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Large sample estimation and prediction for explosivegrowth curve modelsPandurang M. Kulkarni aa Department of Statistics , La Trobe University , Melbourne, AustraliaPublished online: 27 Jun 2007.

To cite this article: Pandurang M. Kulkarni (1987) Large sample estimation and prediction for explosive growth curvemodels, Communications in Statistics - Theory and Methods, 16:9, 2677-2696, DOI: 10.1080/03610928708829532

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COMMUN. STATIST.-THEORY METH., 1 6 ( 9 ) , 2 6 7 7 - 2 6 9 6 ( 1 9 8 7 )

LARGE SAMPLE ESTIMATION AND PREDICTION FOR EXPLOSIVE GROWTH CURVE MODELS

Pandurang M. Kulkarni

Department of S t a t i s t i c s La Trobe Universi ty

Melbourne, Aus t ra l i a

Key Words and Phrases: asymptotic prediction error; regression wi th come Zated errors; maximwn l i k e Zihood estimation; growth curves as time-series; exp Zosive grouth.

ABSTRACT

Asymptotic d i s t r i b u t i o n s of maximum l ikel ihood es t imators f o r

the parameters i n explosive growth curve models a re derived.

Limit d i s t r i b u t i o n s of predic t ion e r r o r s when the parameters a r e

est imated a re a l s o obtained. The growth curve models are viewed

a s mul t ivar ia te t ime-series models, and t h e usual t ime-series

methods a r e used f o r p red ic t ion . Estimation constrained by a

hypothesis of homogeneity of growth r a t e s is a l so considered.

1. INTRODUCTION

Large sample p roper t i e s of es t imators and t e s t s f o r the para-

meters i n l i n e a r growth curve (or response curve) models with

co r re l a t ed e r r o r s a r e now well documented, see Sandland and

McGilchrist (1979) and Hudson (1983). Hudson (1983), i n p a r t i c u l a r ,

has s tudied the asymptotic p rope r t i e s of c e r t a i n t e s t s t a t i s t i c s

using a t@e-ser ies approach.

The main purpose of t h i s paper i s t o study large sample

p roper t i e s of p red ic t ions when the parameters a r e est imated, and

Copyright O 1987 by Marcel Dekker, Inc.

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2678 KULKARNI

i n p a r t i c u l a r , t o examine t h e e f f e c t of est imation of t h e para-

meters on t h e asymptotic variance of the predic t ion e r ro r . It w i l l

be shown t h a t f o r t h e s t a t iona ry models (non-explosive regressors)

t h e e f f e c t of est imation i s asymptotically negl ig ib le whereas f o r

t h e explosive growth curve models the est imation e r r o r cannot be

ignored. Large sample p roper t i e s of predic t ion e r r o r s when t h e

parameters a re estimated a r e studied by Basawa (1986) in a

d i f f e r e n t context f o r non-ergodic models when s imi lar phenomenon

regarding t h e e f f e c t of est imation on predic t ion e r r o r variance has

been observed.

In t h i s paper we study two a l t e r n a t i v e growth curve models

based on (a) a regression model wi th autoregressive e r r o r s (Model I)

and (b) an autoregressive model with a regression component (Model

11). Model I was a l so considered by Hudson (1983) in a d i f f e r e n t

context. The two models exh ib i t s imi l a r asymptotic explosive

behaviour while they i n f a c t d i f f e r i n severa l r e spec t s a s regards

parametr iza t ion, asymptotic p rope r t i e s of t h e es t imates , e t c . We

emphasize t h e case when the regression ( x . ) a t time j increase 3

t o , a t e i t h e r a l i n e a r o r an exponential r a t e a s j -t , and

r e f e r t o such models a s explosive. Explosiveness of regressors

e i t h e r l i n e a r l y o r exponential ly in economic analys is is not un-

common. Simi lar types of models have been s tudied by Granger

(1986) who a l s o d iscusses severa l p r a c t i c a l appl ica t ions of such

models.

