − ft

(θn)εt−1)(

et − ft(θn)et−1)

=n∑t=2

(et − ft

(θn)et−1)2 −

n∑t=2

(εt − ft

(θn)εt−1)2.

(4.16)


By (4.14) and (4.16), we have

n∑t=2

(εt − ft

(θn)εt−1)2

=n∑t=2

(et − ft

(θn)et−1)2

−n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

.

(4.17)

By (4.15), we have

n∑t=2

(εt − ft

(θn)εt−1)2

=n∑t=2

(et − ft

(θn)et−1)2

+n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

+ 2n∑t=2

(et − ft

(θn)et−1)((

g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))

.

(4.18)

Thus, by (4.17) and (4.18), we have

n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

+n∑t=2

(et − ft

(θn)et−1)((

g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))

= 0.

(4.19)

Since ηt = et − ft(θ0)et−1, we have

n∑t=2

(et − ft

(θn)et−1)2

=n∑t=2

(ηt + ft(θ0)et−1 − ft

(θn)et−1)2

=n∑t=1

η2t +

n∑t=2

(ft(θ0) − ft

(θn))2

e2t−1 + 2(ft(θ0) − ft

(θn))

ηtet−1.

(4.20)

Thus, by (4.17), (4.20) and mean value theorem, we have

(n − 1)σ2n =

n∑t=2

(εt − ft

(θn)εt−1)2

=n∑t=1

η2t +

n∑t=2

(ft(θ0) − ft

(θn))2

e2t−1 + 2(ft(θ0) − ft

(θn))

ηtet−1


−n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

=n∑t=1

η2t +(θ0 − θn

)2 n∑t=2

f ′t2(θ)e2t−1 + 2

(θ0 − θn

) n∑t=2

f ′t

(θ)et−1ηt

−n∑t=2

((g ′(xTt β

∗)xTt − ft


∗∗)xTt−1)(

β0 − βn))2

,

(4.21)

where θ = aθ0 + (1 − a)θn for some 0 ≤ a ≤ 1.It is easy to know that

(βn − β0

)TXn

(ϕ0)(

βn − β0)

=

(n∑t=2

ηtX−1/2n

(ϕ0)(


(xTt−1β0

)xt−1))2

+ op(1).

(4.22)

By Lemma 4.2 and (4.22), we have

(n − 1)σ2n =

n∑t=1

η2t +(θ0 − θn

)2 n∑t=2

f ′t2(θ)e2t−1 + 2

(θ0 − θn

) n∑t=2

f ′t

(θ)et−1ηt

−(

n∑t=2

ηtX−1/2n

(β∗, β∗∗, θn

)(g ′(xTt β0)xt − ft(θ0)g ′

(xTt−1β0

)xt−1))2

+ oP (1)

=n∑t=1

η2t +(θ0 − θn

)2 n∑t=2

f ′t2(θ)e2t−1 + 2

(θ0 − θn

) n∑t=2

f ′t

(θ)et−1ηt

−(

n∑t=2

ηtX−1/2n

(ϕ0)(


(xTt−1β0

)xt−1))2

+ oP (1).

(4.23)

Hence, by (4.11), we have

θn − θ0 =∑n

t=2 f′t(θ0)ηtet−1∑n

t=2 f′t2(θ0)e2t−1

+ oP

( √Δn(θ0, σ0)∑n

t=2 f′t2(θ0)e2t−1

)

=∑n


t=2 f′t2(θ0)e2t−1

+ oP

⎛⎜⎝ 1√∑n

t=2 f′t2(θ0)e2t−1

⎞⎟⎠.

(4.24)


By (4.24), we have

(θ0 − θn

)2 n∑t=2

f ′t2(θ)e2t−1 + 2

(θ0 − θn

) n∑t=2

f ′t

(θ)et−1ηt

=

⎛⎜⎝∑n


t=2 f′t2(θ0)e2t−1

+ oP

⎛⎜⎝ 1√∑n

t=2 f′t2(θ0)e2t−1

⎞⎟⎠

⎞⎟⎠

2n∑t=2

f ′t2(θ)e2t−1

+ 2

⎛⎜⎝∑n


t=2 f′t2(θ0)e2t−1

+ oP

⎛⎜⎝ 1√∑n

t=2 f′t2(θ0)e2t−1

⎞⎟⎠

⎞⎟⎠

n∑t=2

f ′t

(θ)et−1ηt + oP (1)

