Maximum likelihood estimation for a fractionally differenced autoregressive model on a two-dimensional lattice

ELSEVIER

Journal of Statistical Planning and Inference 44 (19951 219-235

journal of statistical planning and inference

M a x i m u m likelihood estimation for a fractionally differenced autoregressive model on a two-dimensional lattice S. Sethuraman, I.V. Basawa* D~Tartment ~['Statistics, The University ~?fGeorgia, 204, Statistics Buihling, Athens, GA 30602-1952. ['SA

Received 20 July 19!)2; revised 14 October 1993

Abstract A two-dimensional time series model with long-memory dependence is introduced. The model is based on a fractionally differenced autoregressive process (long-memory) combined with a standard pth order stationary autoregressive process (short-memory). The maximum likelihood estimators for the parameters in the model are derived and their asymptolic distributions are obtained. A novel feature of the derivations is that a new two-dimensional martingale representation is used. A M S Subject ClassOqcation: 62M 10, 62M05 Key words: Time series; Lattice processes; Long-memory dependence; Maximum likelihood estimation; Asymptotic inference

1. Introduction A n d e r s o n (1978) has discussed p r o b l e m s of inference b a s e d on several i n d e p e n d e n t autoregressive processes. B a s a w a et al. (1984) a n d B a s a w a a n d Billard (1989) have studied large s a m p l e inference for regression m o d e l s with multiple o b s e r v a t i o n s a n d a u t o c o r r e l a t e d errors. K i m a n d B a s a w a (1992) c o n s i d e r e d e m p i r i c a l Bayes e s t i m a t i o n based on d a t a from several i n d e p e n d e n t autoregressive processes. In all the p a p e r s cited above, the o b s e r v a t i o n s at each time p o i n t are a s s u m e d to be i n d e p e n d e n t , a n d the d e p e n d e n c e extends only a l o n g the time axis. H o w e v e r , in s o m e situations, when i n d i v i d u a l s at each time p o i n t belong to the s a m e family or share the s a m e c o m m o n e n v i r o n m e n t , the o b s e r v a t i o n s on these i n d i v i d u a l s at a n y given time p o i n t w o u l d

* Corresponding author. 0378-3758/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved. SSD! 0 3 7 8 - 3 7 5 8 ( 9 4 ) 0 0 0 4 6 - 8

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S. Sethuraman, I.K Basawa/Journal of Statistical Planning and Inference 44 (1995) 219-235

generally be correlated. It will therefore be useful to consider models which permit dependence between time points as well as within the group of individuals at each time point. In this paper, we consider such a model and study the asymptotic properties of the maximum likelihood estimators of the model parameters. Most of the standard time series models assume that the observations separated by a long time lag are nearly independent. Such processes exhibit a short-memory dependence. In many empirical problems arising in fields such as econometrics, geology, hydrology and oceanography, the autocorrelations with long time-lags are not negligible, thus necessitating the study of long-memory time series models. If p(k) denotes the autocorrelation function with lag k in a stationary process, the process oo is said to have a long memory if Wk=_~lp(k)]=oo and a short memory if Zk°~- -o~ [p(k)[ < ~. See Granger and Joyeux (1980), Lawrance and Kottegoda (1977) and Greene and Fietlitz (1977, 1979) for applications of long-memory models. Fractionally differenced autoregressive models provide a rich class of long-memory models. Hosking (1981), Yajima (1985, 1991) and Fox and Taqqu (1986) have studied various aspects of the asymptotic estimation problem for one-dimensional fractionally differenced models. In this paper, we introduced a two-dimensional process based on a fractionally differenced autoregressive process. In Sections 2 and 3, we introduce the model, derive the likelihood function under the assumption that the errors are normally distributed and establish the asymptotic normality of the maximum likelihood estimators of the parameters. Inference problems for two-dimensional processes with long-range dependence have not as yet been addressed in the literature, to our knowledge. Stationary processes on a plane and related inference problems have been studied by several authors including Whittle (1954), Tjostheim (t978, 1983), Dahlhaus and Kunsch (1987), Guyon (1982) and Martin (1979, 1990). None of these papers, however, involve long-range dependence. Consider a lattice process {Xts, (t, s)~Y 2 }, where 7/denotes the set of all integers, {0, _+ 1, _+2 .... }. It will be assumed that { Xts} is stationary with a linear (quadrant) representation

