Predictive inference on equicorrelated linear regression models

Applied Mathematics and Computation 95 (1998) 205±217

Predictive inference on equicorrelated linear regression models Shahjahan Khan a, M.I. Bhatti a

b,*

Department of Mathematics and Computing, University of Southern Queensland, Qld 4350, Australia b School of International Business, Grith University, Nathan Campus, Queensland 4111, Australia

Abstract Beyond the customary analysis through the estimation and hypothesis testing about the parameters of the multiple regression models, often a natural interest is to predict the responses for a given set of values of the predictors. The main objective of this article is to obtain the prediction distribution for a set of future responses from a multiple linear regression model which follow equicorrelation structure. It derives the marginal likelihood estimate for the equicorrelation parameter, q, and then uses the invariant di€erentials to compute the joint distribution of the unobserved but realized future errors. The prediction distribution is derived by using the structural relation of the model. The main ®nding of this paper is that the prediction distribution turned out to be a Student-t which depends only on the estimated q and is invariant to the degrees of freedom of the original Student-t distribution. Ó 1998 Elsevier Science Inc. All rights reserved. AMS Classi®cation: 62E15; 62M10; 62H10 Keywords: Multiple regression model; Equicorrelated responses; Marginal likelihood function; Invariant di€erential; Structural relation

1. Introduction Recently, Bhatti [1,2] considered an equicorrelated linear regression model for testing nonzero values of the equicorrelation coecient, q, which arises

*

Corresponding author. E-mail: [email protected].

0096-3003/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 7 ) 1 0 1 0 0 - X

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through the disturbance terms of the model. He proposed point optimal invariant (POI) tests for q and found that these tests are approximately uniformly most powerful (UMP) in a wide range of the alternative parameter space. In recent years there appeared to be a series of articles on estimation and testing for q. Examples include, the papers by Bhatti and King [3], Bhatti [4±6], Khan [7], Wu and Bhatti [8], Bhatti and Barry [9] and just recently Bhatti [10], among others. Bhatti and King [3] proposed a beta-optimal test for testing q, if the disturbances follow the standard symmetric multivariate normal distribution. They found that their test is more powerful than that of SenGupta's [11] locally most powerful (LMP) test. Using the same idea in relation to the linear regression model Bhatti [4] found that the POI test leads in power than that of its competitors the LMP and the Durbin±Watson tests. Later, Bhatti [6] suggested an ecient estimation procedure for estimating the parameters of the random regression coecient models, using Bangladesh's agricultural data. Recently, Bhatti and Barry [9] have developed an optimal testing procedure for q, in the presence of the random regression coecients. Khan [7] used the b-expectation tolerance region to infere about q whereas Bhatti [1] have suggested an optimal testing procedures by using two-stage equicorrelated linear regression model. Beyond the customary analysis through the ecient estimation and optimal testing procedures about the parameters of the equicorrelated linear regression model, often a natural interest is to predict the responses for a given set of values of the predictors. Infact, predicting a future value of the response is often much more useful for the practitioners as compared to that of ®nding the estimates of the population parameters and/or performing the tests about them. Therefore, a more general approach from the inferential point of view is to ®nd the prediction distribution which will provide the basis to respond to such an interest. Guttman [12] argued that the derivation of tolerance regions using the prediction distribution would be another way of looking into the estimation and testing problems. The main objective of this paper is to obtain the prediction distribution for a set of future response from a regression model which follows equicorrelation structure along with the nonnormality assumptions. The predictive inference, based on prediction distribution, as opposed to the inference about the parameters of the model, has been emphasized by Aitchison and Dunsmore [13]. They also sighted a number of real world problems with illustrations where it is and can be applied. However, their prediction analysis is only concentrated on the independent responses, and thereby excluded the most of the time series and survey data where the responses are not necessarily independent. Nevertheless, correlation between the successive realization of the response does exist in real life data due to natural and/or technological reasons. In this paper we consider a multiple linear regression model that allows us to incorporate the equicorrelation structure of data. In

