{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/87300"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/87300","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A new estimation procedure for response surface models","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 = β₀ + Σi=1pβixi + Σi=1pβiixi² + Σi=1p-1Σj=1p-1βijxixj. i < j We fit a model ŷ(x) = β̂̂₀ + Σi=1pk̂iβ̂ixi . We show that the ki which minimize Σi=1pE(kiβ̂i - βi)² are of the form ki = (μ₂Nβi²)/(σ² + μ₂Nβi²) We propose estimating ki by k̂i where k̂i = (μ₂Nβ̂i²)/(σ̂² + μ₂Nβ̂̂i²) 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 k̂i is derived and the probability that k̂i is closer to the optimal ki 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.","abstract_html":"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.","abstract_has_math":false,"creators":["Malone, Linda C."],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Statistics","degree_department":"Statistics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1975,"date_issued":"1975","date_published":"1975","updated_at":"2026-07-22T22:19:12Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/87300","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Statistics"]},{"key":"dc:creator","label":"Author","values":["Malone, Linda C."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-01-31T19:03:46Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-01-31T19:03:46Z"]},{"key":"dc:date.issued","label":"Date","values":["1975"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/87300"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 = β₀ + Σi=1pβixi + Σi=1pβiixi² + Σi=1p-1Σj=1p-1βijxixj. i < j We fit a model ŷ(x) = β̂̂₀ + Σi=1pk̂iβ̂ixi . We show that the ki which minimize Σi=1pE(kiβ̂i - βi)² are of the form ki = (μ₂Nβi²)/(σ² + μ₂Nβi²) We propose estimating ki by k̂i where k̂i = (μ₂Nβ̂i²)/(σ̂² + μ₂Nβ̂̂i²) 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 k̂i is derived and the probability that k̂i is closer to the optimal ki 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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A new estimation procedure for response surface models"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Malone, Linda C."],"dc:date.accessioned":["2019-01-31T19:03:46Z"],"dc:date.available":["2019-01-31T19:03:46Z"],"dc:date.issued":["1975"],"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 = β₀ + Σi=1pβixi + Σi=1pβiixi² + Σi=1p-1Σj=1p-1βijxixj. i < j We fit a model ŷ(x) = β̂̂₀ + Σi=1pk̂iβ̂ixi . We show that the ki which minimize Σi=1pE(kiβ̂i - βi)² are of the form ki = (μ₂Nβi²)/(σ² + μ₂Nβi²) We propose estimating ki by k̂i where k̂i = (μ₂Nβ̂i²)/(σ̂² + μ₂Nβ̂̂i²) 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 k̂i is derived and the probability that k̂i is closer to the optimal ki 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."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/87300"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["A new estimation procedure for response surface models"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Statistics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:12Z"}