=

⎛⎜⎝∑n


t=2 f′t2(θ0)e2t−1

+ oP

⎛⎜⎝ 1√∑n

t=2 f′t2(θ0)e2t−1

⎞⎟⎠

⎞⎟⎠

2n∑t=2

(f ′t(θ0) + o(1)

)2e2t−1

+ 2

⎛⎜⎝∑n


t=2 f′t2(θ0)e2t−1

+ oP

⎛⎜⎝ 1√∑n

t=2 f′t2(θ0)e2t−1

⎞⎟⎠

⎞⎟⎠

·n∑t=2

(f ′′t (θ0) + o(1)

)et−1ηt + oP (1)

=

(∑nt=2 f

′t(θ0)ηtet−1

)2∑n

t=2 f′t2(θ0)e2t−1

− 2(∑n

t=2 f′t(θ0)ηtet−1

)2∑n

t=2 f′t2(θ0)e2t−1

+ oP (1)

= −(∑n


)2∑n

t=2 f′t2(θ0)e2t−1

+ oP (1).

(4.25)

By Lemma 4.2, we have

(n − 1)σ2n =

n∑t=1

η2t −(∑n


)2∑n

t=2 f′t2(θ0)e2t−1

−(

n∑t=2

ηtX−1/2n

(β∗, β∗∗, θn

)(g ′(xTt β0)xt − ft(θ0)g ′

(xTt−1β0

)xt−1))2

+ oP (1)

=n∑t=1

η2t −(∑n


)2∑n

t=2 f′t2(θ0)e2t−1

−(

n∑t=2

ηtX−1/2n

(ϕ0)(


(xTt−1β0

)xt−1))2

+ oP (1).

(4.26)


Now, we prove (3.8). By (4.12), we have

εt(1) = yt − g(xTt β1n

)= g ′(xTt β

∗1n

)xTt

(β0 − β1n

)+ et. (4.27)

Note that

εt − ft(θ0)εt−1 =(g ′(xTt β

∗)xTt − ft(θ0)g ′

(xTt−1β

∗∗)xTt−1)(

β0 − β)+ ηt. (4.28)

From (4.28), we have

εt(1) − θ1nεt−1(1) =(g ′(xTt β

∗1n

)xTt − θ1ng

′(xTt−1β

∗∗1n

)xTt−1)(

β0 − β1n)+ ηt. (4.29)

By (2.8) and (2.10), we have

0 =n∑t=2

(εt(1) − θ1nεt−1(1)

)(g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1)

=n∑t=2

(g ′(xTt β

∗1n

)xTt − θ1ng

′(xTt−1β

∗∗1n

)xTt−1)(

β0 − β1n)(

g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1)

+n∑t=2

ηt(g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1)

=(β0 − β1n

)TX1n

(β∗1n, β

∗∗1n, θ1n

)+

n∑t=2

ηt(g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1).

(4.30)

From (4.30), we obtain that

β1n − β0 = X−11n

(β∗1n, β

∗∗1n, θ1n

) n∑t=2

ηt(g ′(xTt β1n

)xt − θ1ng

′(xTt−1β1n

)xt−1). (4.31)

By (4.29), (4.31) and Lemma 4.2, we have

(n − 1)σ21n =

n∑t=2

(εt(1) − θ1nεt−1(1)

)2

=n∑t=1

η2t +(β0 − β1n

)TX1n

(β∗1n, β

∗∗1n, θ1n

)(β0 − β1n

)

+ 2(β0 − β1n

)T n∑t=2

ηt(g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1)


=n∑t=1

η2t −(

n∑t=2

ηtX−1/21n

(β∗1n, β

∗∗1n, θ1n

)(g ′(xTt β

∗1n

)xt − θ1ng

′(xTt−1β

∗∗1n

)xt−1))2

+ op(1)

=n∑t=1

η2t −(

n∑t=2

ηtX−1/21n

(ϕ0)(

g ′(xTt β0)xt − θ0g

′(xTt−1β0

)xt−1))2

+ op.

(4.32)

By (3.3)–(3.5), we have

d1n = L1n − Ln = (n − 1) ln

(σ21n

σ2n

)= (n − 1)

((σ21n

σ2n

)− 1

)+ oP (1). (4.33)

Under the H01, and by (4.26), (4.32) and (4.33), we have

(n − 1)(σ21n − σ2

n

)

σ2n

=

(∑nt=2 ηtet−1

)2σ2n

∑nt=2 e

2t−1

+ oP (1)

=

(∑nt=2 ηtet−1

)2σ20∑n

t=2 e2t−1

+ oP (1).