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our model, errors are assumed to follow the multivariate Student-t distribution with equicorrelated structure among each other. As it can be observed from the above discussion that most of the authors have used the normality assumption in relation to the disturbances of their models. The validity of this assumption has been questioned by many authors, like Je€reys [14], Huber [15], Zelner [16], Prucha and Kelejian [17], Lawrence and Arthur [18], and more recently, Khan [7]. Following their arguments we hereby assume that the disturbances follows the Student-t distribution. Because its degrees of freedom takes care of the shape of the distribution including the thicker tail. It also includes the Cauchy distribution as a special case when there is only one degree of freedom, and it yields the normal distribution when the degrees of freedom tends to in®nity. Therefore, the assumption of the Student-t distribution for the disturbances is more realistic and versatile in regression analysis without losing much of the normality charm. Haq [19] derived the prediction distribution using the structural relation for a location-scale model with equicorrelated responses. He assumed that q is known. In this article we will consider multiple regression model and use the marginal likelihood estimate for q. This can be done by ®rst deriving the joint density of the errors using invariant di€erentials (for details on invariant di€erentials, see [20], pp. 207±217). Then, ®nd the marginal likelihood estimator of q for each of the clusters using the conditional distribution approach. Finally, following Haq's [21] approach we obtain the prediction distribution of a set of future responses from the regression model by exploiting the structural relation of the model. The plan of the rest of this article is as follows. In Section 2 the equicorrelated model is introduced and the prediction problem is discussed. Section 3 contains the procedure for the marginal likelihood estimation of q which will be used to obtain the prediction distribution of the set of future responses. Moreover, the distribution of the realized and the future errors are given in Section 4 and the structural relation of the model is developed in Section 5. Section 6 presents the distribution of the future responses which are based on the Student-t distribution. The ®nal section contains some discussions and the concluding remarks.

2. The regression model Consider the following linear regression model yj ˆ

p X iˆ1

bi xij ‡ j ;

j ˆ 1; 2; . . . ; m

or equivalently, in vector notation

…1†

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y ˆ bX ‡ ;

…2†

where y is the 1  m vector of observed response, X is a p  m order design matrix,  is a 1  m realized but unobserved error vector, and b is a 1  p vector of regression parameters. Assume that m > p ‡ 1, and …XX 0 † is a nonsingular matrix so that its inverse exists. Also assume that the responses are equicorrelated among themselves. This type of situation arises in many applications of panel and/or survey data (e.g., Bhatti [1,2] and time series- cross sectional and economic data). Further we assume that the errors are jointly distributed as multivariate Student-t with E…j † ˆ 0; and var…j † ˆ r2

for all j:

…3†

It is convenient to introduce a scale factor to the error term in the model, that simpli®es the notation of the variance±covariance matrix for the error vector  as given below y ˆ bX ‡ ru

with  ˆ ru:

…4†

Note the expectation of u is a 1  m dimensional vector of zeros. Under the assumption of equicorrelation among the elements of the error vector the covariance matrix of the scaled error term can be written as 0 1 1 q q


q B C 1 q


q C B q B C C cov…u† ˆ X ˆ B …5† B














C; B C @














A q q q


1 which is an m  m correlation matrix, where in X above the cov…uj ; uj0 † ˆ q; for all j 6ˆ j0 ; and ÿ1 6 q 6 1: Under Eq. (4), the response matrix becomes cov…y† ˆ r2 X; which is also an m  m order matrix. Assuming that the scaled errors are distributed as standardized Student-t with m degrees of freedom, the joint distribution of the error vector can be written as ÿ
 ÿ…m‡m†=2 C m‡m 1 ÿ1 0 2 1 ‡ uX u ; …6† pq …u† ˆ ÿ
m …mp†ÿm=2 C 2m jXi j1=2 Haq [19] considered a normal location-scale model with similar equicorrelated responses to derive prediction distribution for a known (assumed) value of the correlation coecient, and used the structural model (and/or distribution) as proposed by Fraser [22]. Recently, Bhatti [6] obtained ecient estimators for the equicorrelated normal random e€ect model for survey data through the concentrated likelihood function whereas Bhatti and Barry [9] have suggested an optimal procedure for testing q. In Section 3, our objective is to obtain the prediction distribution for the set of future responses for the model (2), under (4) and (5). As the prediction

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density depends on the equicorrelation parameter, q we need to ®nd an estimate for it. The next section is developed to obtain such an estimate, namely the marginal likelihood estimate of q. 3. The marginal likelihood estimation of q The proposed estimate for q is based on its marginal likelihood function. To obtain the marginal likelihood function, let us de®ne the following statistics which are based on the scaled error vector ÿ1

bu ˆ uX 0 …XX 0 † ; 0

s2 …u† ˆ fu ÿ bX gfu ÿ bX g ;