Glesser and Olkin (1972) have studied a general regression

model s imi l a r t o our Model I. However, i n t h i s model the

regressors do not depend on time ' n ' . For regression depending

on ' n ' ,.Anderson (1971) has considered est imation f o r both types

of models I and 11, discussed i n our paper. Anderson (1971) has

imposed c e r t a i n condit ions on the regression va r i ab les t o

guarantee s t a t i o n a r i t y (non-explosiveness) and t o obta in t h e

asymptotic normality of the es t imators of the parameters. Using

s imi la r techniques we obta in r e s u l t s on asymptotic normality, e t c . ,

f o r explosive growth curve models by su i t ab ly modifying the con-

d i t i o n s imposed on the regressors .

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EXPLOSIVE GROWTH CURVES 2679

A p r e l i m i n a r y d i s c u s s i o n o f t h e two models is g iven i n

S e c t i o n 2 , and S e c t i o n 3 c o n t a i n s t h e r e s u l t s on l i m i t d i s t r i b u t i o n s

o f t h e maximum l i k e l i h o o d e s t i m a t o r s f o r t h e e x p l o s i v e growth

curves . Asymptotic, j o i n t d i s t r i b u t i o n s o f a f i n i t e number o f

s u c c e s s i v e p r e d i c t i o n e r r o r s a f t e r a s u i t a b l e normal iza t ion a r e

d e r i v e d i n S e c t i o n s 4 and 5 f o r t h e two models. S p e c i f i c examples

a r e cons idered a t t h e end o f S e c t i o n s 4 and 5.

2 . FORMULATION O F THE BASIC EXPMSIVE MODELS

L e t Y denote t h e random o b s e r v a t i o n a t t ime j , j

1 S j S n , { x i j , l ~ i S q 1 a s e t of " f i x e d " r e g r e s s o r s cor responding

t o Y and E be independent N (0 , 0 2 ) v a r i a t e s ( i n n o v a t i o n s ) . j j

Consider t h e f o l l o w i n g models:

Model A . Regression w i t h a u t o r e g r e s s i v e e r r o r s :

where t h e e r r o r s Z a r e assumed t o s a t i s f y a f i r s t - o r d e r auto- j

r e g r e s s i o n r e l a t i o n ,

Model B . Autoregress ion w i t h a r e g r e s s i o n component

eixij + p Y . + E Y o = O . 1 j ' (3)

i=l

The parameters p , a , (Bi , lSi%) , p and c r 2 i n t h e above

models a r e assumed unknown. We assume throughout t h e paper t h a t

t h e a u t o r e g r e s s i o n parameter p s a t i s f i e s j p 1 < 1 and 10 1 < .

The "growth" component i n t h e above models is g iven by t h e

r e g r e s s i o n f u n c t i o n E (Y. ) , and we have 3

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KULKARNI

f o r model A

la[&] + 1 1 ~i~k- lx i , j -k+ l f o r model B 1-p i=l k=l

Thus f o r model A t he growth is l i n e a r in the parameters while it is

non-linear f o r model B . 2j

v a r ( ~ . lxij) = 0 2

1 1 - P 2 I , f o r both t h e models A and B 3

It w i l l be assumed i n what follows t h a t f o r each i , 0 < xij I. rn a s j + rn . It i s then c l e a r t h a t , f o r both the

models

where x = max{xij ,l 6 i 5 q} . A growth process s a t i s f y i n g (6) j

and z : t a s j + - w i l l be r e fe r red t o a s an explosive process. 3

I n s p i t e of t h e d i f ference i n the spec i f i ca t ion of t h e two

models both t h e processes exh ib i t t he same asymptotic explosive

behaviour i n t h e sense of (6) . Also, f o r both the models it is

c l e a r from (4 ) t h a t

We s h a l l now use the bas ic models discussed above t o formulate

t h e growth curve models i n the following section.

- Note: 0 i n (6) implies Y when divided by X is bounded i n - P j j p robab i l i ty .