Xts = ~ j=O

~ ~(k,j)~t-k,s-j,

(1.1)

k=O

where { ~ij} is a two-dimensional white noise, and c((k,j) are non-random coefficients which depend on a finite number of parameters. For simplicity, we consider the special case where ~ (k,j) is factorizable, i.e. ~ (k,j) = qk qJj, where {qk } and { ~bj} are determined by a fractionally differenced and a standard autoregressive processes respectively. We further assume that { ~ij} are Gaussian random variables. Let Fij denote the ~-field generated by {Xt~, t<~i and s<~j}. Given any Xt~, we shall visualize the 'past' as F t - 1 , s VFt, s - 1. The likelihood function based on the sample {Xt~, t= 1..... m and s = 1..... n } is then obtained in terms of the prediction errors of Xt, given the 'past' and the corresponding prediction error variances. In the derivations of the asymptotic distributions of the maximum likelihood estimators, we use a two-dimensional

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and lnjerence 44 (1995) 219 235

221

martingale approach which is of interest on its own. Our method of deriving the likelihood and the use of the martingale approach in asymptotics are both general enough to be useful for any general lattice process (not necessarily factorizable) having a quadrant representation. It is to be noted that, even in the context of a onedimensional fractionally differenced process, our martingale technique provides a new powerful tool.

2. The model and the likelihood function

Consider the factorizable model on a two-dimensional lattice specified by

Vd,O(Bs)Xts=¢ts,

0 < d < 0 . 5 , t , s = 0 , + l , + 2 .....

(2.1)

where V~=(1-B~) d= ~ 7r~d)B~ k-O

with P

,,

i

1

Bt and Bs are backward-shift operators defined by BIXts=X~ i.~ and BJXts=X,.~_j, respectively. We assume {~,~} are independently and normally

and

distributed random errors with mean zero and variance ~2. It is also assumed that O(z)=

(2.2)

1 - ~ Oizi 4:0, i=1

/

for [z[~
(2.3)

Y,~= O(B~)X,s.

(2.4)

where

From (2.2) and (2.4), we have X , , = ~ _jO (°) Y,,_i,, j=o

~ 10}°)[< ~c,

(2.5)

j=o

where q ' ( z ) = l / O ( z ) = ~ Oj(o)zj j=o

and

0=(01, .... Op)r.

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S. Sethuraman, I.V. Basawa/Journal of Statistical Plannin9 and Inference 44 (1995) 219-235

Equation (2.3) implies that

Y,,= ~ r/~a)~,-k.s,

~ [r/~a']2< o0,

k=0

(2.6)

k=O

where q(#~=F(k+d)/{F(k+1)F(d)} and F ( . ) denotes the gamma function. Note that, from (2.5) and (2.6),

X,s=~

~,(~),,,< "lk "?'jo) ~ t - k , s - j ,

(2.7)

j=Ok=O

We have thus shown that Xt~ has a linear and quadrant representation. See Tjostheim (1983). In addition, the model is factorizable (see e.g. Martin (1979)). Consequently, Cov(X,-X,:,) =°~

~o \j=O

"~{ V

.~").~")

q'j v ' j + l s - s l l / ~ k % "lk "tk+lt-t~l

.(d) ~,~-sllYl,-t,

_ ~ 2 ~(0)

-u

,~,~0~,/,~0~

(say).

