数理解析研究所講究録
916 巻 1995 年 169-179

169

Discriminant Analysis for Regression Models
with Long-memory Linear Disturbances
阪大基礎工

張国強 (G. Zhang)

We shall consider the problems of classifying an observation from regression model withstationary long-memory linear disturbances into one of two populations described by themean functions of the model. We mainly use the -likelihood ratio(LR) as discriminantstatistic which is optimal in the sense of minimizing the misclassification probabilities.
Then we propose a new discriminant statistic related to the Best Linear Unbiased Estimator(BLUE) and the Least Squares Estimator(LSE). Finally, we discuss the asymptotics ofquatratic discriminant statistics in the non-Gaussian case.
$\log$

Keywords: Discriminant analysis, Regression model, Long-memory linear disturbances,
Misclassification probability, BLUE, LSE.

1 INTRODUCTION
There are a variety of problems related to regression analysis in time series. One is that ofdiscriminanting between two regressions. For simplicity, let us consider the two hypotheses$H_{j}$:$Y_{t}= \sum_{i=1}^{k}X_{i}(t)\beta_{i}^{(j)}+\epsilon_{t}$,$j=1,2$ .

(1.1)

are vectors of rewhere $X_{1}(t),$ $\cdots,X_{k}(t)$ are nonstochastic regressors,
is assumed to be a zero mean Gaussian stationary process withgression parameters, andis nonsin. Assume that the covariance matrix of the disturbances
spectral density
areandgular. Then the two p.d.f.s corresponding to$\beta^{(j)}=(\beta_{1}^{(j)}, \cdots,\beta_{k}^{(j)})’$$\{\epsilon_{t}\}$$\Sigma$$f(\lambda)$$\{\epsilon_{t}\}$$H_{2}$$H_{1}$$p_{j}( \mathrm{Y})=\frac{1}{(2\pi)^{\frac{l}{2}}|\Sigma|^{\frac{1}{2}}},\exp\{-\frac{1}{2}(\mathrm{Y}-X\beta^{(j)})’\Sigma^{-1}(\mathrm{Y}-X\beta^{(j)})\}$,$j=1,2$

(1.2)

where
$\mathrm{Y}=$ ,$X_{r}=$ ,$X=(X_{1}, \cdots, X_{k})$fromIn this paper, we consider the problem of classifying
and. We assign toiffalls into region
populations, whereandare exclusive and exhaustive regions into$\mathrm{Y}$$H_{1}$$H_{2}$$R_{1}$$H_{2}$$R_{2}$$\mathrm{Y}$$H_{1}$$\mathrm{Y}$orinto one of two; otherwise we assign it. Let the probability of$H_{1}$$R_{1}$$R^{n}$.

$H_{2}$

170

from
into
be
. We choose$P(2|1)+P(1|2)$ . Then it is known that the regions based on LR

misclassifying$\mathrm{Y}$$H_{i}$$H_{j}$$R_{j}=\{\mathrm{Y}$$P(j|i)= \int_{R_{j}}p_{i}(\mathrm{Y})d\mathrm{Y}$:$(-1)^{j-1}\mathrm{L}\mathrm{R}>0\}$,$R_{1}$$j=1,2$

and$R_{2}$

to minimize(1.3)

give the optimal classification, where$\mathrm{L}\mathrm{R}=(\beta^{(2)}-\beta^{(1)})’X’\Sigma^{-1}(\mathrm{Y}-\frac{1}{2}X(\beta^{(2)}+\beta^{(1)}))$

.

(1.4)