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EXPLOSIVE GROWTH CURVES

3 . GROWTH CURVE: MODELS AND ESTIMATION OF P W T E R S

Let Y U v ( j ) denote the observation a t time j , on the v t h

indiv idual belonging t o uth group, 1 6 u < m , 1 < v < N , and

1 6 j 6 n . The t o t a l number of indiv iduals is given by

N = N1 + ... + Nm , and on each of these N indiv iduals observ-

a t i o n s a t n time p o i n t s a r e made. Assume t h a t ';yij and {ykk1

a r e independent for i f R or j # k . The growth curve models

based on t h e bas ic models A and B of t h e previous sec t ion a re then

defined by the following.

Model I

where

and E ( j 1 a r e independent N (0 ,a t ) var i a t e s . uv

Model I1

YUv(0) = 0 , and E ( j ) a r e a s inModel I . uv

We now consider t he maximum l ike l ihood es t imators of t he

parameters, and ob ta in t h e i r l i m i t d i s t r i b u t i o n s a s n + " .

3.1. Estimation f o r Model I

The log-l ikelihood function based on the observations f o r t h e

u t h group f o r Model I is given by, ( ignoring constant te rms) ,

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KULKARNI

where

The estimating equations f o r Put Biu , 1 S i 5 q , and pU are

given by

where E ( j ) i s defined by (12). F ina l ly a 2 can be estimated uv u

by

where E ( j ) denotes cUV ( j ) with a l l the parameters replaced by uv

t h e i r estimators.

and

Assume t h a t the following condition i s s a t i s f i e d : Dow

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E X P L O S I V E GROWTH CURVES

(h) lirn{cil/(bibl) '~ = fiL, 1 5 i, k ' q . n*

Remark: Condition l ( a ) ensures t h e exis tence of t he l imi t ing

covariance between t h e mle 's of t h e parameters vU and AU T

fl = (0 Lu , . . . , 8 ) , and condit ion l ( b ) t h a t of t h e covariance Nu qu between

Oiu and 0 . ( 1 , 1 . These condit ions a re

xu needed because of the explosiveness of the regressors . I t can be

seen t h a t even when t h e r eg res so r s a re not explosive these

condit ions a r e enough t o ensure t h e asymptotic normality of t h e

mle 's .

Also, l e t I (n) = diag{n,b b ,bq,n}, where IU(n ) i s U 2 'T. -

a (q+2) x(q+2) diagonal matr ix , and & = (!-I 1€J1ute2u,. . - 1 0 ,P I u qu u

where & i s (q+2) x1 vector of t h e parameters and T denotes t h e

transpose. Let F denote a qxq matrix with ( i , k ) t h element given

by fig . Note t h a t t he diagonal elements of F a re a l l equal t o T

uni ty . Denote M_ = ( f l , . . . , f ) where M i s a q x l vector. The 9

Fisher information matrix corresponding t o the parameter i s

then given by J,(&) ,

where

LA'(&) denotes the matrix of t h e second de r iva t ives of LU wi th

respect t o the components of . It i s e a s i l y v e r i f i e d t h a t

Let $(n) denote t h e es t imate of & obtained by solv ing t h e

equation (13). The following theorem can be v e r i f i e d v i a standard

asymptotics.

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2684 KULKARNI

Theorem 3.1. Under c o n d i t i o n 1, and Model I , a s n + , and f o r e a c h u , 1 < u < m ,

4 Iu (n) (&in) -&) + Nqi2 (0,~;' (h) ) , i n d i s t r i b u t i o n ,

p rov ided JU is non-singular .

3.2. Es t imat ion f o r Model I1

The log- l ike l ihood f u n c t i o n f o r model I1 based on observa-

t i o n s from t h e u t h group i s g iven by LU

a s d e f i n e d i n (11) , where

t h e e r r o r terms E ( j ) a r e now d e f i n e d by uv

The l i k e l i h o o d e s t i m a t i n g e q u a t i o n s f o r u U r O i u , l S i S q ,

and pU a r e g iven by

1 1 e U V ( j ) = 0 , 1 1 c U V ( j v j v j

and 1 x ~ ~ ~ ( j ) Y ~ ~ ( j - l ) = 0 v j

where E. ( j ) a r e g iven by (15) uv

x ( i , j ) = O , 1 5 i S q , uv

(16)

F o r e a s e of p r e s e n t a t i o n we now

t a k e q = l s o t h a t t h e r e i s o n l y one r e g r e s s o r x U v ( j ) , and (15) is

r e p l a c e d by

Denote a; (n) = x xUv ( j ) , b: (n) = x x2 (j) , uv

v j v j

and suppose t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d .

