{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/71281"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/71281","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Some nonparametric tests for constancy of regression relationships over time","abstract":"Let Y₁, Y₂... be a sequence of random variables obeying the law Y<sub>i</sub> = β’<sub>i</sub> + ε<sub>i</sub>, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = β<sub>m</sub> = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y<sub>m</sub>. This study develops nonparametric procedures for testing H₀: = β = β<sub>m+1</sub> = β<sub>m+2</sub> = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y<sub>m+1</sub> - β̂’₀ x<sub>m+1</sub>, …, Y<sub>m+N</sub> - β̂’₀ x<sub>m+N</sub> is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained.","abstract_html":"Let Y₁, Y₂... be a sequence of random variables obeying the law Y&lt;sub&gt;i&lt;/sub&gt; = β’&lt;sub&gt;i&lt;/sub&gt; + ε&lt;sub&gt;i&lt;/sub&gt;, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = β&lt;sub&gt;m&lt;/sub&gt; = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y&lt;sub&gt;m&lt;/sub&gt;. This study develops nonparametric procedures for testing H₀: = β = β&lt;sub&gt;m+1&lt;/sub&gt; = β&lt;sub&gt;m+2&lt;/sub&gt; = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y&lt;sub&gt;m+1&lt;/sub&gt; - β̂’₀ x&lt;sub&gt;m+1&lt;/sub&gt;, …, Y&lt;sub&gt;m+N&lt;/sub&gt; - β̂’₀ x&lt;sub&gt;m+N&lt;/sub&gt; is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained.","abstract_has_math":false,"creators":["Roller, William Frederick"],"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":1977,"date_issued":"1977","date_published":"1977","updated_at":"2026-07-24T05:56:05Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/71281","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":["Roller, William Frederick"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-05-23T18:29:20Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-05-23T18:29:20Z"]},{"key":"dc:date.issued","label":"Date","values":["1977"]},{"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/10919/71281"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let Y₁, Y₂... be a sequence of random variables obeying the law Y<sub>i</sub> = β’<sub>i</sub> + ε<sub>i</sub>, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = β<sub>m</sub> = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y<sub>m</sub>. This study develops nonparametric procedures for testing H₀: = β = β<sub>m+1</sub> = β<sub>m+2</sub> = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y<sub>m+1</sub> - β̂’₀ x<sub>m+1</sub>, …, Y<sub>m+N</sub> - β̂’₀ x<sub>m+N</sub> is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained."]},{"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":["Some nonparametric tests for constancy of regression relationships over time"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Roller, William Frederick"],"dc:date.accessioned":["2016-05-23T18:29:20Z"],"dc:date.available":["2016-05-23T18:29:20Z"],"dc:date.issued":["1977"],"dc:description.abstract":["Let Y₁, Y₂... be a sequence of random variables obeying the law Y<sub>i</sub> = β’<sub>i</sub> + ε<sub>i</sub>, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = β<sub>m</sub> = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y<sub>m</sub>. This study develops nonparametric procedures for testing H₀: = β = β<sub>m+1</sub> = β<sub>m+2</sub> = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y<sub>m+1</sub> - β̂’₀ x<sub>m+1</sub>, …, Y<sub>m+N</sub> - β̂’₀ x<sub>m+N</sub> is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/71281"],"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":["Some nonparametric tests for constancy of regression relationships over time"],"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":"2026-07-24T05:56:05Z"}