In most of time series, the dependence between distant observations is quite weak, and weusually use ARMA and other stationary short-memory processes to discribe their characteristics,
and a series of works for the discriminant analysis of such time series have been done. In the
problems of discriminanting two Gaussian processes by linear fultering, Shumway and Unger(1974) gave certain spectral approximations of Kullback-Leibler information and J-divergence.
Zhang and Taniguchi (1994) used an approximation of LR as a classification statistic for the
non-Gaussian vector time series classification problems. Zhang (1994) discussed the higher-order
asymptotic theories of discriminant analysis for stationary ARMA processes. Shumway (1982)
gave an extensive review of various discriminant problems in time series.
However, time series classification problems are not restricted to the physical sciences, but
occur under many and varied circumstances in the other fields. For instance, in many empirical time series, especially those of economics and geophysics, the dependence between distantobservations is so strong that ARMA models are unable to express the spectral densities oflow frequencies adequately. For such long-memory(or strongly dependent) processes, Adenstedt(1974) found that for alarge class of spectral densities, the variance of the BLUE for the mean isseen to depend asymptotically only on the spectral density near the origin. Giraitis and Surgailis(1990) gave a central limit theorem for quadratic forms in strongly dependent linear variables
and applied it to prove the asymptotic normality of Whittle’s estimator of parameters of strongly
dependent linear sequences. Yajima $(1988, 1991)$ considered estimation of a regression modelwith long-memory stationary errors by LSE and BLUE, and gave their asymptotic properties.
In Section 2, we use LR as a discriminant statistic for classifying two regression models(1.1) with short-memory stationary distubances. We show that LR is a consistent classification
statistic under Grenander’s conditions on.
In Section 3, we use LR as a classification statistic for regression models (1.1) with stationary
Gaussian long-memory disturbances and discuss its asymptotic properties of misclassification
probabilities. We study a classification statistic which is based on a linear combination of LSE
and BLUE.
In Section 4, we propose a discriminant statistic of quadratic form for a simple regression
model with stationary “non-Gaussian” long-memory disturbances. Then we elucidate someresults which are different from those for short-memory disturbances.
$X_{t}$

171

2 SHORT-MEMORY CLASSIFICATIONIn this section, we summarize the discriminant theory of regression model (1.1) whenis a zero mean short-memory stationary process with positive and continuous spectral density$\{\epsilon_{t}\}$$f(\lambda),$ $\lambda\in[-\pi, \pi]$.

Following Grenander (1954) we assume that the nonstochastic regressors $\{X_{r}(t)\}$ possess thefollowing properties:
(G.1)
(G.2)
(G.3)

$\chi_{r}(n)=\sum_{t=1}^{n}|X_{r}(t)|^{2}arrow\infty$$\lim_{narrow\infty}\frac{\chi_{r}(n+1)}{\chi_{r}(n)}=1$, as, for $r=1,$$narrow\infty$$\cdots,$$\lim_{narrow\infty}\frac{\sum_{--}^{n}X_{r}(t+h)X_{\epsilon}(t)}{\{\chi_{r}(n)\chi \mathrm{Q}(n)\}^{1/2}}.=R_{rs}(h)$$k$, for $r=1,$$\cdots,$

$k$

,

,

,

(say)

, and all integers $h\geq 0$ . Defining $X_{r}(t)=0,$ $t<0$ , (for eachexists for each $r,s=1,$$k)$ , the above definition of $R_{rs}(h)$ can now be extended to all integer values of$r=1,$, and we can write$\cdots,$$k$$\cdots,$$h$$R(h)=\{R_{rs}(h)\}$i.e. $R(h)$ is
(G.4)

$a$

$q\cross q$

The matrix,$h=0,$ $\pm 1,$ $\pm 2,$

matrix with entry $R_{rs}(h)$ in the$R(\mathrm{O})$$r\mathrm{t}\mathrm{h}$$\cdots$,

row and sth column.

is nonsingular.

is positive
It is not difficult to show that $R’(h)=R(-h)$ (for each ), and thatdefinite(cf Grenander and Rosenblatt (1957)). It then follows that, there exists a Hermitianmatrix function, $M(\lambda)=\{M_{rs}(\lambda)\}$ such that $R_{rs}(h)$ admits a spectral representation of the
form,
$h$

$R(\mathrm{O})$$R_{rs}(h)= \int_{-\pi}^{\pi}e^{ih\lambda}dM_{rs}(\lambda)$

where matrix form

$M(\lambda)$

has positive semi-definite matrix increments. Let$D_{n}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\chi_{1}^{1/2}(n), \cdots, \chi_{k}^{1/2}(n)\}$i.e. the$D_{n}$

is the diagonal matrix with entry$\chi_{r}^{1/2}(n)$in the$r\mathrm{t}\mathrm{h}$row and column.

is a positive and continuous function[Theorem 2.1] Suppose that the spectral density, we use the classification
andinto one of two hypotheses
on $[-\pi, \pi]$ . For classifying, the misclassification probabilities. Then under Assumptions (G. )rule (1.3) based on
satisfy
.
(2.1)
, for
$f(\lambda)$$H_{2}$$H_{1}$$\mathrm{Y}$$1$$\mathrm{L}\mathrm{R}$$-(\mathrm{G}.4)$$\beta^{(1)}\not\equiv\beta^{(0)}$$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=0$