Remark: Condi t ion 2 i s t o o b t a i n t h e l i m i t i n g covar iance between

u and (Burpu) . It can be no ted t h a t c o n d i t i o n 2 i s s i m i l a r u

t o c o n d i t i o n 1 (a) i n S e c t i o n 3.1.

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EXPLOSIVE GROWTH CURVES

Now, denote t h e m a t r i x o f norms a s fo l lows .

1: (n) = d i a g { n , b: (n) , b: (n) 1 , where I; (n) i s a (3x3) d i a g o n a l

m a t . l e t = (au ,6u ,pu)T be t h e (3x1) v e c t o r o f parameters .

It can be shown, u s i n g T o e p l i t z r s l e m , t h a t t h e F i s h e r in format ion

m a t r i x p e r o b s e r v a t i o n cor responding t o t h e e s t i m a t i o n o f is

where f* i s d e f i n e d by c o n d i t i o n 2 ,

c = l i m i I : a (n) /I: xUV (n) 1 , d = l i m ( 1 a2 (n) /I: x:v (n) 1 , and uv

n* v v n* v uv v n-1

e = lim{E a uv

7 n-@x (R). uv (n) xuv (n) f i x i v (n) 1 where a (n) = e U L P U uv n* v v !L=1

Under t h e assumptions, 0 < xUv (n) + , and c o n d i t i o n 2 , it can

e a s i l y be v e r i f i e d t h a t a l l t h e l i m i t s f*, c , d , and e i n J* u

above e x i s t wi thout f u r t h e r c o n d i t i o n s .

S p e c i f i c examples f o r t h e s e models w i l l be d i s c u s s e d i n t h e

fo l lowing s e c t i o n .

W e s t a t e t h e fo l lowing theorem, which can be proved us ing t h e

u s u a l asympto t ic t h e o r y .

Theorem 3.2. Under Model 11, and c o n d i t i o n 2 , a s n -+ f o r e a c h

u , 1 S u S m , we have

i n d i s t r i b u t i o n , where $ * (n) i s t h e s o l u t i o n of (16) . '-4

We now c o n s i d e r parameter e s t i m a t i o n f o r a s p e c i a l c a s e of

Model I1 under t h e hypothes i s t h a t t h e r e g r e s s i o n parameters f o r

d i f f e r e n t groups are equa l . Denote by Model IIH , t h e model I1

under t h e n u l l h y p o t h e s i s s p e c i f i e d above.

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KULKARNI

3.3. Model 11, : Parameter E s t i m a t i o n

Consider Model I1 w i t h m=2 ( i . e . two groups) , q = l , a =0, U

02=1, u=1,2, and l e t t h e n u l l hypothes i s 11 s p e c i f y t h a t 8 =8 u 1 2

(homogeneity r e g a r d i n g t h e r e g r e s s i o n p a r a m e t e r s ) . Then, under H ,

t h e parameters o f t h e model a r e 0 , (where O =O = 8 ) , pl and 1 2 P2 '

We s h a l l r e f e r t o t h e model under H by model IIH . It is t o be

noted t h a t t h e e s t i m a t e s of 8 , pl and p2 under H use

o b s e r v a t i o n s from both t h e groups u n l i k e i n t h e case of t h e models

I and 11 , where o b s e r v a t i o n s o n l y from a s i n g l e group a r e needed

t o e s t i m a t e t h e parameters o f t h a t group.