(2.8)

It is to be noted that the model has a long-memory in t and short memory in s. Our goal is to estimate d, 0 and a z based on the sample X = ( X t s , t = 1..... m; s= 1..... n). The likelihood function of X is seen to be L = (2r~)-,."/2 [Sx [- 1/2 exp { -- 2-1 X r Sx 1X },

(2.9)

where Sx is the variance-covariance matrix of X determined by (2.8). The direct calculation of ISxl and Sx 1 can be avoided by expressing Sx in terms of the one-step predictors and their mean squared errors which are easily calculated recursively using the algorithms given in Brockwell and Davis (1987). To this end, we proceed as follows. Let F u denote the a-field generated by the random variables {Xt~, t ~
E(X,~Ipast)=E(X,~IF,_I.s)+E(X,~IF,,~_I) -E{E(X,,IF,_L,)IF,_I,,_a

}.

(2.10)

It is seen that the last term on the right-hand side of (2.10) is equal to E(Xts I Ft- 1,~- 1) since Ft-~,~-1 ~ F, a,~. Note that the conditional expectations on the right-hand side of (2.10) can be carried out using one-dimensional recursive 'innovations algorithms' described by Brockwell and Davis (1987). Let b(a. ts 0) = Xt~ - E (Xt~ I past)

(2.11)

denote the prediction error. The prediction error variance is given by Var(bts) = trZu~ 1 r~°)-a,

(2.12)

S. Sethuraman, I.V. Basawa/Journal of Statistical Planniny and In[erence 44 (1995) 219 235 223

where ul d~a and r~°2 ~ are as defined in the Appendix. It can further be verified that { bt~ ] and {bc~,} are o r t h o g o n a l for t # t ' or s#s'. We can finally rewrite the likelihood function in (2.9) in terms of the prediction errors {b,~} and their variances, as follows: - n~2

× exp

} - m/2

~'~j ~ /q3 ~ t_(2o.2)_1i_~1 ~ /[-fh(d,O),2 / ,,~-~/ i" j=l [_"i l'j 1~)

(2.1 3)

Let L1 = ( n m ) 1/nL. The m a x i m u m likelihood (ML) estimators (d, (~, 0 2) are the values of d, 0 and a 2 which maximize L1 and are obtained as a solution of~L1/O[3=O, where [3r = (d, 0 r, cr2 ). In the next section, we shall study the asymptotic properties of the M L estimators.

3. The joint asymptotic distribution of maximum likelihood estimators Let fl=(fll,flT,f13)T=(d, OT,~72)T and fl = (ill, fiT,/~3)T = ( 6~,0T, 0"2)T. Let flo = ( d o , 0oT, ~roZ)w be the true value of ft. In order to simplify the notation, we shall use the same notation [3 to represent the true value flo. Consider the likelihood equation: "

O=(mn)l/2~

_

--(mn)

1/2cL[

~2L1

gfl + ( (~f12 ~.)(mn)

1/2

(fi--fl),

(3. l,

where Ifl*-fll<~bfi-[31, OL1/tg[3 refers to the vector of derivatives of L1 with respect to [3 and ~2L~/~[32 I~* denotes the matrix of second derivatives evaluated at [3=[3*. Equation (3.1) yields

lmnt

[31;

Ir

3.:'-t

Let C be the p a r a m e t e r set defined by C={0e~P:O(z)#0

forlzl~
where 0 = ( 0 1 , 0 2 . . . . . 0p). Note that 0 can be expressed as a continuous function (O(zl,z 2. . . . . zp)) of the zeros zl,z2 ..... z~ of O(z). The p a r a m e t e r set C is therefore the image under 0 of the set {(z~ . . . . . zp): Iz~l>l, i = 1 . . . . . p}. Let deD where D=[6,½-6] with 0 < 6 < ¼ and 0 < a 2 < ~. The following t h e o r e m gives the limit distribution of ft.

Theorem 3.1. As m ~

and n --* ~, we have

( x / ~ ) ( d _ d , ( O _ O ) V , o 2 _ 6 2) ~-~ Np+ 2(O,~),

224

S. Sethuraman, I.V. B a s a w a / J o u r n a l o f Statistical Planning and Inference 44 (1995) 2 1 9 - 2 3 5

]

where

V6/~2

o

=LOo

V-I 0

20-4

and V is the p × p variance-covariance matrix of a stationary AR(p) process, P

Zt= ~ OiZt-i+et, t = p + l . . . . . 2p, i=1

with {e,} i.i.d. N(0, a2). Proofi We shall show that as m ~ ~ and n ~ ~ ,

( aeL1 ) .... ,Z#*

(a) \ - ~ 5 -

(b) x / ~ a L 1 / a f t

d

N(0, Z -

1).