…7†

where b…u† is the regression vector of u on X and s2 …u† is the squared residual sum. Now we can de®ne the standardized residual vector as follows: 1 …8† d…u† ˆ fu ÿ bX g; s where s ˆ s…u† and b ˆ b…u† for notational briefness. This allows us to express the error vector in terms of above de®ned statistics as u ˆ bX ‡ sd;

…9†

where d ˆ d…u†: It is well known that d…u† ˆ d…y†; where d…y† is de®ned exactly the same way as d…u† but by replacing u by y. Using the properties of invariant di€erential (see, e.g., [20], pp. 207±217), as well as the relation in Eq. (9), and following Haq [21], we can ®nd the joint distribution of b and s, conditional on d from the density in Eq. (5) as follows:  ÿ…m‡m†=2 1 ÿ1 0 0 ÿ1 0 mÿpÿ1 2  1 ‡ uX X u ‡ s dX d : …10† pq …b; sjd† ˆ U…m; p†s m Note that dX 0 ˆ O as d and X are orthogonal, and U…m; p† is the normalizing constant. The density in Eq. (10) does not factor, and hence b and s are not independently distributed. Applying appropriate integration we can obtain the normalizing constant as h i…mÿp†=2 ÿ
ÿ1 0 1=2 ÿ1 0 2C m‡m X j dX d jX X 2 : …11† U…m; p† ˆ
ÿm
p=2 ÿ …mp† C mÿp C 2 2 Clearly, the joint density of b and s depends on q. The distribution of d for given q is found by applying the rule of conditional probability under Eqs. (5) and (10), as follows: h iÿ…mÿp†=2 ; …12† pq …djq† ˆ U …m; p†sÿ…mÿpÿ1†  Xÿ1=2 jX Xÿ1 X 0 jÿ1=2 dXÿ1 d0 where U …m; p† is a constant of proportionality and does not depend on q.

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Since q is the only unknown quantity in the above density, the marginal likelihood function of q can be expressed in the following from: L…qjd† ˆ U …m; p†  jXi j

1=2

jX Xÿ1 X 0 j

ÿ1=2

‰dXÿ1 d0 Š

ÿ…mÿp†=2

:

…13†

It may be noted that X is a nonsingular pattern matrix its determinant and inverse are given by (see, [9]), jXj ˆ …1 ÿ q†mÿ1 ‰1 ‡ …m ÿ 1†qŠ and sq ˆ Xÿ1 ˆ …f ÿ g†  Im ‡ g  Nm ;

…14†

respectively. In Eq. (14) above, Im is the identity matrix of order m; Nm is an m  m matrix of ones; and 1 f ˆ f1 ‡ …m ÿ 2†qg; k 1 g ˆ fÿqg k in which

…15†

k ˆ …1 ÿ q†f1 ‡ …m ÿ 1†qg: Furthermore, we can write

m m m X X X ÿ1 0 0 0 lq ˆ jX X X j ˆ f det xj xj ‡ g det xj xj ; jˆ1 jˆ1 j6ˆj0 ˆ1

and ®nally, ÿ1 0

jq ˆ dX d ˆ f ÿ g ‡ g

m X

!2 dj

ˆ f ‡ 2gr;

…16†

jˆ1

P where r ˆ mÿ1 jˆ1 dj dj‡1 is the sample correlation coecient between the successive component of the sample responses, and det; denotes the determinant operator of the matrix. Note that dj2 ˆ 1 as dj ˆ …uj ÿ bxj †=s. Putting together the results in Eqs. (14)±(16) in Eq. (13) we get L…q† ˆ …sq †

ÿ1=2

 …lq †

ÿ1=2

ÿ…mÿp†=2

 …jq †

…17†

up to a multiplicative constant. Clearly, L…q† depends on d only through s, the sample estimate of q. Since all the terms in Eq. (17) are known, given a set of observed responses, except for the q, therefore, the value of q that maximizes the marginal likelihood function is considered to be the marginal likelihood estimate for the equicorrelation parameter. We would denote such an estimate by q^. This estimate

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of q appears in the prediction distribution of a set of future responses in the subsequent sections. 4. The distribution of realized and future errors We begin this section by considering a sample of future responses of size q P 1; an arbitrary integer number, from the model. The future responses can be expressed by the following equation: z ˆ bX f ‡ rv;