Thus, model 1111 is s p e c i f i e d by

The log- l ike l ihood f u n c t i o n f o r Model I1 i s g iven by H

( i g n o r i n g t h e te rms f r e e from t h e parameters )

T Denote = (p1,p2,8) , and c o n s i d e r t h e ( 3 x 3 ) m a t r i x of norm w

I (n) = d i a g i b * (n) ,b* (n) ,b* (n) + b* (n) 1 where b* ( n ) = 1 Z x k v ( j ) , U 1 2 1 2 U v j

u=1,2 . With E ( j ) s p e c i f i e d by (19) , t h e e s t i m a t i n g e q u a t i o n s uv

a r e

1 ~ ~ ~ ( j ) Y ~ ~ ( j - l ) = 0 , u = l r 2 , and 1 1 1 ~ ~ ~ ( j ) x ~ ~ ( j ) = 0. (20) V I u v j

Le t

n-1 n-1-R where a (n) = 0

uv L P u x (9.) , and l e t u s suppose t h a t uv

Condi t ion 3 is s a t k $ i e d .

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EXPLOSIVE GROWTH CURVES 2687

Condi t ion 3 1imC.E 1x2 ( j ) / (b; (n)+b; ( n ) ) 1 = a* i s f i n i t e , u=1,2 uv u

n- v j

where b; (n) i s d e f i n e d a s above.

Remark: Note t h a t c o n d i t i o n (3) reduces t o u n i t y f o r t h e c a s e

of s i n g l e group i . e . u=l . We need c o n d i t i o n (3) because it i s

c l e a r t h a t e s t i m a t i o n f o r model I1 needed o b s e r v a t i o n s on ly from

a s i n g l e group, whereas here f o r model 11 e s t i m a t i o n of H

pa rameters i n v o l v e s o b s e r v a t i o n s from a l l groups.

Now, u s i n g T o e p l i t z ' s lemma one can show t h a t t h e F i s h e r

in format ion m a t r i x p e r o b s e r v a t i o n is g iven by

where R* (u=1,2) a r e a s s p e c i f i e d i n (21) and c o n d i t i o n dul u r U

( 3 ) .

The f o l l o w i n g theorem can be v e r i f i e d us ing s t a n d a r d

asympto t ics .

Theorem 3.3. Under t h e Model II and c o n d i t i o n 3, a s n + OD we

have

where (n ) i s t h e e s t i m a t e of $ obta ined by s o l v i n g (20) . N U N U

4 . PREDICTION AND ASYMPTOTIC DISTRIBUTION OF

PREDICTION ERRORS FOR MODEL I

4.1. L i m i t d i s t r i b u t i o n of p r e d i c t i o n e r r o r s

I n t h i s s e c t i o n we s h a l l d i s c u s s t h e p r e d i c t i o n f o r Model I , assuming t h a t t h e f i r s t n o b s e r v a t i o n s a r e known, and t h e n o b t a i n

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2688 KULKARNI

t h e asympto t ic d i s t r i b u t i o n o f t h e p r e d i c t i o n e r r o r w i t h e s t i m a t e d

parameters .

L e t us denote t h e t - s t e p ahead p r e d i c t o r (1 < t 5 k) o f

Y . ( n + t ) given o b s e r v a t i o n s YUv ( j ) , 1 5 j < n , by uv

& ( n + t ) = E [Y ( n + t ) I h(1) , . . .flU(n) I -u

= E [Y ( n + t ) 1 (n ) 1 , where 5 ( n + t ) i s a N x l v e c t o r -u 4 U

w i t h i t s v t h element given by

L e t (n-kt) dermte & ( n + t ) wi.th a l l t h e parameters r e p l a c e d by

t h e i r e s t i m a t e s . Also , l e t e (11-kt) = 5 (ni . t ) - Y ( n + t ) -u NU hZ1

and z U ( n + t ) = 5 (n+t ) - Yw ( n + t ) be t h e p r e d i c t i o n e r r o r s when t h e W

parameters a r e lcnown and when they a r e es t imated r e s p e c t i v e l y , 2 t

and denote n ( n + t ) = v a r (Y (n+k) ]Y_U (n) ) = L (1-p,, ) a{/ (1-pt) I I , u RU

where . I is an (N xN ) i d e n t i t y mat r ix . Then t h e r e s p e c t i v e u u

s t a n d a r d i z e d p r e d i c t i o n e r r o r s a r e def ined by

Now

Using T a y l o r ' s expansion f o r E; (n+t ) we o b t a i n W

where F;: (n+t ) denotes t h e (N ) x (q+2) m a t r i x of p a r t i a l d e r i v a t i v e s u

of L ( n + t ) w i t h r e s p e c t t o t h e components of and I U b ) i s

t h e m a t r i x of norms d e f i n e d i n S e c t i o n 3.1.