Then using Slutsky's theorem, in conjunction with (3.2), we get the required result. F r o m now on, for notational convenience, let us suppress the dependence on d and 0 and denote ~t~' td 0) , hta, ~t~ o), u~)- 1 and r~°-) 1 by ~t~, bts, ut- 1 and r~_ l, respectively. We shall prove part (a) for one term only. The proofs for the convergences of the other terms follow along similar lines.

o2L, ,.~,

[Or~_,][Or~_,]" , O~r~_,}

ao~=2-ni~,(~T-,_,L~JL 1

L aO J

mno'2 t = 1 s = 1

1

+

ao

bit- 1 rs- 1

J r~_, ~ ao .[.

vts tit- 1 rs- 1

~ ¢[ ( -_,~ k abt~ so F ars_ 17 T

t ab's [Or~_ll T b,s voa0 J

J +.,-lr.-,L

1

2 02r,_a

+~b,, ~

bt2 [Or,_,lFOr,_,lT t ao jL-~- j ."

u,_,r~_,L

(3.3)

By Lemmas A. 1-A.6, in the Appendix, it follows that the second term of (3.3) converges to 1

02411

and the other terms converge to zero as m --*~ and n ~ , O~C. Hence, as m - - * ~ and n ~ ,

002

0-2

uniformly in d~D and

002 J ) '

(3.4)

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S. Sethuraman, I.V. Basawa/Journal ol Statistical Planning and lnlerence 44 (1995; 219 235

Since E[(~{ll/~O)((~ll/Gqo)T]=~2V --*m and n --+m,

and

E[~11(82~11/(~02)3~-0,

we

obtain

as

m

(?2L 1

....

(3.:5)

902 Similarly, as m --+ oc and n -+ <~, [ e I(0~11~2\~/ + E (

....

i}2Ll~d 2

82911~J-n2%

since E [ { l l ?,2¢1ffSd2] =0 and E({3~11/8d) 2 =0.2 n2/6, see, for instance, Yajima (1985). Also, t~2L1

('~( 0.2 )2

a.s.

' (20"4)- 1 -- 0.- 6 E ( ~ 2 1 ) = -- (2°'4) - 1 ,

and t'~2L 1

~2L1

80 8d'

tgd (?0.2

and

~2L 1 8 0 ?G 2

all converge almost surely to zero. Consequently, as m --+ oo and n -+ oc, ?2L I a* a.s. __S_1

(3.6)

~f12 Hence part (a) is proved. T o prove part (b), let us c o n s i d e r

X~T(?L1/@fl,

where

,/I,T=[,~.l,/~T,23]

with

/ ] ' 2 = ( 2 2 1 ,/~22 . . . . . 22p) T" ~

,,~X (")'L 1

eft

= _(mn)- U2

~ ,_q~,t~,~-2l + t = 1 s = 1 (0.211 t _ 1 Fs 1

-I-232

10.-211--O" 2 Ut

20.2~

t= l s= l

1

b2sl rs- 11}

j

-t

,~2,,t, ,~0 (72 bit- 1 rs

u2_tr,

lrj

1

~nO

T (?rs- 1 1

(?W~- q

l l t - l-77.2 l's 1

(3,7) '

Using L e m m a s A.1 and A.2 and the fact that E(b2,,)<~E(X2,), it can be shown that the second and third terms of (3.7) converge to zero in probability as m --+ m and n ---,:c~.