…18†

f

where X is the p  q future design matrix, and v is the 1  q vector of future scaled errors associated with the future response vector, z. The assumptions about v are identical to those on u in Section 2. Now, we have realized but unobserved scaled error u as well as the future scaled error v. To derive the joint distribution of these two error vectors, let us de®ne w ˆ …u; v†, the combined error vector of order 1  n, where n ˆ m ‡ q. Since the expected value of the combined error vector is an n-dimensional row vector of zeros, and the components of w are equicorrelated so its covariance matrix would be of the same pattern as X, but its order would be n  n. Note that u and v are neither independent nor uncorrelated, rather they are correlated. Therefore, their joint density cannot be written as the product of the marginals. If the realized but unobserved error vector is uncorrelated to the future error vector it is easier to write the quadratic form in the exponent of the density as the sum of two uncorrelated quadratic forms. Unfortunately, it is not the case with the present model. Khan [7] has considered a similar multiple regression model with dependent but uncorrelated errors to construct a b-expectation tolerance region where the errors follow heteroscedastic multivariate Student-t model. To write the joint density of the n components of w we have to use the following notations. The mean vector and the covariance matrix can be expressed as E…w† ˆ O, row vector of n zeros, and 0 1 1 q q


q B q 1 q


q C B C cov…w† ˆ R ˆ B C; @














A q

q

q






1

respectively, where R is a square matrix of order n, and has the same pattern as X. Secondly, the inverse of the covariance matrix can be partitioned as follows: ! R11 R12 ÿ1 ; …19† R ˆ R21 R22

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where R11 , and R22 are square matrices of order m and q, respectively and 0 R12 ˆ …R21 † is an m  q matrix. The quadratic form in the exponent part of the density function of w can be expressed as wRÿ1 w0 ˆ uR11 u0 ‡ 2uR12 v0 ‡ vR22 v0 :

…20†

Therefore, the joint density function of the unobserved but realized and the future error vectors becomes   ÿ…m‡m‡q†=2 1 …21† p…w† ˆ W…
†  1 ‡ uR11 u0 ‡ 2uR12 v0 ‡ vR22 v0 m where W…
† is the normalizing constant. To evaluate W…
†, one may note that uR11 u0 ‡ 2uR12 v0 ‡ vR22 v0 ˆ u ‡ vR21 …R11 †ÿ1 Š…R11 †ÿ1 ‰u ‡ vR21 …R11 †ÿ1 Š0 ÿ1

‡ v‰R22 ÿ R21 …R11 † R12 Šv0 :

…22†

We do not need to compute to the normalizing constant explicitly at this point. In fact, we would require to evaluate the normalizing constant for a modi®ed form of the density function. However, in Section 5 we need to establish the structural relation of the model that enables us to ®nd the prediction density of the future responses from the joint density of the errors. 5. The structural relation of the model The future responses described in Eq. (18) can be expressed in the following way: z ˆ hv; …23† where h ˆ ‰b; rŠ is an element of a transformation group, G that connect the error with the response by acting through X f as follows: ‰b; rŠv ˆ bX f ‡ v: The inverse of the group element is given by hÿ1 ˆ ‰b; rŠÿ1 ˆ ‰ÿrÿ1 b; rÿ1 Š: De®nition 5.1. A function t…
† that maps the sample space into the group G is a transformation variable if, t…gv† ˆ gt…v† for all g 2 G; and v in the sample space. For further detail about the transformation variable as well as transformation group, please refer to Fraser ([22], p. 25). It can be easily veri®ed that t…u† ˆ ‰b…u†; s…u†Š; and t…y† ˆ ‰b…y†; s…y†Š are transformation variables where b…y† and s…y† are de®ned exactly the same way as b…u† and s…u† by replacing u by y in Eq. (7).

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Now from the de®nition of the transformation variable the structural relation of the model, the regression equation in Eq. (1) can be written as t…y† ˆ ht…u†; which gives h ˆ t…y†tÿ1 …u†;



 b…u† 1 ; : ˆ ‰b…y†; s…y†Š ÿ s…u† s…u†

Substituting this value of h in Eq. (23) we obtain   b 1 z ˆ ‰b…y†; s…y†Š ÿ ; v ˆ ‰b…y†; s…y†Šsÿ1 fv ÿ bX f g; s s

…24†

…25†

where b ˆ b…u† and s ˆ s…u†, as de®ned earlier in the previous section. Further simpli®cation and rearrangement yield for following structural relation for the model sÿ1 …y†fz ÿ b…y†X f g ˆ sÿ1 fv ÿ bX f g:

