{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/101276"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/101276","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Minimum bias designs for an exponential response","abstract":"For the exponential response ηu = α + βeγZu (u = 1,2,…,N) where α and β lie on the real line (-∞,∞), and γ is a positive integer; the designs are given which minimize the bias due to the inherent inability of the approximation function ŷu = Σj=0dbjejZu to fit such a model. Transformation to ηu = α + βxuγ and ŷu = Σj=0dbjxuj facilitates the solution for minimum bias designs. The requirements for minimum bias designs follow along lines similar to those given by Box and Draper (J. Amer. Stat. Assoc., 54, 1959, p. 622). The minimum bias designs are obtained for specific values of N with a maximum protection level, γd*(N), for the parameter γ and an approximation function of degree d. These designs obtained possess several degrees of freedom in the choice of the design levels of the xu or the Zuu , which may be used to satisfy additional design requirements. It is shown that for a given N, the same designs which minimize bias for approximation functions of degree one also minimize bias for general degree d, with a decrease in γd*(N) as d increases. In fact γd*(N) = γ1*(N) - d + 1, but with the decrease in γd*(N) is a compensating decrease in the actual level of the minimum bias. Furthermore, γd*(N) increases monotonically with N, thereby allowing the maximum protection level on 1 to be increased as desired by increasing N. In the course of obtaining solutions, some interesting techniques are developed for determining the nature of the roots of a polynomial equation which has several known coefficients and several variable coefficients.","abstract_html":"For the exponential response η<sub>u</sub> = α + βe<sup>γZ<sub>u</sub></sup> (u = 1,2,…,N) where α and β lie on the real line (-∞,∞), and γ is a positive integer; the designs are given which minimize the bias due to the inherent inability of the approximation function ŷ<sub>u</sub> = Σ<sub>j=0</sub><sup>d</sub>b<sub>j</sub>e<sup>jZ<sub>u</sub></sup> to fit such a model. Transformation to η<sub>u</sub> = α + βx<sub>u</sub><sup>γ</sup> and ŷ<sub>u</sub> = Σ<sub>j=0</sub><sup>d</sub>b<sub>j</sub>x<sub>u</sub><sup>j</sup> facilitates the solution for minimum bias designs. The requirements for minimum bias designs follow along lines similar to those given by Box and Draper (J. Amer. Stat. Assoc., 54, 1959, p. 622). The minimum bias designs are obtained for specific values of N with a maximum protection level, γ<sub>d</sub>*(N), for the parameter γ and an approximation function of degree d. These designs obtained possess several degrees of freedom in the choice of the design levels of the x<sub>u</sub> or the Z<sub>u</sub>u , which may be used to satisfy additional design requirements. It is shown that for a given N, the same designs which minimize bias for approximation functions of degree one also minimize bias for general degree d, with a decrease in γ<sub>d</sub>*(N) as d increases. In fact γ<sub>d</sub>*(N) = γ<sub>1</sub>*(N) - d + 1, but with the decrease in γ<sub>d</sub>*(N) is a compensating decrease in the actual level of the minimum bias. Furthermore, γ<sub>d</sub>*(N) increases monotonically with N, thereby allowing the maximum protection level on 1 to be increased as desired by increasing N. In the course of obtaining solutions, some interesting techniques are developed for determining the nature of the roots of a polynomial equation which has several known coefficients and several variable coefficients.","abstract_has_math":false,"creators":["Manson, Allison Ray"],"institution":"Virginia Polytechnic Institute","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Statistics","degree_department":"Statistics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1965,"date_issued":"1965","date_published":"1965","updated_at":"2026-07-22T22:19:01Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/101276","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":["Manson, Allison Ray"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-12-14T17:35:57Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-12-14T17:35:57Z"]},{"key":"dc:date.issued","label":"Date","values":["1965"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"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. 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Transformation to ηu = α + βxuγ and ŷu = Σj=0dbjxuj facilitates the solution for minimum bias designs. The requirements for minimum bias designs follow along lines similar to those given by Box and Draper (J. Amer. Stat. Assoc., 54, 1959, p. 622). The minimum bias designs are obtained for specific values of N with a maximum protection level, γd*(N), for the parameter γ and an approximation function of degree d. These designs obtained possess several degrees of freedom in the choice of the design levels of the xu or the Zuu , which may be used to satisfy additional design requirements. It is shown that for a given N, the same designs which minimize bias for approximation functions of degree one also minimize bias for general degree d, with a decrease in γd*(N) as d increases. In fact γd*(N) = γ1*(N) - d + 1, but with the decrease in γd*(N) is a compensating decrease in the actual level of the minimum bias. Furthermore, γd*(N) increases monotonically with N, thereby allowing the maximum protection level on 1 to be increased as desired by increasing N. In the course of obtaining solutions, some interesting techniques are developed for determining the nature of the roots of a polynomial equation which has several known coefficients and several variable coefficients."]},{"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":["Minimum bias designs for an exponential response"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Manson, Allison Ray"],"dc:date.accessioned":["2020-12-14T17:35:57Z"],"dc:date.available":["2020-12-14T17:35:57Z"],"dc:date.issued":["1965"],"dc:description.abstract":["For the exponential response ηu = α + βeγZu (u = 1,2,…,N) where α and β lie on the real line (-∞,∞), and γ is a positive integer; the designs are given which minimize the bias due to the inherent inability of the approximation function ŷu = Σj=0dbjejZu to fit such a model. Transformation to ηu = α + βxuγ and ŷu = Σj=0dbjxuj facilitates the solution for minimum bias designs. The requirements for minimum bias designs follow along lines similar to those given by Box and Draper (J. Amer. Stat. Assoc., 54, 1959, p. 622). The minimum bias designs are obtained for specific values of N with a maximum protection level, γd*(N), for the parameter γ and an approximation function of degree d. These designs obtained possess several degrees of freedom in the choice of the design levels of the xu or the Zuu , which may be used to satisfy additional design requirements. It is shown that for a given N, the same designs which minimize bias for approximation functions of degree one also minimize bias for general degree d, with a decrease in γd*(N) as d increases. In fact γd*(N) = γ1*(N) - d + 1, but with the decrease in γd*(N) is a compensating decrease in the actual level of the minimum bias. Furthermore, γd*(N) increases monotonically with N, thereby allowing the maximum protection level on 1 to be increased as desired by increasing N. In the course of obtaining solutions, some interesting techniques are developed for determining the nature of the roots of a polynomial equation which has several known coefficients and several variable coefficients."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/101276"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Minimum bias designs for an exponential response"],"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"]},"updated_at":"2026-07-22T22:19:01Z"}