A preliminary test estimator for multivariate response functions

If y₁, y₂, ... , y _p represent vectors of independent observations, the generalized multivariate regression model is of the form y_j = X_1j β_1j + X_2j β_2j + ε_j , j = 1, 2, …, p, where X_1j and X_2j are general linear model regression matrices, β_1j and β_2j are vectors of unknown coefficients, and the ε_j are error vectors such that cov(ε_i,ε_j) = σ_ijI. If X_1j = X₁ and X_2j = 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 n_j is n_j = x_1j^’ β_1j + x_2j^’ β_2j where x_1j^’ x_2j^’ are typical row vectors in the matrices X_1j and X_2j. For x_j^* = [x_1j^’, x_2j^’] and β_j^* = [ß_1j^’, β_2j^’], the n_j are to be estimated either by ŷ_j = x_1j^’β̂_1j or ŷ_j* = x_j^*’ β̂_j^* where β̂_1j and β̂_j^* are the least squares estimators of β_1j and β_j^* obtained from the full multivariate regression model. The estimators for the n_j are determined by a test of the hypothesis H_o: J₁ ≤ J₂ where J₁ and J₂ denote the integrated mean squared errors of a linear combination of the ŷ_j and ŷ_j* respectively. Rejection of H_o results in selection of the ŷ_j*; otherwise the ŷ_j are chosen. A test statistic is developed to test H_o with consideration extending to several important special cases. Distinctions are drawn between the preliminary test estimator constructed around H_o, and that based on the usual hypothesis β_2j = 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 J_o, the integrated mean squared error of the preliminary test estimator, is obtained, and difficulties in its evaluation are discussed. An estimator of J_o is presented along with a special case in which J_o can be evaluated exactly. Graphical comparisons are made on the relative performance of the estimators based on H_o , and those constructed around the standard hypothesis. An operating range of type I error probabilities is also discussed.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Statistics

Department dc:contributor.department

Statistics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1974

Author dc:creator

• Blackmon, Paul W.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10919/76071

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/76071