We put the proof of theorems and propositions in Section 5 if they are not straightforward.
, we consider the case when
Next, to evaluate the goodness of classification statistic:
is contiguous to
;
:
(2.2)
$\mathrm{L}\mathrm{R}$$\beta^{(1)}$$H_{1}$$H_{2}$$\beta^{(1)}=\beta$: $\beta^{(2)}=\beta+D_{n}^{-1}H$ ,$\beta^{(2)}$172

where $H=(H_{1}, \cdots, H_{k})’$ is a constant vector. Then
[Theorem 2.2] Suppose thatis a positive and continuous function on $[-\pi, \pi]$ and that(G. )hold. Then under contiguous assumption (2.2), the misclassification probabilities
of LR are evaluated as$f(\lambda)$$1$$-(\mathrm{G}.4)$.

$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=\Phi(-\frac{1}{2}\sqrt{H’\Omega H})$

(2.3)

where
$\Omega=\frac{1}{2\pi}\int_{-\pi}^{\pi}f^{-1}(\lambda)dM(\lambda)$.

Let us consider the case of polynomial regression.
[Example 2.1] Suppose that $X_{r}(t)=t^{r-1},$ $r=1,$
andis a stationary process withzero mean and spectral densitythat is a positive and continuous function on $[-\pi, \pi]$ . Then$\cdots,$$k$$\{\epsilon_{t}\}$$f(\lambda)$$M(\lambda)$

has only ajump at$\lambda=0$

of $M_{0}= \{\frac{[(2r-1)(2s-1)]:}{(r+s-1)}\}_{r,s=1,\cdots,k}$ (e.g. Anderson(1971)). Thus

under contiguous assumption (2.2), the misclassification probabilities of LR are evaluated as$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=\Phi(-\frac{1}{2}\sqrt{H’\Omega_{0}H})$

.

(2.4)

where
$\Omega_{0}=\{\frac{[(2r-1)(2s-1)]^{\frac{1}{2}}}{2\pi f(0)(r+s-1)}\}_{r,s=1,\cdots,k}$

3 LONG-MEMORY CLASSIFICATIONIn this section, we consider the discriminant problem of regression model (1.1) whena Gaussian long-memory linear process of the form$\epsilon_{i}=\sum_{j=0}^{\infty}\alpha_{j}\omega_{t-j}$,$\omega_{t}\sim N(0,1)$,$\sum_{j=0}^{\infty}\alpha_{j}^{2}<\infty$.

$\{\epsilon_{t}\}$

is

(3.1)

We introduce an important Lemma that plays key role in the following discussion. Then weshow the discriminant properties of LR and a new classification statistic based on LSE andBLUE. Let$g_{jl}^{n}( \lambda)=\frac{(\sum_{t=1}^{n}X_{j}(t)e^{-it\lambda})(\sum_{t=1}^{n}X_{l}(t)e^{it\lambda})}{2\pi\chi_{r}(n)\chi_{s}(n)}$,$M_{jl}^{n}( \lambda)=\int_{-\pi}^{\lambda}g_{jl}^{n}(\omega)d\omega$

Besides (G. ), we add the following assumptions:$X_{r}(t)=t^{r-1}$(G.5), for $r=1,$ $\cdots,p,$ $1\leq p\leq k$ ,$0<d< \frac{1}{2}$ , where(G.6)
has the spectral density$1$$\{\epsilon_{t}\}$

,

(3.2)

$-(\mathrm{G}.4)$$f( \lambda)=\frac{f^{*}(\lambda)}{|1-\mathrm{e}^{i\lambda}|^{2d}},$$f^{*}(\lambda)$

is a positive and173

continuous function on $[-\pi, \pi]$ .
(G.7)
For every $\delta>0$ there exists,$j=p+1,$$\cdots,$

(G.8)

$c$

such that, for every$\int_{|\lambda|\leq c}f(\lambda)dM_{jj}^{n}(\lambda)<\delta$

and

$n$

$k$

$\frac{\max_{1<t<n}X_{r}^{2}(t)}{\chi_{r}^{1/2}(n)}=o(n^{-\mu}),$$r=p+1,$ $\cdots,k$ , for some $\mu>1-2d$ .

