Prediction intervals for the realization of a random variable under a general mixed linear model

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1985
Authors
Jeske, Daniel
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Altmetrics
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Statistics
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Statistics
Abstract

Let y represent an n x 1 observable random vector and w an unobservable random variable, whose joint distribution is of the multivariate normal form with E(y) = X(beta) and E(w) = (lamda)'(beta), where X is a given n x k matrix, (lamda) is a given k x 1 vector and (beta) is a k x 1 vector of unknown parameters. The elements of the variance-covariance matrix of y and w are assumed to be known functions of a q x 1 vector (theta) of unknown parameters;The problem considered is that of constructing an approximate 100(1 - (alpha))% prediction interval for the realized value of w, that is, an interval P(,1)(y),P(,2)(y) such that the probability (with respect to the joint distribution of y and w) of the event P(,1)(y) < w < P(,2)(y) approxi- mates 1 - (alpha), for all (beta) and (theta);There exist functions p(,1)*((.);(alpha),(theta)) and p(,2)*((.);(alpha),(theta)), depending on (alpha) and (theta) such that P P(,1)*(y;(alpha),(theta)) < w < P(,2)*(y;(alpha),(theta)) = 1 - (alpha), for all (theta) and (beta). If (theta) were known, P(,1)*(y;(alpha),(theta)),P(,2)*(y;(alpha),(theta)) would be an exact 100(1 - (alpha))% prediction interval for the realization of w. When (theta) is unknown, an exact 100(1 - (alpha))% prediction interval for the realization of w does not exist (except in relatively simple special cases);Five types of approximate intervals were investigated: (1) naive intervals P(,1)*(y;(alpha),(')(theta)),P(,2)*(y;(alpha),(')(theta)) , where (')(theta) is an even translation-invariant estimator of (theta); (2) modified naive intervals (eta)(,B)(y;(')(theta)) (+OR-) t(,(alpha)/2) (nu)((')(theta)) M*((')(theta)) ('1/2), where (nu)((.)) is a specified function, t(,(alpha)/2) (nu)((')(theta)) is the upper (alpha)/2 point of the t-distribution with (nu)((')(theta)) degrees of freedom;and where M*((')(theta)) is an approximation to E (eta)(,B)(y;(')(theta))-w ('2); (3) conserva- tive intervals of the general form;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);where R represents a confidence set for (theta); (4) parametric bootstrap intervals (eta)(,B)(y;(')(theta)(,m)) (+OR-) k(,(alpha)/2)((')(theta)(,m)) M(,m)*((')(theta)(,m)) ('1/2), where k(,(alpha)/2)((theta)) is the upper (alpha)/2 point of the distribution of (eta)(,B)(y;(')(theta)(,m))-w / M*((')(theta)(,m)) ('1/2) and (')(theta)(,m) is the maximum likelihood estimator of (theta); (5) Bayesian credibility intervals;Results on the properties of these intervals are given for the general case and for the following two special cases: (1) the comparison of the means of two independent normal populations with unknown and unequal variances (the Behrens-Fisher problem), and (2) the prediction of a group mean in the balanced one-way random model. The modified naive intervals, the conservative intervals and the parametric bootstrap intervals seem to provide satisfactory approximate prediction intervals.

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