A preliminary test estimator for multivariate response functions
| dc.contributor.author | Blackmon, Paul W. | en |
| dc.contributor.department | Statistics | en |
| dc.date.accessioned | 2017-03-10T15:15:03Z | en |
| dc.date.available | 2017-03-10T15:15:03Z | en |
| dc.date.issued | 1974 | en |
| dc.description.abstract | If y₁, y₂, ... , y <sub>p</sub> represent vectors of independent observations, the generalized multivariate regression model is of the form y<sub>j</sub> = X<sub>1j</sub> β<sub>1j</sub> + X<sub>2j</sub> β<sub>2j</sub> + ε<sub>j</sub> , j = 1, 2, …, p, where X<sub>1j</sub> and X<sub>2j</sub> are general linear model regression matrices, β<sub>1j</sub> and β<sub>2j</sub> are vectors of unknown coefficients, and the ε<sub>j</sub> are error vectors such that cov(ε<sub>i</sub>,ε<sub>j</sub>) = σ<sub>ij</sub>I. If X<sub>1j</sub> = X₁ and X<sub>2j</sub> = 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<sub>j</sub> is n<sub>j</sub> = x<sub>1j</sub><sup>’</sup> β<sub>1j</sub> + x<sub>2j</sub><sup>’</sup> β<sub>2j</sub> where x<sub>1j</sub><sup>’</sup> x<sub>2j</sub><sup>’</sup> are typical row vectors in the matrices X<sub>1j</sub> and X<sub>2j</sub>. For x<sub>j</sub><sup>*</sup> = [x<sub>1j</sub><sup>’</sup>, x<sub>2j</sub><sup>’</sup>] and β<sub>j</sub><sup>*</sup> = [ß<sub>1j</sub><sup>’</sup>, β<sub>2j</sub><sup>’</sup>], the n<sub>j</sub> are to be estimated either by ŷ<sub>j</sub> = x<sub>1j</sub><sup>’</sup>β̂<sub>1j</sub> or ŷ<sub>j</sub>* = x<sub>j</sub><sup>*</sup>’ β̂<sub>j</sub><sup>*</sup> where β̂<sub>1j</sub> and β̂<sub>j</sub><sup>*</sup> are the least squares estimators of β<sub>1j</sub> and β<sub>j</sub><sup>*</sup> obtained from the full multivariate regression model. The estimators for the n<sub>j</sub> are determined by a test of the hypothesis H<sub>o</sub>: J₁ ≤ J₂ where J₁ and J₂ denote the integrated mean squared errors of a linear combination of the ŷ<sub>j</sub> and ŷ<sub>j</sub>* respectively. Rejection of H<sub>o</sub> results in selection of the ŷ<sub>j</sub>*; otherwise the ŷ<sub>j</sub> are chosen. A test statistic is developed to test H<sub>o</sub> with consideration extending to several important special cases. Distinctions are drawn between the preliminary test estimator constructed around H<sub>o</sub>, and that based on the usual hypothesis β<sub>2j</sub> = 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<sub>o</sub>, the integrated mean squared error of the preliminary test estimator, is obtained, and difficulties in its evaluation are discussed. An estimator of J<sub>o</sub> is presented along with a special case in which J<sub>o</sub> can be evaluated exactly. Graphical comparisons are made on the relative performance of the estimators based on H<sub>o</sub> , and those constructed around the standard hypothesis. An operating range of type I error probabilities is also discussed. | en |
| dc.description.degree | Ph. D. | en |
| dc.format.extent | iv, 85 leaves | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/10919/76071 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Polytechnic Institute and State University | en |
| dc.relation.isformatof | OCLC# 34239050 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.lcc | LD5655.V856 1974.B56 | en |
| dc.title | A preliminary test estimator for multivariate response functions | en |
| dc.type | Dissertation | en |
| dc.type.dcmitype | Text | en |
| thesis.degree.discipline | Statistics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Ph. D. | en |
Files
Original bundle
1 - 1 of 1