From Assumption (G.5), we have$\chi_{r}(n)=\sum_{t=1}^{n}t^{2(r-1)}\sim\frac{n^{2r-1}}{2r-1’}$$r=1,$ $\cdots,p$ ,(3.3)

so that Assumptions (G.1) and (G.2) are certainly satisfied. Define$\frac{\chi_{p}^{1/2}(n)}{n^{d}},\chi_{p+1}^{1/2}(n),$$\overline{D}_{n}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\frac{\chi_{1}^{1/2}(n)}{n^{d}},$$\cdots,$$\cdots,\chi_{k}^{1/2}(n))$

,

(3.4)

The following Lemma follows from Yajima (1991).
,[Lemma 3.1] Under Assumptions (G. )$-(\mathrm{G}.8)$$1$$\lim_{narrow\infty}\overline{D}_{n}^{-1}X’\Sigma^{-1}X\overline{D}_{n}^{-1}=$(3.5)$\lim_{narrow\infty}D_{n}^{-2}\overline{D}_{n}X’\Sigma X\overline{D}_{n}D_{n}^{-2}=$

(3.6)

and

where

$W_{1},$$B_{1}$

are$p\cross p$

matrix with$(i,j)\mathrm{t}\mathrm{h}$entry$w_{1}^{ij}= \frac{\Gamma(i-d)\Gamma(j-d)\{(2i-1)(2j-1)\}^{\frac{1}{2}}}{2\pi f^{*}(0)\Gamma(i-2d)\Gamma(j-2d)(i+j-1-2d)}$$b_{1}^{ij}= \frac{2\pi f^{*}(0)\Gamma(1-2d)}{\Gamma(d)\Gamma(1-d)}[\{(2i-1)(2j-1)\}^{\frac{1}{2}}\int_{0}^{1}\int_{0}^{1}x^{i-1}y^{j-1}|x-y|^{2d-1}dxdy]$

and$W_{2},$$B_{2}$

are $(k-p)\cross(k-p)$ matrix with$w_{2}^{ij}= \frac{1}{2\pi}\int_{-\pi}^{\pi}f^{-1}(\lambda)dM_{i+pj+p}(\lambda)$$(i,j)\mathrm{t}\mathrm{h}$

,

entry

$b_{2}^{ij}=2 \pi\int_{-\pi}^{\pi}f(\lambda)dM_{i+pj+p}(\lambda)$

respectively.
(with long-memory disturbances) into one of two hypotheses
Now, to classify, we propose a new classification criterion. Let
besides using criterion$\mathrm{Y}$$H_{1}$

and$H_{2}$,$\mathrm{L}\mathrm{R}$$\hat{\beta}_{\alpha}=\alpha\hat{\beta}_{L}+(1-\alpha)\hat{\beta}_{B}$

,

(3.7)

$(0\leq\alpha\leq 1)$

where$\hat{\beta}_{L}=(X’X)^{-1}X’\mathrm{Y}$

,

(3.8)

$\hat{\beta}_{B}=(X’\Sigma^{-1}X)^{-1}X’\Sigma^{-1}\mathrm{Y}$are the LSE and BLUE of , respectively. We use the discriminant rule: assign$\beta$$(\hat{\beta}_{\alpha}-\beta^{(1)})’(\hat{\beta}_{\alpha}-\beta^{(1)})<(\hat{\beta}_{\alpha}-\beta^{(2)})’(\hat{\beta}_{\alpha}-\beta^{(2)})$

$Y$

to

$H_{1}$

if

174

to

$H_{2}$, otherwise. That is, assign

$Y$

to

$H_{1}$

if$I_{\alpha}=(\beta^{(2)}-\beta^{(1)})’[2\hat{\beta}_{\alpha}-(\beta^{(2)}+\beta^{(1)})]<0$

to$H_{2}$,(3.9), otherwise.

[Theorem 3.1]$(\mathrm{i}\mathrm{i})(\mathrm{i})$Under Assumptions (G. )$1$$-(\mathrm{G}.8)$,$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{\lim_{narrow\infty}=}narrow\infty P_{\mathrm{L}\mathrm{R}}(1|2)=0\lim_{narrow\infty}P_{\mathrm{I}_{\alpha}}(2|1)P_{\mathrm{I}_{\alpha}}(1|2)=0\}$

Now, let the regression parameters associated with

$H_{1}$$H_{1}$:$\beta^{(1)}=\beta$$H_{2}$:$\beta^{(2)}=\beta+\overline{D}_{n}^{-1}H$

and$H_{2}$

for$\beta_{1}\not\equiv\beta_{2}$(3.10)

be;(3.11).