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EXPLOSIVE GROWTH CURVES

Let 442 (n) be a (KNUxl) v e c t o r w i t h i t s e lements

(n) -+ N [E,B (&) ] i n d i s t r i b u t i o n ,

*u

whpre B ( 4 ) = ( (gts)) is a (KNU X mu) mat r ix w i t h hZ1

I is (NUxNU) i d e n t i t y mat r ix .

The l i m i t d i s t r i b u t i o n o f t h e p r e d i c t i o n e r r o r w i t h parameters

known is t h u s given by (24) .

F u r t h e r , i n o r d e r t o o b t a i n t h e d i s t r i b u t i o n of p r e d i c t i o n

e r r o r w i t h e s t i m a t e d p a r a m e t e r s , denote t h e tth column of GU(n)

by

and assume t h a t t he c o n d i t i o n 4 holds.

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2690 KULKARNI

Condi t ion 4. lim[w ( i , n ) ] = g U v ( i , t ) i s f i n i t e uv

Remark: Condit ion (4) i s needed t o o b t a i n t h e convergence i n

d i s t r i b u t i o n o f p r e d i c t i o n e r r o r , whereas c o n d i t i o n (1) f o r t h e

convergence i n d i s t r i b u t i o n of t h e e s t i m a t e s o f t h e parameters .

I n g e n e r a l , e i t h e r o f t h e s e c o n d i t i o n s does n o t imply t h e o t h e r .

It is c l e a r i n examples i n t h e fo l lowing s e c t i o n s t h a t t h e s e

c o n d i t i o n s a r e n o t hard t o ach ieve .

Then we have

'I' and & ( i , t ) = ( g u l ( i , t ) ,... g U r N ( i , t ) ) ,

Nuxl u

w i t h guv (i,t) d e f i n e d in Condit ion 4 .

L e t GU = [& , . . . G (K) I and ;$ (n) be a (lUi x l ) v e c t o r w i t h -u -u u

its e lements

Now, us ing (25) , Theorem (3.1) and n o t i n g t h a t t h e two te rms

on t h e r i g h t o f (24) a r e u n c o r r e l a t e d , it is easy t o v e r i f y t h a t

n + N (2, ( ) i n d i s t r i b u t i o n (28) u

- 1 w i t h B (k) , GU and JU a s s p e c i f i e d i n (24) , (26) -(27) and

Theorem 3 . 1 r e s p e c t i v e l y .

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EXPLOSIVE GROWTH CURVES 2691

Thus, (28) g i v e s t h e asymptot ic d i s t r i b u t i o n o f t h e

p r e d i c t i o n e r r o r w i t h e s t i m a t e d parameters .

4.2. Examples

TO i l l u s t r a t e t h e above r e s u l t s , we c o n s i d e r some s p e c i f i c

examples, f o r Model I .

I n each c a s e we o b t a i n 2, F and G ( t ) , which can r e s p e c t -

i v e l y be used t o g e t J (2) d e f i n e d i n (15) and $ (iU) d e f i n e d

i n (28) . Le t h = [ ( l - ~ ~ ~ ) 0 2 ] - * [1 -p2 l f .

Example 1. Take m=l, N =1 i n model I . u

Case (i). (Exponential growth) x . = A' , A > 1 . I

It can be shown t h a t

- 1 J (A) = cr-2 d i a g [ (1-p ) ', 1, (1-p * ) ] which i s a ( 3 x 3 ) d i a g o n a l

m a t r i x and

T t t

(t) f w

-' ( A 2 - 1 ) . = [ O , g ( t ) , O ] h , where g ( t ) =----- ( A - P )

Case (ii) . (Linear growth) x . = j . I

6 (1- ) T We g e t M = - , F=l , and G ( t ) = [0,0,01 h . 2 l + p

T h i s l e a d s t o t h e conc lus ion t h a t t h e e s t i m a t i o n o f parameters

does n o t c o n t r i b u t e t o t h e asymptot ic v a r i a n c e of t h e p r e d i c t i o n

e r r o r .