(4.34)

It is easily proven that

∑nt=2 ηtet−1

σ0

√∑nt=2 e

2t−1

−→ N(0, 1). (4.35)

Thus, by (4.33)–(4.35), we finish the proof of (3.8).Next we prove (3.9). Under H02 : ft(θ) = θ, g(u) = u, and yt = xT

t β0 + et, we have

εt(2) = yt − xTt β2n = xT

t β0 − xTt β2n + et = xT

t

(β0 − β2n

)+ et. (4.36)

Hence

εt(2) − θ2nεt−1(2) = xTt

(β0 − β2n

)+ et − θ2n

(xTt−1(β0 − β2n

)+ et−1

)

=(xTt − θ2nx

Tt−1)(

β0 − β2n)+ ηt.

(4.37)

By (2.8), (2.10), we have

0 =n∑t=2

(εt(2) − θ2nεt−1(2)

)(xt − θ2nxt−1

)

=n∑t=2

(xTt − θ2nx

Tt−1)(

β0 − β2n)(

xt − θ2nxt−1)+

n∑t=2

ηt(xt − θ2nxt−1

).

(4.38)


From (4.38), we obtain,

β2n − β0 = X−12n

(θ2n) n∑

t=2


). (4.39)

Thus, by (4.37), (4.39) and Lemma 4.2, we have

(n − 1)σ22n =

n∑t=2

(εt(2) − θ2nεt−1(2)

)2

=n∑t=1

η2t +(β0 − β2n

)TX2n

(θ2n)(

β0 − β2n)+ 2(β0 − β2n

)T n∑t=2


)

=n∑t=1

η2t −(

n∑t=2


)T)X−1

2n

(θ2n)( n∑

t=2


))

=n∑t=1

η2t −(

n∑t=2

ηtX−1/22n (θ0)(xt − θ0xt−1)

)2

+ op(1).

(4.40)

By (3.3)–(3.5), we have

d2n = L2n − Ln = (n − 1) ln

(σ22n

σ2n

)= (n − 1)

((σ22n

σ2n

)− 1

)+ oP (1). (4.41)

Under the H02, by (4.26), (4.40), and (4.41), we obtain

(n − 1)(σ22n − σ2

n

)

σ2n

=

(∑nt=2 ηtet−1

)2σ2n

∑nt=2 e

2t−1

+ oP (1)

=

(∑nt=2 ηtet−1

)2σ20∑n

t=2 e2t−1

+ oP (1).

(4.42)

Thus, by (4.35), (4.42), (3.9) holds.Finally, we prove (3.10). Under H03, we have

εt(3) = yt − exTt β

∗3n

1 + exTt β

∗3n

=ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xTt

(β0 − β3n

)+ et. (4.43)


Thus

εt(3) − θ3nεt−1(3) =ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xTt

(β0 − β3n

)+ et

− θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xTt−1(β0 − β3n

)− θ3net−1

=

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xTt − θ3n

exTt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xTt−1

⎞⎟⎠(β0 − β3n

)+ ηt.

(4.44)

By (2.8) and (2.10), we have

0 =n∑t=2

(εt(3) − θ3nεt−1(3)

)⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠

=n∑t=2

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xTt − θ3n

exTt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xTt−1

⎞⎟⎠(β0 − β3n

)

×

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠

+n∑t=2

ηt

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠

=(β0 − β3n

)TX3n

(β∗3n, β

∗∗3n, θ3n

)+

n∑t=2

ηt

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠.

(4.45)

From (4.45), we obtain

β3n − β0 = X−13n

(β∗3n, β

∗∗3n, θ3n

) n∑t=2

ηt

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠. (4.46)


By (4.44), (4.46) and Lemma 4.2, we have

(n − 1)σ23n =

n∑t=2

(εt(3) − θ3nεt−1(3)

)2

=n∑t=1

η2t +(β0 − β3n

)TX3n

(β∗3n, β

∗∗3n, θ3n

)(β0 − β3n

)

+ 2(β0 − β3n

)T n∑t=2

ηt

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xt − θ3nex

Tt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xt−1

⎞⎟⎠

=n∑t=1

η2t −

⎛⎜⎝

n∑t=2

ηtX−1/23n

(β∗3n, β

∗∗3n, θ3n

)

×

⎛⎜⎝ ex

Tt β

∗3n

(1 + ex

Tt β

∗3n

)2xTt − θ3n

exTt−1β

∗∗3n

(1 + ex

Tt−1β

∗∗3n

)2xTt−1

⎞⎟⎠

⎞⎟⎠

2

=n∑t=1

η2t −

⎛⎜⎝

n∑t=2

ηtX−1/23n

(ϕ0)⎛⎜⎝ ex

Tt β0

(1 + ex

Tt β0)2xt − θ0

exTt−1β0

(1 + ex

Tt−1β0)2xt−1

⎞⎟⎠

⎞⎟⎠

2

+ op(1).