where $H=(H_{1}, \cdots, H_{k})’$ is a constant vector, then
[Theorem 3.2] Under Assumptions (G. )$1$$-(\mathrm{G}.8)$

and under contiguous condition (3.11),$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=\Phi(-\frac{1}{2}\{H’H\}^{\frac{1}{2}})$$\Phi(-\frac{1}{2\sqrt{k_{1}}})\geq\lim_{narrow\infty}P_{I_{\alpha}}(2|1)=\lim_{narrow\infty}P_{I_{\alpha}}(1|2)\geq\Phi(-\frac{1}{2\sqrt{k_{2}}})$

where$k_{1}$

and$k_{2}$,(3.12)(3.13)are chosen so that $k_{1}HH’\geq F_{\alpha}\geq k_{2}HH’$ . Here

$F_{\alpha}=\alpha^{2}AA+(1-\alpha^{2})$ ,where$A= \lim_{narrow\infty}\overline{D}_{n}(X’X)^{-1}\overline{D}_{n}^{-1}D_{n}^{2}$[Example 3.1].

In Theorem 3.2, if $X_{r}(t)=1,$ $r=1$ , Then$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=\lim_{narrow\infty}P_{I_{0}}(2|1)=\lim_{narrow\infty}P_{I_{0}}(1|2)$$= \Phi(-\frac{|H|\Gamma(1-d)}{2\sqrt{(1-2d)2\pi f^{*}(0)}\Gamma(1-2d)})$(3.14),

and$\lim_{narrow\infty}P_{I_{1}}(2|1)=\lim_{narrow\infty}P_{I_{1}}(1|2)=\Phi(-\frac{|H|}{2\sqrt{G_{f}}})$

where$G_{f}= \frac{2\pi f^{*}(0)\Gamma(1-2d)}{d(1+2d)\Gamma(d)\Gamma(1-d)}$.

,(3.15)

175

4 NON-GAUSSIAN CLASSIFICATIONIn this section, we consider the following stochastic model:$H_{j}$$Y_{t}=m^{(j)}+\epsilon_{t},$where $m^{(j)},j=1,2$ is the constant mean ofsatisfy$\{Y_{t}\}$$\epsilon_{t}=\sum_{j=0}^{\infty}\alpha_{j}\omega_{t-j}$,$j=1,2$ ,under$H_{j}$(4.1), and the linear disturbances$\sum_{j=0}^{\infty}\alpha_{j}^{2}<\infty$$\{\epsilon_{t}\}$

(4.2)

,

is a sequence of i.i.d. random variables with zero mean andwhere
has the spectral density
Suppose that$\{\omega_{t}\}$$\mathrm{v}\mathrm{a}\mathrm{r}\{\omega_{t}\}=1$.

$\{\epsilon_{t}\}$$f(\lambda)=f^{*}(\lambda)|1-e^{i\lambda}|^{-2d}$

Since we do not assume the Gaussianity of(G.9)

For every $r=1,2,$$\cdots$, the$r\mathrm{t}\mathrm{h}$$\{\omega_{t}\}$,$0<d<1/2$ .

(4.3), the following condition is imposed.

order cumulant$\kappa_{r}^{\omega}$of$\{\omega_{t}\}$exists.

Next we introduce a Toeplitz matrix $A=\{a_{s-l}\}$ with$a_{\tau}= \int_{-\pi}^{\pi}ea(i\tau x\wedge x)dx$

,

(4.4)

and assume that
(G.10)(G.11)$\wedge a(x)<c|x|^{-2\gamma}$$\gamma+d<\frac{1}{4}$,$\gamma<\frac{1}{2}$.

.

The following lemma is due to Giraitis and Surgailis (1990),
be the[Lemma 4.1] Let
Assumptions(4.1) satisfy (4.2). Then under$\epsilon=(\epsilon_{1}, \cdots,\epsilon_{n})’$$n$consective stretch of,$\{\epsilon_{t}\}$in regression models$(\mathrm{G}.9)-(\mathrm{G}.11)$$\frac{\epsilon’A\epsilon-E_{j}\{\epsilon’A\epsilon\}}{\sqrt{n}}arrow N(\mathcal{L}0,$$\sigma_{a}^{2}),$(4.5)$j=1,2$where$\sigma_{a}^{2}=16\pi^{3}\int_{-\pi}^{\pi}(a(\wedge x)f(x))^{2}dx+\kappa_{4}^{\omega}(2\pi\int_{-\pi}^{\pi}a(\wedge x)f(x)dx)^{2}$and$\mathrm{L}\mathrm{R}$$\kappa_{4}^{\omega}$is the fourth cumulant of$\{\omega_{t}\}$.