Example 2. Le t m=l., N =1, q=2 i n Model I . Let h be a s u

d e f i n e d i n example 1.

j Case (i) . (Both growth components e x p o n e n t i a l ) x . . = 1 . , hi > 1, 11 1

i=1 ,2 we g e t = (0,O) ,

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KULKARNI

T and G ( ~ ) = [O, g ( i , t ) ,0] h ,

N lSi.52 t t f - 1

where . g ( i , t ) = (Ai-p ) (A:-1) (Ai-p) .

Case ( i i ) . (Exponential and l i n e a r growth components) x =A: , 1 j

A > 1 and x = j . 1 2 j

We o b t a i n = [ 0 , 6 / 2 1 , F = I = i d e n t i t y m a t r i x , and 2x2

t t f -1 G ( ~ ) = [0 , g ( t ) , 0 , O I T h , where g ( t ) = (A-p )(A2-1) (A-p) . N

We may n o t e t h a t t h e c o n t r i b u t i o n s due t o t h e e s t i m a t i o n o f t h e

parameter O2 (corresponding t o t h e l i n e a r growth) i n t h e

asympto t ic v a r i a n c e o f p r e d i c t i o n e r r o r is z e r o , while t h e

e s t i m a t i o n o f O1 corresponding t o t h e e x p o n e n t i a l growth r e s u l t s

i n an i n c r e a s e i n t h e p r e d i c t i o n e r r o r var iance .

5. PREDICTION AND ASYMPTOTIC DISTRIBUTION O F

PREDICTION ERRORS FOR MODEL 11

The l i m i t i n g d i s t r i b u t i o n of p r e d i c t i o n e r r o r s w i t h e s t i m a t e d

parameters f o r Models I1 can be ob ta ined on t h e same l i n e s a s f o r

Model I d i s c u s s e d i n S e c t i o n 4.

5.1. Limit d i s t r i b u t i o n o f p r e d i c t i o n e r r o r s

,We s h a l l use t h e same n o t a t i o n a s i n S e c t i o n 4 , f o r d e f i n i n g

t h e p r e d i c t i o n e r r o r , e r r o r v a r i a n c e , e t c . For s i m p l i c i t y l e t us

t a k e p = l i n Model I1 , and proceed ing a s i n S e c t i o n 4 , we g e t ,

A.

f ;* ( n i t ) = (n+t ) + ( n ) ( t ) I (n) J I n ( n - (29) h21

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EXPLOSIVE GROWTH CURVES 2693

where e * ( n t t ) = ( n + t ) k ( n + t ) , % ( n + t ) = & (n+t ) - & ( n + t ) , CZ1

5: (n+t ) denotes t h e (NUx3) m a t r i x o f p a r t i a l d e r i v a t i v e s o f

5 ( n + t ) w i t h r e s p e c t t o t h e components of . And , I:(n) -Il

a r e a s d e f i n e d i n Sec t ion 3 . 2 .

Also,

where

L e t us assume t h a t t h e c o n d i t i o n (5) ho lds

Condit ion 5.

S i m i l a r remark a s i n Sec t ion 4 ho lds f o r c o n d i t i o n s ( 2 ) and (5 ) .

Then we have,

where oT = ( 0 , . . -0) , hU a s d e f i n e d above, G N U

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) = lirnl.2 ( n ) ] , v=1, ... N n-%o uv u -

Assuming t h a t c o n d i t i o n s (3) and (5)

same arguments a s i n S e c t i o n 4 , it can be

h o l d , and us ing t h e

shown t h a t

(1) = r T 1 , - . . n I N B ) i n d i s t r i b u t i o n KN

ICNU u (32)

where B (z) = B (k), B (&) d e f i n e d i n (24) .

Hence t h e asymptot ic v a r i a n c e of p r e d i c t i o n e r r o r s w i t h parameters

known, is t h e same f o r bo th t h e Models I and 11 .

