{"id":{"repo_id":"njit","oai_identifier":"oai:digitalcommons.njit.edu:dissertations-1405"},"canonical_url":"https://search.dev.ndltd.org/etd/njit/oai:digitalcommons.njit.edu:dissertations-1405","repository":{"repo_id":"njit","name":"NJIT","base_url":"https://digitalcommons.njit.edu/do/oai/"},"display":{"title":"The goodness-of-fit tests for geometric models","abstract":"We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF's maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for the second type of test by calculating the empirical critical points and constructing the density curves. We compare the proposed tests with Chi-square and the empirical distribution function (EDF) related tests proposed in the literature in terms of significance level and power. Based on the comparison results, we recommend the second type of goodness-of fit test for both geometric distribution and BGD(B&D) because of its robustness, efficiency in computation and no need for selecting t. Real data sets are used for illustration.","abstract_html":"We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF's maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for the second type of test by calculating the empirical critical points and constructing the density curves. We compare the proposed tests with Chi-square and the empirical distribution function (EDF) related tests proposed in the literature in terms of significance level and power. Based on the comparison results, we recommend the second type of goodness-of fit test for both geometric distribution and BGD(B&D) because of its robustness, efficiency in computation and no need for selecting t. Real data sets are used for illustration.","abstract_has_math":false,"creators":["Chen, Feiyan"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematical Sciences - (Ph.D.)","degree_level":null,"degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Sunil Kumar Dhar","Sundarraman Subramanian","Aridaman Kumar Jain"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-01-31T08:00:00Z","date_published":"2013-01-31T08:00:00Z","updated_at":"2026-07-24T03:22:14Z","subjects":["Bivariate geometric distribution","Hypothesis test","Probability generating function","Parametric bootstrap","Power","Empirical probability generating function","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.njit.edu/dissertations/350","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sunil Kumar Dhar","Sundarraman Subramanian","Aridaman Kumar Jain"]},{"key":"dc:creator","label":"Author","values":["Chen, Feiyan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematical Sciences - (Ph.D.)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Bivariate geometric distribution","Hypothesis test","Probability generating function","Parametric bootstrap","Power","Empirical probability generating function","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.njit.edu/dissertations/350"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF's maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for the second type of test by calculating the empirical critical points and constructing the density curves. We compare the proposed tests with Chi-square and the empirical distribution function (EDF) related tests proposed in the literature in terms of significance level and power. Based on the comparison results, we recommend the second type of goodness-of fit test for both geometric distribution and BGD(B&D) because of its robustness, efficiency in computation and no need for selecting t. Real data sets are used for illustration."]},{"key":"dc:title","label":"Title","values":["The goodness-of-fit tests for geometric models"]}]}],"canonical_facts":{"dc:contributor":["Sunil Kumar Dhar","Sundarraman Subramanian","Aridaman Kumar Jain"],"dc:creator":["Chen, Feiyan"],"dc:description.abstract":["We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF's maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for the second type of test by calculating the empirical critical points and constructing the density curves. We compare the proposed tests with Chi-square and the empirical distribution function (EDF) related tests proposed in the literature in terms of significance level and power. Based on the comparison results, we recommend the second type of goodness-of fit test for both geometric distribution and BGD(B&D) because of its robustness, efficiency in computation and no need for selecting t. Real data sets are used for illustration."],"dc:identifier":["https://digitalcommons.njit.edu/dissertations/350"],"dc:subject":["Bivariate geometric distribution","Hypothesis test","Probability generating function","Parametric bootstrap","Power","Empirical probability generating function","Mathematics"],"dc:title":["The goodness-of-fit tests for geometric models"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_name":["Doctor of Philosophy in Mathematical Sciences - (Ph.D.)"]},"updated_at":"2026-07-24T03:22:14Z"}