A new estimation procedure for response surface models
| dc.contributor.author | Malone, Linda C. | en |
| dc.contributor.department | Statistics | en |
| dc.contributor.department | Statistics | en |
| dc.date.accessioned | 2019-01-31T19:03:46Z | en |
| dc.date.available | 2019-01-31T19:03:46Z | en |
| dc.date.issued | 1975 | en |
| dc.description.abstract | Several attempts have been made to find an estimator of a response which will have a smaller integrated mean square error than existing procedures. In this work another such attempt is made by introducing a shrinkage procedure. Suppose the true functional relationship between a response η and ρ independent variables is η<x> = β₀ + Σ<sub>i=1</sub><sup>p</sup>β<sub>i</sub>x<sub>i</sub> + Σ<sub>i=1</sub><sup>p</sup>β<sub>ii</sub>x<sub>i</sub>² + Σ<sub>i=1</sub><sup>p-1</sup>Σ<sub>j=1</sub><sup>p-1</sup>β<sub>ij</sub>x<sub>i</sub>x<sub>j</sub>. i < j We fit a model ŷ(x) = β̂̂₀ + Σ<sub>i=1</sub><sup>p</sup>k̂<sub>i</sub>β̂<sub>i</sub>x<sub>i</sub> . We show that the k<sub>i</sub> which minimize Σ<sub>i=1</sub><sup>p</sup>E(k<sub>i</sub>β̂<sub>i</sub> - β<sub>i</sub>)² are of the form k<sub>i</sub> = (μ₂Nβ<sub>i</sub>²)/(σ² + μ₂Nβ<sub>i</sub>²) We propose estimating k<sub>i</sub> by k̂<sub>i</sub> where k̂<sub>i</sub> = (μ₂Nβ̂<sub>i</sub>²)/(σ̂² + μ₂Nβ̂̂<sub>i</sub>²) and where β̂̂ = (X₁′X₁)⁻¹X₁′y and σ̂² = (y′y - β̂′X′y)/(N-p) are the usual least squares estimators. A A The distribution of k̂<sub>i</sub> is derived and the probability that k̂<sub>i</sub> is closer to the optimal k<sub>i</sub> than a k using upper bounds on the parameters is computed. Also an expression for the integrated mean square error of the proposed procedure is found. Various comparisons among least squares estimation minimum variance and minimum bias designs and optimal least squares and shrink.age estimation are made for the one, two, and three variable cases. | en |
| dc.description.degree | Ph. D. | en |
| dc.format.extent | v, 77 pages, 2 unnumbered leave | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/10919/87300 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Polytechnic Institute and State University | en |
| dc.relation.isformatof | OCLC# 40153313 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.lcc | LD5655.V856 1975.M34 | en |
| dc.title | A new estimation procedure for response surface models | 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 |
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