…26†

The right-hand side of the structural relation is a function of the errors and the future design matrix, X f only. Given a set of sample data, y is the only random variable on the left-hand side of Eq. (26) which is the future response vector, z (see [21], for further details on structural relation). It is evident from the above relation that the distribution of the future responses, z can be derived from the joint density function of b; s and v by making appropriate transformations. Therefore, to derive the prediction density of z, we ®rst require to ®nd the joint density function of b; s and v from Eq. (21). Then making a suitable transformation, we can obtain the density function of the statistic on the right-hand side of Eq. (26), which yields the prediction distribution for the set of future responses, z, given in Eq. (18). 6. The distribution of future responses This section would use the same arguments as were used in Eq. (10) to obtain the derivation of the density. Thus the joint density of b; s and v is obtained from the density function of w, as  ÿ…m‡m‡q†=2 1 mÿpÿ1 1 ‡ fQg …27† pq …b; s; vjd† ˆ !…m; q; p†s m where !…m; q; p† is the normalizing constant, and Q is the following quadratic form: Q ˆ bX R11 X 0 b0 ‡ s2 dR11 d0 ‡ 2bR12 v0 ‡ vR22 v0 :

…28†

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Note the following representation of the quadratic form in the exponent of Eq. (27) to evaluate the normalizing constant 0

Q ˆ …b ‡ vR21 X 0 Eÿ1 †‰EŠ…b ‡ vR21 X 0 Eÿ1 † ‡ vF v0 ‡ s2 dR11 d0 ;

…29†

where E ˆ X R11 X 0 and F ˆ R22 ÿ R21 X 0 Eÿ1 X R12 are two p  p and q  q positive de®nite matrices, respectively. Since the joint density function in Eq. (27) does not factor in to the marginals, the normalizing constant is obtained by applying the multivariate Student-t integral for both b and v and the Gamma integral for s, and then multiplying them together. This yields !…m; q; p† ˆ

jEj

1=2

jF j

1=2

‰dR11 d0 Š

…mÿp†=2

…30†

…2†……q‡p†=2†ÿ1 …p†…q‡p†=2 C…m ÿ p=2†

Consider the following set of transformations to ®nd the density function of the statistic on the right-hand side of Eq. (26) from the above density function: / ˆ b;

w ˆ sÿ1 fv ÿ bX f g:

w ˆ s;

…31†

Since w is the same as the right-hand side of the structural relation, and we want to derive its density function from the joint density of the transformed variables. The transformation is one-to-one and the Jacobean turns out to be J f‰b; s; vŠ ! ‰/; w; wŠg ˆ …w†q :

…32†

Nothing that the inverse transformation is v ˆ /X f ‡ ww; the quadratic form in Eq. (28) can be expressed as Q ˆ f/ ‡ wxH 0 M ÿ1 g‰MŠf/ ‡ wwH 0 M ÿ1 g0 ‡ w2 fdR11 d0 ‡ xRx0 g; where H ˆ X R12 ‡ X f R22 ; a p  q matrix; 11

f

22

f

0

12

f

0

…33† …34†

22

0

ÿ1

M ˆ X R ‡ X R X ‡ 2X R X ; a p  p matrix; and R ˆ R ÿ H M H a q  q matrix. The representation of the quadratic form Q ; in Eq. (33) along with the Jacobian factor in Eq. (32) make it easier to evaluate the normalizing constant for the joint density function of the new statistics /; w and w: Thus the joint density function of the transformed statistics becomes  ÿ…v‡m‡q†=2 1 ; …35† pq …/; w; wjd† ˆ ! …m; q; p†wm‡qÿpÿ1 1 ‡ fQ g m where ! …m; q; p† is the new normalizing constant. Integrating / over Rp , the qdimensional real space; w over R‡ , the positive half of the real line; and w over Rq , the q-dimensional real space, the normalizing constant is found to be statistics becomes

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ÿ
1=2 1=2 11 0 …mÿp†=2 2C m‡m‡q jMj jRj ‰dR d Š  2 : ! …m; q; p† ˆ …m‡q†=2 …p‡q†=2 ÿ mÿp
ÿ m
…m† …p† C 2 C 2

