A preliminary test estimator for multivariate response functions

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Virginia Polytechnic Institute and State University


If y₁, y₂, ... , y p represent vectors of independent observations, the generalized multivariate regression model is of the form yj = X1j β1j + X2j β2j + εj , j = 1, 2, …, p, where X1j and X2j are general linear model regression matrices, β1j and β2j are vectors of unknown coefficients, and the εj are error vectors such that cov(εij) = σijI. If X1j = X₁ and X2j = X₂ , j = 1, 2, …, p, the above is a standard multivariate regression model .

Insofar as can be determined, the true relationship between the design variables and a response nj is nj = x1j β1j + x2j β2j where x1j x2j are typical row vectors in the matrices X1j and X2j. For xj = [x1j, x2j] and βj = [ß1j, β2j], the nj are to be estimated either by ŷj = x1jβ̂1j or ŷj* = xj’ β̂j where β̂1j and β̂j are the least squares estimators of β1j and βj obtained from the full multivariate regression model.

The estimators for the nj are determined by a test of the hypothesis Ho: J₁ ≤ J₂ where J₁ and J₂ denote the integrated mean squared errors of a linear combination of the ŷj and ŷj* respectively. Rejection of Ho results in selection of the ŷj*; otherwise the ŷj are chosen.

A test statistic is developed to test Ho with consideration extending to several important special cases. Distinctions are drawn between the preliminary test estimator constructed around Ho, and that based on the usual hypothesis β2j = 0, j = 1, 2, ..., p.

Under the assumption of error normality, an approximation to the distribution of the test statistic is developed in order to determine type I and type II error probabilities.

An explicit expression for Jo, the integrated mean squared error of the preliminary test estimator, is obtained, and difficulties in its evaluation are discussed. An estimator of Jo is presented along with a special case in which Jo can be evaluated exactly.

Graphical comparisons are made on the relative performance of the estimators based on Ho , and those constructed around the standard hypothesis. An operating range of type I error probabilities is also discussed.