Now, to classify $\mathrm{Y}=(Y_{1}, \cdots,Y_{n})’$ into one of two hypothesestoif, we adopt the following rule: assign$\mathrm{Y}$$H_{1}$

and$H_{2}$, instead of using$H_{1}$$[\mathrm{Y}’A\mathrm{Y}-E_{1}(\mathrm{Y}’A\mathrm{Y})]^{2}<[\mathrm{Y}’A\mathrm{Y}-E_{2}(\mathrm{Y}’A\mathrm{Y})]^{2}$, otherwise. Setting $m_{j}=(m^{(j)}, \cdots, m^{(j)})’$ and $\nabla_{j}=m_{j}’Am_{j}$ , without loss of generality,totoif, then the classification rule can be written as: assign
we assume that
$H_{2}$$\nabla_{1}>\nabla_{2}$$D=2\epsilon’A\epsilon+4m_{j}’A\epsilon+2\nabla_{j}-(\nabla_{1}+\nabla_{2})-2\mathrm{t}\mathrm{r}(A\Sigma)>0$$\mathrm{Y}$$H_{1}$

176

to$H_{2}$, otherwise.

[Theorem 4.1] Suppose that the stationary linear disturbancesin regression models (4.1)
satisfy (4.2). Then under Assumptions,$0\leq\rho<1$ , (i.e., $\wedge a(x)=g(x)$ is a short-memory spectral density), then(i) If$\{\epsilon_{t}\}$$(\mathrm{G}.9)-(\mathrm{G}.11)$$|a_{7}\cdot|=O(\rho^{7^{-}}),$$\frac{D-E_{j}D}{n^{\frac{1}{2}+d}}arrow N\mathcal{L}(0,16(m^{(j)})^{2}G_{fg})$, $j=1,2$

(4.6)

where

.

$c_{fg}= \frac{2\pi f^{*}(0)g^{2}(0)\Gamma(1-2d)}{d(1+2d)\Gamma(d)\Gamma(1-d)}$

(ii)

If

$\gamma<-\frac{d}{2}$, then$\frac{D-E_{j}D}{n^{\frac{1}{2}}}-^{\mathcal{L}}arrow N(0,4\sigma_{a}^{2}),$$j=1,2$(4.7)[Remark 4.1] Theorem 4.1 holds only for the long-memory $(d\neq 0)$ processes. From thistheorem, we can see that the distribution of $D$ depends heavily on the selection of 7. When$\gamma=0$ , that is
is a positive and continuous function on $[-\pi, \pi],$ $D$ becomes a robust statisticwith respect to non-Gaussianity of.
The following theorem and proposition follow from Theorem 4.1.
$\wedge a(x)$$\{\epsilon_{t}\}$[Theorem 4.2] Suppose that the stationary linear disturbancesin regression models (4.1)
satisfy (4.2). Then under Assumptions,(i) If $\wedge a(x)=g(x)$ , where $g(x)$ is a positive and continuous function on $[-\pi, \pi]$ , then$\{\epsilon_{t}\}$$(\mathrm{G}.9)-(\mathrm{G}.11)$,$P_{D}(2|1)+P_{D}(1|2)=1+ \Phi(-\frac{n^{-\frac{1}{2}-d}(\nabla_{1}-\nabla_{2})}{4|m^{(1)}|\sqrt{G_{jg}}})-\Phi(\frac{n^{-\frac{1}{2}-d}(\nabla_{1}-\nabla_{2})}{4|m^{(2)}|\sqrt{c_{fg}}})+o(1)$

(ii)

If

$\gamma<-\frac{d}{2}$(4.8), then$P_{D}(2|1)=P_{D}(1|2)= \Phi(-\frac{n^{-\frac{1}{2}}(\nabla_{1}-\nabla_{2})}{2\sigma_{a}})+o(1)$

.

(4.9)

Next, let us consider the classification effects of $D$ under contiguous condition$H_{1}$:$m^{(1)}=m_{0}$$H_{2}$:$m^{(2)}=m_{0}+Hn^{-\frac{1}{2}+d}$;(4.10)$\{$.

[Proposition 4.1] Suppose that the stationary linear disturbancesin regression models(4.1) satisfy (4.2). Then under Assumptions
and contiguous condition (4.10), if$\wedge a(x)=g(x)$ , then$\{\epsilon_{t}\}$$(\mathrm{G}.9)-(\mathrm{G}.11)$$\lim_{narrow\infty}P_{D}(2|1)=\lim_{narrow\infty}P_{D}(1|2)=\lim_{narrow\infty}P_{I_{1}}(2|1)=\lim_{narrow\infty}P_{I_{1}}(1|2)=\Phi(-\frac{|H|}{2\sqrt{c_{f}}})$.