226

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and Inference 44 (1995) 219-235

Hence, consider gts ,

%~t=ls=l

where

gt~=

--21

'~t~ :a

~'t~-~2_,~0

0"2 U t - 1 r s _ 1

0"2Ut_lrs_ 1

1

(3.8)

20"2\tYZUt_srs_l

Set Hm,,= ~

~ gts-

(3.9)

t=ls=l

Let o~,.. be the a-algebra generated by { X,s, t -%
)

E(H,../~ij)=HI~= ~ ~ gt~,

where

(i,j)<~(m,n).

(3.10)

t=ls=l

We shall prove this result for the case i = m - 1 and j = n - 1. The proof of the result for other values of i and j follows along similar lines. Note that H,.. can be written as m--1

H,..=H.,-1,.-I+

g,.~+ ~. gt.. s-1

(3.11)

t=l

Consequently

,.

)

31.,

The proof is complete once we show that m--1

=

o~

,. ,)__o

(3.13)

We shall first show that

E(s~=lgms/o~m-l,n-, ) = 0 . Since ~ , . _ 1 , . ~ m _ 1 , . _ 1 ,

(3.14)

we have

E( ~=lgm./~m-l,.-1)=E[E( ~=lgmJ~m-l,.)/~m_l,._, 3. Ob,.JOdis

Observe that Also, note that

~ . , _ a , . - m e a s u r a b l e and

E(b~s/~m- x,.) = 0"2Urn- 1 r~_ 1

E(bms/~m-l,.)=O, for

for s = 1. . . . . n.

(3.15) s = 1. . . . . n.

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and ln/erence 44 (1995) 219 235

2127

Consequently, by the definition of gins in (3.25), we have E

gm~/.Y',,-1,n s

x2 -bms2 -

= --E

1

s = l (72blrn

lrs

(?bins / ~ r n 1 ~ 0 /'

1,n

•

(3.16)

F r o m (3.15) and (3.16), we have E s 1

Oms/~m_l,n_l = E

Since ~ , . , . - 1 ~ . , - 1 , n - 1 , we have

Fgms/g,, s 1 0-2 Um - 1 Fs 1

(3.17)

~,# ~

(~0

"

by the s m o o t h i n g p r o p e r t y of conditional expectation.

crrj ./~??/

l,n

l

~ E

Next observe that 6bms/O0 is ~m,,._l-measurable s = 1. . . . . n. Consequently,

and

(72/'1m 1G 1 ~ v / / ~ " ' =

s=l

for

E(bmj~m,n-1)=O.

{3.19)

F r o m , (3.17) (3.19), we arrive at the result stated in Eq. (3.14). Similarly, we can show that E

gt~/~,,-1,,-i

=0.

\t=l

Hence { Hm,, ~mn } is a martingale on a two-dimensional lattice. N o w define K mn--L~t=l~n=l 2 --Win E [ g t2,,/~~ ' ~ _ l , n _ l ] 2 2 and Smn=E(Kmn). Using facts that E( ? In L/~d) 2 = -- E( ~,2 In Lfi~d 2 )

the

(3.20)

and [ ? l n L aln L ]

EL

[-a21n L 7

a,Bj

it can be readily shown that as m -0 m and n --+ o-,:, p - l i m ( m n ) - 1 K i2n ,--- lira (ran) -1 Sran 2 = ) , 2 ( 7 - 2 E ( ( ? ~ 1 lloyd)2 + f c T V ~ 2 +22(2(74) 1.

(3.21)

228

S. Sethuraman, l.F. Basawa/Journal of Statistical Plannin9 and Inference 44 (1995) 219-235

We shall verify Eq. (3.21) for one term only. That is, we shall show that as m ~ o e and n ----~o o ,

a-2(mn)-' ~ L E[ -2,bu ~bul2 2~r_2E(~_) 2.

(3.22)

LUt-lrs-1 ad A

,=1,=1

The proofs for the convergences of the other terms follow along similar lines. Using the result in (3.20), we have

a_Z(mn)-i 2- E. e / -F- ---- 2 abt, 0b,sl2 ,=1,=1 LU~_lr,_l ad J n V E'ab"'z

g

/ "'

=2~(mn~r2) - ' 2-.. ~

/

L as°t"

--

+ E { v,,~

t=l,---_aLU,-lr,-x

~-) au,_ I _E(b,, _ ao,__

\ut-lr~-l/

U2- 1 r s - 1

]

.

