In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and the asymptotic variance of a MELE will not increase as the number of UGEE's increases. We give several important examples and demonstrate that efficient estimates of moment based distribution characteristics in the presence of side information can be obtained using JEL for U-statistics.
In the second part, we propose several JEL goodness-of-fit tests for spherical symmetry, rotational symmetry, antipodal symmetry, coordinatewise symmetry and exchangeability. We employ the jackknife empirical likelihood for vector U-statistics to incorporate side information. We use estimated constraint functions and allow the number of constraints and the dimension to grow with the sample size so that these tests can be used to test hypotheses for high dimensional symmetries. We demonstrate that these tests are distribution free and asymptotically chisquare distributed. We conduct extensive simulations to evaluate the performance of these tests.
","abstract_html":"<p>In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and the asymptotic variance of a MELE will not increase as the number of UGEE's increases. We give several important examples and demonstrate that efficient estimates of moment based distribution characteristics in the presence of side information can be obtained using JEL for U-statistics.</p> <p>In the second part, we propose several JEL goodness-of-fit tests for spherical symmetry, rotational symmetry, antipodal symmetry, coordinatewise symmetry and exchangeability. We employ the jackknife empirical likelihood for vector U-statistics to incorporate side information. We use estimated constraint functions and allow the number of constraints and the dimension to grow with the sample size so that these tests can be used to test hypotheses for high dimensional symmetries. We demonstrate that these tests are distribution free and asymptotically chisquare distributed. We conduct extensive simulations to evaluate the performance of these tests.</p>","abstract_has_math":false,"creators":["Li, Lingnan"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hanxiang Peng","Benzion Boukai","Guang Cheng","Wei Zheng"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-08-01T07:00:00Z","date_published":"2016-08-01T07:00:00Z","updated_at":"2026-07-24T03:53:55Z","subjects":["Pure sciences","Jackknife pseudo value","Maximum empirical likelihood estimate","U-statistics","U-statistics based general estimating equations","Mathematics","Statistics and Probability"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/796","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hanxiang Peng","Benzion Boukai","Guang Cheng","Wei Zheng"]},{"key":"dc:creator","label":"Author","values":["Li, Lingnan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Jackknife pseudo value","Maximum empirical likelihood estimate","U-statistics","U-statistics based general estimating equations","Mathematics","Statistics and Probability"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/796"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and the asymptotic variance of a MELE will not increase as the number of UGEE's increases. We give several important examples and demonstrate that efficient estimates of moment based distribution characteristics in the presence of side information can be obtained using JEL for U-statistics.
In the second part, we propose several JEL goodness-of-fit tests for spherical symmetry, rotational symmetry, antipodal symmetry, coordinatewise symmetry and exchangeability. We employ the jackknife empirical likelihood for vector U-statistics to incorporate side information. We use estimated constraint functions and allow the number of constraints and the dimension to grow with the sample size so that these tests can be used to test hypotheses for high dimensional symmetries. We demonstrate that these tests are distribution free and asymptotically chisquare distributed. We conduct extensive simulations to evaluate the performance of these tests.
"]},{"key":"dc:title","label":"Title","values":["Maximum empirical likelihood estimation in U-statistics based general estimating equations"]}]}],"canonical_facts":{"dc:contributor":["Hanxiang Peng","Benzion Boukai","Guang Cheng","Wei Zheng"],"dc:creator":["Li, Lingnan"],"dc:description.abstract":["In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and the asymptotic variance of a MELE will not increase as the number of UGEE's increases. We give several important examples and demonstrate that efficient estimates of moment based distribution characteristics in the presence of side information can be obtained using JEL for U-statistics.
In the second part, we propose several JEL goodness-of-fit tests for spherical symmetry, rotational symmetry, antipodal symmetry, coordinatewise symmetry and exchangeability. We employ the jackknife empirical likelihood for vector U-statistics to incorporate side information. We use estimated constraint functions and allow the number of constraints and the dimension to grow with the sample size so that these tests can be used to test hypotheses for high dimensional symmetries. We demonstrate that these tests are distribution free and asymptotically chisquare distributed. We conduct extensive simulations to evaluate the performance of these tests.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/796"],"dc:subject":["Pure sciences","Jackknife pseudo value","Maximum empirical likelihood estimate","U-statistics","U-statistics based general estimating equations","Mathematics","Statistics and Probability"],"dc:title":["Maximum empirical likelihood estimation in U-statistics based general estimating equations"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:55Z"}