(4.47)

By (3.3)–(3.5), we know that

d3n = L3n − Ln = (n − 1) ln

(σ23n

σ2n

)= (n − 1)

((σ23n

σ2n

)− 1

)+ oP (1). (4.48)

Under the H03, by (4.26), (4.47) and (4.48), we have

(n − 1)(σ23n − σ2

n

)

σ2n

=

(∑nt=2 ηtet−1

)2σ20∑n

t=2 e2t−1

+ oP (1). (4.49)

Thus, (3.10) follows from (4.48), (4.49), and (4.35). Therefore, we complete the proof ofTheorem 3.1.

5. Conclusions and Open Problems

In the paper, we consider the generalized linear mode with FCA processes, which includesmany special cases, such as an ordinary regression model, an ordinary generalized regressionmodel, a linear regression model with constant coefficient autoregressive processes, time-dependent and function coefficient autoregressive processes, constant coefficient autore-gressive processes, time-dependent or time-varying autoregressive processes, and a linear


regression model with functional coefficient autoregressive processes. And then we obtainthe QML estimators for some unknown parameters in the generalized linear modemodel andextend some estimators. At last, we use pseudo LR method to investigate three hypothesistests 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 afractal time series can be regarded as the solution to a differential equation of fractional orderwith a white noise in the domain of stochastic processes (see [25]). In the paper, {εt} is aconventional nonlinear time series. We may investigate some hypothesis tests by pseudo LRmethod when the {εt} is a fractal time series (the idea is given by an anonymous reviewer).In particular, we assume that

p∑i=0

ap−iDviεt = ηt, (5.1)

where vp, vp−1, . . . , v0 is strictly decreasing sequence of nonnegative numbers, ai is a constantsequence, and Dv is the Riemann-Liouville integral operator of order v > 0 given by

Dvh(t) =1

Γ(v)

∫ t

0(t − u)v−1h(u)du, (5.2)

where Γ is the Gamma function, and h(t) is a piecewise continuous on (0,∞) and integrableon 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 andtechnologies (see [25, 27–32]). Thus it is very significant to investigate various regressionmodels with fractal time series errors, including regression model (1.1)with (5.1).

(2)We maybe investigate the others hypothesis tests, for example:

H04: ft(θ) = 0, g(u) = u, σ20 > 0;

H05: ft(θ) = θ, g(u) = 0, σ20 > 0;

H06: ft(θ) = 0, g(u) = eu/(1 + eu), σ20 > 0;

H07: ft(θ) = at and g(u) is a continuous function, σ20 > 0;

H08: ft(θ) = at, g(u) = u, σ20 > 0;

H09: ft(θ) = at, g(u) = eu/(1 + eu), σ20 > 0.

Acknowledgments

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


References

[1] Z. D. Bai and M. Guo, “A paradox in least-squares estimation of linear regression models,” Statistics& Probability Letters, vol. 42, no. 2, pp. 167–174, 1999.

[2] Y. Li and H. Yang, “A new stochastic mixed ridge estimator in linear regression model,” StatisticalPapers, vol. 51, no. 2, pp. 315–323, 2010.

[3] A. E. Hoerl and R. W. Kennard, “Ridge regression: Biased estimation for nonorthogonal problems,”Technometrics, vol. 12, pp. 55–67, 1970.

[4] X. Chen, “Consistency of LS estimates of multiple regression under a lower order moment condition,”Science in China Series A, vol. 38, no. 12, pp. 1420–1431, 1995.

[5] T. W. Anderson and J. B. Taylor, “Strong consistency of least squares estimates in normal linearregression,” The Annals of Statistics, vol. 4, no. 4, pp. 788–790, 1976.

[6] G. Gonzalez-Rodrıguez, A. Blanco, N. Corral, and A. Colubi, “Least squares estimation of linearregression models for convex compact random sets,” Advances in Data Analysis and Classification, vol.1, no. 1, pp. 67–81, 2007.