(3.23)

ad

Since E [ b,s(~?2bt,/ dd 2 )] = E [ b,,( Ob,,/8d)] = 0, Eq. (3.23) becomes

"

"

F 2,b,s

abul 2

~r-2(mn)-' ~=,~=,ELU-_,r,----_,ad J = - -

~ , ~d ,

m n o "2 t = l

bit-Its-1

s=l

, ~,~-2 WEaGl/aCl]~,

(3.24)

by monotone convergence theorem. Next observe that

L

-'

t=ls=l

[

= e2(SZ""] z kmn/

1-1

(ran) -z ~ L E(94) • t=~,=l

L

t=ls=l

(3.25)

Since the process { Xt,} is stationary and ergodic, by ergodic theorem, we have as m--,oo and n ~ o o ,

( mn)-a ~ L 9~

....

) E(gll )

t=ls=l

and hence (ran) -2 ~', L 34

.... , 0.

(3.26)

t=ls=l

By monotone convergence theorem, in conjunction with (3.24)-(3.26), we conclude that as m ---,oo and n --.oe,

t=l

s=l

S. Sethuraman, I.V. Basawa/dournal of Statistical Planning and Inference 44 (1995) 219-235

229

Since { H,.,, ~ , . . } is a martingale on a two-dimensinal lattice and (i) K2,S~, 2 L 1

/=ls=l

it follows that S~, 1

gt~--*N(0,1)

as m ~ o o

and n--.oc.

(3.27)

t=ls=l

The results stated in (3.27) can be proved via a direct extension of the martingale central limit t h e o r e m of Brown (1971, p. 60). Hence we obtain the desired result.

Appendix: Some properties of predictors and prediction error variances Let

X,~=E(X,~IF, 1,s), ~Gs=E(X,~IF,,~-I), u ~ ) = ~ o ( ~ / ! a ) ) 2, and 7 ~ ) = ~ = o

(~!°)) 2. Define

u~=E(X,~-R,,)2/(a2~,~?),

t~2

(0)l =E(Xts_~ts)2/(a2 rs-

2<~s<~p

g(Od)),

and r~°~= l for all j ~>p. It can then be shown that Var(b,~)

=

G2u~d_) 1 r ~ 1,

where b,~ is as defined in (2.11). The following technical lemmas are needed in the p r o o f of the a s y m p t o t i c normality of the M L estimators ( T h e o r e m 3.1). L e m m a A.1. We have, ul~ ~ rs-l"(°)__. 1

as t ~ oo and s --* ~x),

uniformly in d~D and OEC. Proof. First note that tr- z E { [ b~a' 0)] 2 ~f -_,,~a) - ~ t - l r s - - 1,~o) Yajima (1985, p. 309) showed that

•

U~d)-1 = { r ( t + 1 -d)}-2 { F ( t + 1 ) r ( t + 1 - 2 d ) } ,

230

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and Inference 44 (1995) 219-235

and 0 ~>.p. Hence, it readily follows (see, Remark 3 of Brockwell and Davis (1987), p. 169) that r~l ~1

as s - ~ ,

uniformly in OeC.

Hence we have the assertion.

Lemma A.2. (i) O(i)u~d)_1/~dti) = O ( t - 1), i= 1, 2, uniformly in d~D.

(ii) ~t°r~°)_ 1/(;300) ---~o( 1 ), as s--*~, uniformly in OeC, for i= 1, 2. Proof. Since u~a)_1 = [ F ( t + l - d ) ] (~=2u~1

-2 { F ( t + 1 ) F ( t + 1 - 2 d ) } , we have

{G(t+ 1-d)-G(t+

1-2d)}

where G(z)=~lnx/Oz.

We have

c~u~a)-~ <~Mt-~

where M is a constant independent of d.