(4.11)

177

5 APPENDIX: PROOFS OF THEOREMS
Since the mean and variance of LR are[Proof of Theorem 2.1]

$E_{j} \mathrm{L}\mathrm{R}=\frac{(-1)^{j}}{2}(\beta^{(2)}-\beta^{(1)})’X’\Sigma^{-1}X(\beta^{(2)}-\beta^{(1)})=(-1)^{j}\mu$

Varj
where

$E_{j}$,$\mathrm{L}\mathrm{R}=(\beta^{(2)}-\beta^{(1)})’X’\Sigma^{-1}X(\beta^{(2)}-\beta^{(1)})=v^{2}$and Varj stand for the expectation and variance under$\mathrm{L}\mathrm{R}\sim N((-1)^{j}\mu, v^{2}),$$H_{j}$$j=1,2$ ,,. Hence$j=1,2$ ,we have$P_{\mathrm{L}\mathrm{R}}(2|1)=P_{\mathrm{L}\mathrm{R}}(1|2)= \Phi(-\frac{\mu}{v})=\Phi(-\frac{1}{2}\{(\beta^{(2)}-\beta^{(1)})’X’\Sigma^{-1}X(\beta^{(2)}-\beta^{(1)})\}^{\frac{1}{2}})$.

From Grenander and Roseblatt (1957), we have$\lim_{narrow\infty}D_{n}^{-1}X’\Sigma^{-1}XD_{n}^{-1}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f^{-1}(\lambda)dM(\lambda)=\Omega$,that is$P_{\mathrm{L}\mathrm{R}}(2|1)=P_{\mathrm{L}\mathrm{R}}(1|2) \sim\Phi(-\frac{1}{2}\{(\beta^{(2\rangle}-\beta^{(1)})’D_{n}\Omega D_{n}(\beta^{(2)}-\beta^{(1)})\}^{\frac{1}{2}})$.

By Assumption (G.1),, for$\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(2|1)=\lim_{narrow\infty}P_{\mathrm{L}\mathrm{R}}(1|2)=0$[Proof of Theorem 2.2]$\beta^{(2)}\not\equiv\beta^{(1)}$.

From the proof of Theorem 1, we have

$P_{\mathrm{L}\mathrm{R}}(2|1)=P_{\mathrm{L}\mathrm{R}}(1|2) \sim\Phi(-\frac{1}{2}\{(\beta^{(2)}-\beta^{(1)})’D_{n}\Omega D_{n}(\beta^{(2)}-\beta^{(1)})\}^{\frac{1}{2}})$,hence, putting contiguous condition (2.2) into the above, we get (2.3) immediately.
[Proof of Theorem 3.1]

(i)

From Lemma 3.1, we have

$P_{\mathrm{L}\mathrm{R}}(2|1)=P_{\mathrm{L}\mathrm{R}}(1|2) \sim\Phi(-\frac{1}{2}\{(\beta^{(2)}-\beta^{(1)})’\overline{D}_{n}\overline{D}_{n}(\beta^{(2)}-\beta^{(1)})\}^{\frac{1}{2}})$

,

since
$\frac{\chi_{r}^{1/2}(n)}{n^{d}}\sim\frac{n^{r-\frac{1}{2}-d}}{(2r-1)^{1/2}}arrow\infty$, as$narrow\infty$

for $r=1,$ $\cdots,p$ , and considering Assumption (G.1) for $r=p+1,$,$\cdots,$$k$, we have the result.

178

(ii)

We first calculate the mean and variance of$I_{\alpha}$

under hypotheses$E_{j}I_{\alpha}=(-1)^{j}(\beta^{(2)}-\beta^{(1)})’(\beta^{(2)}-\beta^{(1)})$$H_{j}$,, $j=1,2$

and$\mathrm{V}\mathrm{a}\mathrm{r}_{j}I_{\alpha}=4(\beta^{(2)}-\beta^{(1)})’[\alpha^{2}(X’X)^{-1}X’\Sigma X(X’X)^{-1}$$+(1-\alpha^{2})(X’\Sigma^{-1}X)^{-1}](\beta^{(2)}-\beta^{(1)})$$\sim 4(\beta^{(2)}-\beta^{(1)})’\overline{D}_{n}^{-1}F_{\alpha}\overline{D}_{n}^{-1}(\beta^{(2)}-\beta^{(1)})$$P_{I_{\alpha}}(2|1)=P_{I_{\alpha}}(1|2) \sim\Phi(-\frac{(\beta^{(2)}-\beta^{(1)})’(\beta^{(2)}-\beta^{(1)})}{2\{(\beta^{(2)}-\beta^{(1)})’\overline{D}_{n}^{-1}F_{\alpha}\overline{D}_{n}^{-1}(\beta^{(2)}-\beta^{(1)})\}^{\frac{1}{2}}})$

since all the elements of$\overline{D}_{n}^{-1}$tend to,under (G.1) and (G.5) as $narrow\infty$ , we get the result.