[7] H. Cui, “On asymptotics of t-type regression estimation in multiple linear model,” Science in ChinaSeries A, vol. 47, no. 4, pp. 628–639, 2004.

[8] M. Q. Wang, L. X. Song, and X. G. Wang, “Bridge estimation for generalized linear models with adiverging number of parameters,” Statistics & Probability Letters, vol. 80, no. 21-22, pp. 1584–1596,2010.

[9] L. C. Chien and T. S. Tsou, “Deletion diagnostics for generalized linear models using the adjustedPoisson likelihood function,” Journal of Statistical Planning and Inference, vol. 141, no. 6, pp. 2044–2054,2011.

[10] L. Fahrmeir and H. Kaufmann, “Consistency and asymptotic normality of the maximum likelihoodestimator in generalized linear models,” The Annals of Statistics, vol. 13, no. 1, pp. 342–368, 1985.

[11] Z. H. Xiao and L. Q. Liu, “Laws of iterated logarithm for quasi-maximum likelihood estimator ingeneralized linear model,” Journal of Statistical Planning and Inference, vol. 138, no. 3, pp. 611–617,2008.

[12] Y. Bai, W. K. Fung, and Z. Zhu, “Weighted empirical likelihood for generalized linear models withlongitudinal data,” Journal of Statistical Planning and Inference, vol. 140, no. 11, pp. 3446–3456, 2010.

[13] S. G. Zhao and Y. Liao, “The weak consistency of maximum likelihood estimators in generalizedlinear models,” Science in China A, vol. 37, no. 11, pp. 1368–1376, 2007.

[14] P. Pere, “Adjusted estimates and Wald statistics for the AT(1) model with constant,” Journal ofEconometrics, vol. 98, no. 2, pp. 335–363, 2000.

[15] R. A. Maller, “Asymptotics of regressions with stationary and nonstationary residuals,” StochasticProcesses and Their Applications, vol. 105, no. 1, pp. 33–67, 2003.

[16] W. A. Fuller, Introduction to Statistical Time Series, JohnWiley & Sons, New York, NY, USA, 2nd edition,1996.

[17] G. H. Kwoun and Y. Yajima, “On an autoregressive model with time-dependent coefficients,” Annalsof the Institute of Statistical Mathematics, vol. 38, no. 2, pp. 297–309, 1986.

[18] J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case,”Annalsof Mathematical Statistics, vol. 29, pp. 1188–1197, 1958.

[19] J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case—II,”Annals of Mathematical Statistics, vol. 30, pp. 831–834, 1959.

[20] J. D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, NJ, USA, 1994.[21] F. Carsoule and P. H. Franses, “A note on monitoring time-varying parameters in an autoregression,”

International Journal for Theoretical and Applied Statistics, vol. 57, no. 1, pp. 51–62, 2003.[22] R. Azrak and G. Melard, “Asymptotic properties of quasi-maximum likelihood estimators for ARMA

models with time-dependent coefficients,” Statistical Inference for Stochastic Processes, vol. 9, no. 3, pp.279–330, 2006.

[23] R. Dahlhaus, “Fitting time series models to nonstationary processes,” The Annals of Statistics, vol. 25,no. 1, pp. 1–37, 1997.

[24] H. Hu, “QML estimators in linear regression models with functional coefficient autoregressiveprocesses,”Mathematical Problems in Engineering, vol. 2010, Article ID 956907, 30 pages, 2010.

[25] M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, ArticleID 157264, 26 pages, 2010.

[26] Y. S.Mishura, Stochastic Calculus for Fractional BrownianMotion and Related Processes, vol. 1929 of LectureNotes in Mathematics, Springer, Berlin, Germany, 2008.


[27] M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” MathematicalProblems in Engineering, vol. 2012, Article ID 302786, 13 pages, 2012.

[28] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011.

[29] C. Cattani, “Fractals and hidden symmetries in DNA,”Mathematical Problems in Engineering, vol. 2010,Article ID 507056, 31 pages, 2010.

[30] J. Levy-Vehel and E. Lutton, Fractals in Engineering, Springer, 1st edition, 2005.[31] V. Pisarenko and M. Rodkin, Heavy-Tailed Distributions in Disaster Analysis, Springer, 2010.[32] H. Sheng, Y.-Q. Chen, and T.-S. Qiu, Fractional Processes and Fractional Order Signal Processing, Springer,

2012.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

hypothesis testing in generalized linear models with...

Documents