Similarly we can show that ~2U~d-) 1

0d z - O ( t - 1 ) • Hence the proof of part (i). Part (ii) follows directly from L e m m a A. 1. Hence the proof is completed. L e m m a A.3. Let

~,~("'°)--vdO(Bs)X~=--.

-

~, ~(kd)OjX,-k,~-~ k=oj=o

where 0 o = - 1. Then, (mn)-i ~

~ [~,a~0)]: .... , Ee#(la,'l°)] z

as m --*~ and n --*~,

t=ls=l

uniformly in d~D and O~C. Proof. Note that {Xt~} is regular since it can be expressed in the linear form

/=Ok=O

S. Sethuraman, I.V. Basawa/Journal o[' Statistical Planninq and lnlerence 44 (1995) 219 235

25,1

where q~,e)= n~-a). Since { Xts } is regular and strictly stationary, it follows that [ Xt~I is ergodic and hence {~t~ ~(a, 0)~, is also ergodic. By the ergodic theorem we have as m ~ ~ and n ~ ~,,

t=ls=l

for any d6D and OeC. The uniformity in proved as follows. We observe that

Irc~a)l<~[F(-d)kl+a]-I

d~D and O~C of the

above convergence is

for k = 1 , 2 , 3 . . . . .

Let 1

M -- sup

a~Dr(--d)

where the g a m m a function 'St ~ ' e

F(x)

is defined as

tdt

x>0,

x-'F(l+x)

X=0, x<0.

Y(x):= 3¢o Hence,

I~a)l<~Mk (l+d)<~Mk-"+~)

fork=l,2,3 .....

(A.I)

Next note that O(z) can be expressed as p

O(z)-- [I (1- - Z i i

1 Z),

(A.2)

1

where Izil > 1, for i = 1. . . . . p. Let { Mi, i = 1, 2, 3, 4, 5 } be positive constants being independent of d and 0. F r o m (A.1), (A.2) and the fact that la-bl<
k~orC~a,{i=Q(l_zF1B~)} Xt k,s

~<{ k=L'~ku)'(14-BYlX,-k,~'} p

E

2 IX,-k.s-,ik

i=0

i=l

k=l

and

i=1

i=lk=2

iA.:;t

232

S. Sethuraman, I.V. Basawa/Journal of Statistical Plannin9 and Inference 44 (1995) 219-235

Similarly,

l a~#~°)/a01 ~
where X t - k = ( X t - k , s - 1 . . . . . Xt-k,~_p) t. The rest of the proof of the lemma can be completed by following the arguments in L e m m a 3.1 of Yajima (1985). L e m m a A.4. Let rTAa, o)_ ~h(d, O)" Then rr t , $ o) -_- ~ t;(a, ,8 t,8 l m ~ =1

w(d.O)l 2

....

0,

as m ~ o o and n --.oo, uniformly in d e D and OEC.

Proof. Define t-I W ~ / ' 0) .... = -

,

p

L '~'~i - ( " ' - ~ 'T ~ ' t l l t - l , i J'O I X t - i , s - I

2

i=0/=0

and

Wt,s,d,O)2 --

--

~ ~ ~Id) OiXt_i,s_l. i=tl=O

Then

t=ls=l

t=is=l

First we shall show that as m ---,oo and n --*oo, rr t, S, 21

)0

t=ls=l

uniformly in d e D and OeC. F r o m (A.3), we have

j=tl=O

t-j,s-I J

~-z~t,s

and by the stationarity of the process {Xts}

j-(l+~)

e [ E Zl2)s,] 2F T(2)t.~t2,sz-1121~< Lj=tl

E(X~I)" _l

Hence

[ ira.)-' -

s=l L~t'sA J

<~Mn-lm-4a,

j

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and lnJerence 44 (1995) 219 235

233

which leads to the assertion by the arguments given in Lemma 3.3 of Yajima (1985). Next, we shall show that (mn) -1