$0$[Proof of Theorem 3.2] Under contiguous condition (2.2), the fact for LR is clear, we only
prove (3.13). From the proof of Theorem 3.1, we know that under (2.2),$P_{I_{\alpha}}(2|1)=P_{I_{\alpha}}(1|2) \sim\Phi(-\frac{1}{2}\frac{H’\overline{D}_{n}^{-2}H}{\{H’\overline{D}_{n}^{-2}F_{\alpha}\overline{D}_{n}^{-2}H\}^{1/2}})$.

Since $k_{1}HH’\geq F_{\alpha}\geq k_{2}HH’$ , we have.

$k_{1}\{H’\overline{D}_{n}^{-2}H\}^{2}\geq H’\overline{D}_{n}^{-2}F_{\alpha}\overline{D}_{n}^{-2}H\geq k_{2}\{H’\overline{D}_{n}^{-2}H\}^{2}$

Hence$\Phi(-\frac{1}{2\sqrt{k_{1}}})\geq\lim_{narrow\infty}P_{I_{\alpha}}(2|1)=\lim_{narrow\infty}P_{I_{\alpha}}(1|2)\geq\Phi(-\frac{1}{2\sqrt{k_{2}}})$[Proof of Theorem 4.1]

(i)

Under Assumptions (G.9) and (G.10), we$E_{j}[\epsilon’A\epsilon-\mathrm{t}\mathrm{r}(A\Sigma)]=0$$\mathrm{V}\mathrm{a}\mathrm{r}_{j}[n^{-\frac{1}{2}-d}\epsilon’A\epsilon]=n^{-2d}\sigma_{a}^{2}+o(n^{-2d})$,.

$\mathrm{h}a\mathrm{v}\mathrm{e}$,(from Lamma 4.1).

Thus$n^{-\frac{1}{2}-d}(D-E_{j}D)=n^{-\frac{1}{2}-d}4m_{j}’A\epsilon+o_{p}(1)$$:=4m^{(j)}n^{-\frac{1}{2}-d}U’A\epsilon+o_{p}(1)$where $U=(1, \cdots , 1)$ ’. Define$\epsilon_{t}\sim=\sum_{k=-\infty}^{\infty}a_{k}\epsilon_{t+k}$. Then we can show that$n^{-\frac{1}{2}-d;\sim}E|U’A \epsilon-U\epsilon|=n^{-\frac{1}{2}-d}\sum_{t=1}^{n}\sum_{|l|>n}|a_{l-t}|E|\epsilon_{l}|arrow 0$

where$\overline{\epsilon}=$

.

,

as

$narrow\infty$

Note thathas the spectral density. Thus applying Yajima(1991) to, we obtain$(\overline{\epsilon}_{1}, \cdots , \overline{c.})’n$$f(\lambda)g^{2}(\lambda)$$\{\overline{\epsilon}_{t}\}$$n^{-\frac{1}{2}-d\sim}U’\epsilon$$\frac{1}{2\pi}\sum_{-\infty}^{\infty}a_{\eta^{-}}e^{-i\tau\lambda}$$n^{-\frac{1}{2}-d}(D-E_{j}D)-^{\mathcal{L}}arrow N(0,16(m^{(j)})^{2}G_{fg}),$$j=1,2$ .

with $g(\lambda)=$179

In this case, since(ii)$n^{-1}E_{j}|U’A\epsilon|^{2}=n^{-1}U’A\Sigma AU=O(n^{2\gamma+d})=o(1)$

,

(From

$2\gamma+d<0$ ).

Then$n^{-\frac{1}{2}}(D-E_{j}D)=2n^{-\frac{1}{2}}[\epsilon’A\epsilon-\mathrm{t}\mathrm{r}(A\Sigma)]+o_{p}(1)$$-^{\mathcal{L}}N(0,4\sigma_{a}^{2})$, (from Lemma 4.1).

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