215

…36†

Then integrating the density function in Eq. (35) with respect to / and w, we obtain the marginal density function of x as follows:  m ‡ q ÿ p  m ÿ p 1=2 11 0 …mÿp†=2 …p†q=2 C pq …wjd† ˆ jRj ‰R d Š C 2 2 h iÿ…m‡qÿp†=2 : …37†  dR11 d0 ‡wRw0 This density function is expressed as the standard multivariate Student-t distribution in the following way: jRj1=2 C

ÿ m‡qÿp


…m ÿ p†…m‡qÿp†=2 pq …w=d† ˆ ÿ
11 0 q=2 …p†q=2 C mÿp ‰dR d Š 2  ÿ…m‡qÿp†=2 mÿp 0  …m ÿ p† ‡ wRw dR011 2

Now it is clearly observed that w has a q-dimensional multivariate Student-t distribution with …m ÿ p† degrees of freedom whose expected value and the covariance matrix become E…w† ˆ O, a row vector of q zeros, and cov…w† ˆ

dR11 d 0  Rÿ1 ; mÿpÿ2

a matrix of order q  q, respectively. Since w ˆ sÿ1 …y†fz ÿ b…y†X f g; the joint prediction distribution of the q future responses becomes " mÿp  fz ÿ b…y†X f g…R† pq …zjy† ˆ Ki …m; q; p† …m ÿ p† ‡ 2 s …y†dR11 d0 #ÿ…m‡qÿp†=2  fz ÿ b…y†X f g0 where

;

…39†

ÿ
…m‡qÿp†=2 C m‡qÿp …m ÿ p† 2 ; ÿ
…p†q=2 C mÿp ‰R011 Šq=2 sq …y† 2 1=2

K…m; q; p† ˆ

jRj

which depends on d only through y. Therefore, the prediction distribution of the future responses is a multivariate Student-t with …m ÿ p† degrees of freedom, whose location and scale parameters are given by E…z† ˆ b…y†X f ; a row vector of q elements, and

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s2 …y†dR11 d0  Rÿ1 ; mÿpÿ2 a matrix of order q  q; respectively. cov…z† ˆ

7. Discussions and conclusions In the foregoing sections the general derivation of the prediction distribution for the multiple regression model having equi-correlated responses has been provided. The correlation parameter q appears in the prediction density function Eq. (39). This has to be replaced by its marginal likelihood estimate, q^, as discussed earlier. Thus the matrices like R11 ; R12 ; R22 ; etc. would be function of q^ only. Therefore, for given w everything in the prediction distribution is known except the future responses, the only random variable. From the prediction distribution it is possible to ®nd the value of the probability that any set of future responses will be bounded by some given numbers by using the appropriate multivariate Student-t distribution, Furthermore, tolerance regions (or interval in one dimension), like some of those in Geisser [23], Khan [24], Aitchison and Dunsmore [13], and Guttman [12], could be constructed with desired con®dence level from the prediction distribution for a set of future responses. If q ˆ 1, that is, we want to make predictive inference about a single future response from the model, then the prediction distribution is a univariate Student-t distribution, but with the same degrees of freedom. References [1] M.I. Bhatti Optimal testing for equicorrelated linear regression models, Statistical Papers 36 (4) (1995a) 299±312. [2] M.I. Bhatti, Testing regression models based on sample survey data, Avebury Publishers, UK, 1995b. [3] M.I. Bhatti, M.L. King, A beta optimal test of the equicorrelation coecient, Aust. J. Statist. 32 (1) (1990) 87±97. [4] M.I. Bhatti, Optimal testing for block e€ects in regression models, The Third Paci®c Area Statistical Conference, Pre-prints, The Paci®c Statistical Institute, Tokyo, Japan. [5] M.I. Bhatti, Optimal testing for serial correlation in large number of small samples, Biometrical J. 34 (1) (1992) 57±66. [6] M.I. Bhatti, Ecient estimation of random coecients model based on survey data, J. Quantitative Economics 9 (1993) 99±110. [7] S. Khan, b-expectation tolerance region for the heteroscedastic multiple regression model with multivariate Student-t errors, Statistical Papers 35 (1994) 127±138. [8] P.X. Wu, M.I. Bhatti, Testing the block e€ects and misspeci®cation in regression models based on survey data, J. Statist. Comp. Sim. 50 (1-2) (1994) 75±90. [9] M.I. Bhatti, M.A. Barry, On testing for equicorrelation in random coecient models, Biometrical J. 37 (4) (1995) 383±391.

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