: ~ [ W(a,o)12 t,s,l_l .... ' 0 t=ls=l

uniformly in dGD and OGC. Note that w~td~O~can be written as t

1

w%:,~ = - Z

p

Y, , ~ ) ~,), 0, x , _ ~,~_,,

k=0/=0

where ;dka) , t --- 1 -.,(a) -r tPt, kLr =(d)lk _l 1 = 1 - - F ( t ) F( t - k - d )

[ F ( t - d ) F( t - k ) ] - 1

Following the arguments given in Lemma 3.3 of Yajima (1985), we have fork<~t ~

:(d)t Il < M k ( t _ k ) _ l I~k,

for t ~ < k < . t - - 2 ,

< ~ M t ~ ( t - k ) -~

where (1 +23) -1 < f l < 1. Consequently, (

ta

IW,(,aL~I
P

Z IX,-k,~-tlk-~(t-k) -'

Lk=O/=O t-3

-4-

p

~

t-1

IX,-k,s-l[tak-(Z+l)(t--k)-a+

k=tt~+ 1 l = 0

~

~

/

IX,-~,s-~lt-'

k = t - 21=O

=M(Z~L I "~ z t,(1)s, 2 ~'

~ t( ". ... . L

(A.4)

(say).

Clearly,

(mn)-I :

:[-7(1)32 L ~ t , s, a d

.... > 0.

t=ls=l

Since the process {Xts} is stationary, from (A.4), it follows that E

(ran) -1

(Zt, s, 1) 2

z 2amE(X,~l )

Mn-lm2(2p

t=ls=l

and E

(mn) 1

(Z~1~,2)2

<
~

2at~)E(X41).

t=ls=l

Since ( 2 f l - 2 - 2 6 f l ) < 0 and (1 - f l - 2 6 f l ) < 0 , we have as m --,oo and n ~oo,

i_Lt,s," t=ls=l

a's'~ 0

234

S. Sethuraman, I.V. Basawa/Journal of Statistical Planning and Inference 44 (1995) 219-235

and (W//'/)-I

~ ~ rZ~ls{2]2 .... ) 0 t=ls=l

Hence we have the assertion. The following lemmas (Lemmas A.5 and A.6) can be obtained by proceeding along similar lines as those in Lemmas A.3 and A.4. The proofs are omitted. Lemma

A.5.

(i) (mn)-' £ ~. {(?~'°)/Od}2 .... ,E[c?~{lnlO,/~3d]2' t=ls=l

(ii) (ran)-' £ £ {a2¢~a'°}/ad2}2

....

} EEcq2~(dio,/ad2]2'

t=ls=l

(iii)

(ran)-' ~ ~

{¢3¢[~'°'/#0} 2 .... ,

E[O¢]ni°'/OO]2,

t=ls=l

(iv) (mn)-' £ £ {02~}a'°)/002} 2

....

) E[~2¢(ldiO,/~0212,

t=ls=l

(v) (mn)-' ~. ~ {(72~'°'/~d~0}2 .... , E[~¢]di°'/c~d¢30]2 t=ls=l

as m ~ Lemma

and n --+oo, uniformly in d~D and O~C. A.6.

(i) (ran)-1 £ £ {OW~ff'°'/3d}2

....

'0,

t=ls=l

(ii) (ran)-' ~ ~, {~2W[d,f}/c~d2}2 .... ,0, l=ls=l

(iii) finn)-' £

£ {dW~a;°'/80} 2 .... ,0,

t=ls=l

(ivt i==l-' ~ ~ {02wL";°'/002)2 .... ,o, t=ls=l

(v} (==t-' ~ ~ {~w:~o,/odoo)~ ....,o l=ls=l

as m ---~ooand n ~oc,, uniformly in d~D and O~C.

S. Sethuraman, I.V. Basawa/Journal (~[ Statistical Planninq and Inference 44 (1995) 219 235

235

Acknowledgement I.V. B a s a w a ' s w o r k w a s p a r t i a l l y s u p p o r t e d Research and from the National

b y g r a n t s f r o m t h e Office o f Naw~l

Science Foundation.

We thank

t h e r e f e r e e s for

a careful reading and helpful suggestions.

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