Also,

j%(n) + N , ) ) i n d i s t r i b u t i o n

T -1 where $ (G) = B (,Q + GU J: GU ,

GU = [GU , . . . G:) 1 , G:) l S t U , d e f i n e d i n (31) . and

J*-I i s a s d e f i n e d i n Theorem 3.2. u

We now c o n s i d e r an example f o r model I1 .

Example 4. L e t m=l, N =1, and q = l , i n Model II . u

Case (i). x. = A' , A > 1 . Then we have, 7

u s i n g t h e s e J* (G) d e f i n e d i n (18) can be ob ta ined .

f t-1 t-1 i t-i-1 And g ( t ) = [ O (,A2-1) l i t (A-p)-l + ( t - i ) A p 1

i=l t t f

= e ( A -p ) ( A ~ - 1 ) ( A - P ) - ~ , - 4 t t f , and g * ( t ) = (A -p ) (,A2-1) (A-p)

- 1

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EXPLOSIVE GROWTH CURVES 2695

By comparing t h i s w i t h Example 1 c a s e ( i ) , it is c l e a r t h a t

t h e asymptot ic v a r i a n c e o f p r e d i c t i o n e r r o r w i t h es t imated

parameter i s d i f f e r e n t f o r t h e Models I and I1 .

Case (ii). L e t x = j . Then c = ~ ( l - p ) - ' , f * = 6 / 2 , d = c 2 , j

e = c . Using t h e s e r e s u l t s J ($$) d e f i n e d i n (18) can be T

o b t a i n e d , and a s expected we g e t g ( t ) = [O ,0,01 h .

Remark: D e t a i l s of o b t a i n i n g t h e asymptot ic p r e d i c t i o n e r r o r

d i s t r i b u t i o n f o r Model 1111 a r e o m i t t e d s i n c e t h e y can e a s i l y be

o b t a i n e d on t h e same l i n e s a s i n S e c t i o n s 4 and 5.

ACKNOWLEDGEMENT

The au thor would l i k e t o thank D r . I . V . Basawa f o r s u g g e s t i n g

t h e t o p i c . Also he would l i k e t o thank t h e e d i t o r and t h e r e f e r e e s

f o r t h e i r h e l p f u l comments.

BIBLIOGRAPFN

Anderson, T.W. (1971). The S t a t i s t i c a l Analys i s of Time S e r i e s New York, John Wiley and Sons, Inc.

Basawa, I . V . (1986). Asymptotic d i s t r i b u t i o n s of p r e d i c t i o n e r r o r s and r e l a t e d t e s t s o f f i t f o r n o n s t a t i o n a r y p r o c e s s e s . To appear i n Annals of S t a t i s t i c s .

Chakravor ty , S.R. (1976). Maxinun l i k e l i h o o d e s t i m a t e s of t h e g e n e r a l growth curve model. Ann. I n s t . S t a t i s t . Math, 28 349-357.

G l e s s e r , L. J. and O l k i n , I. (1972). E s t i m a t i o n f o r a r e g r e s s i o n model w i t h an unknown covar iance mat r ix . Proc . S i x t h Berkeley Symposium, Vol. 1, 541-568.

Granger, C.W.J. (1986). Models t h a t g e n e r a t e t r e n d s . Presen ted a t t h e 26 th summer r e s e a r c h i n s t i t u t e of t h e A u s t r a l i a n Mathe- m a t i c a l S o c i e t y , Canberra , January 1986.

Hudson,, I .L. (1983) . Asymptotic t e s t s f o r growth curve models w i t h a u t o r e g r e s s i v e e r r o r s . A u s t r a l . J. S t a t i s t . , 25, 413-424.

Sandland, R.L. and McGilchr i s t , C.A. (1979). S t o c h a s t i c growth curve a n a l y s i s . B iomet r ics , 2, 255-271.

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Rec&Lved Novemberr, 1985 ; RevAed June, 1 9 t i 7 .

Recommended AnonymowLy.

Redefiecd Anonymounly.

KULKARNI

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