The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). Alternative estimators have been proposed to deal with the problems of non-existence and \"blowing up\" of the estimates. Mittal and Dahiya (1987, 1989) employ the Bayes model estimator of Blumenthal and Marcus (1975) for the normal and Weibull cases and Hegde and Dahiya (1989) apply it to the gamma. Hegde (1986) also studies the harmonic mean estimator of Joe and Reid (1984). Here we prove a sufficient and asymptotically necessary condition for the existence of the ML estimator of the scale parameter of the truncated Cauchy distribution. A modified ML estimator and an estimator based on equating population and sample quantiles are presented as alternatives. These estimators exist with probability one. The performance of these estimators is examined by making use of simulations. Asymptotic variances of the ML estimators are also given.
Finally, an application of truncated distributions is presented. The fit of returns on common stocks to the normal, Cauchy, truncated normal, and truncated Cauchy distributions is compared via the Kolmogorov-Smirnov statistic. The results show that a truncated distribution is a better fitting model in virtually all cases.
","abstract_html":"<p>The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). Alternative estimators have been proposed to deal with the problems of non-existence and "blowing up" of the estimates. Mittal and Dahiya (1987, 1989) employ the Bayes model estimator of Blumenthal and Marcus (1975) for the normal and Weibull cases and Hegde and Dahiya (1989) apply it to the gamma. Hegde (1986) also studies the harmonic mean estimator of Joe and Reid (1984). Here we prove a sufficient and asymptotically necessary condition for the existence of the ML estimator of the scale parameter of the truncated Cauchy distribution. A modified ML estimator and an estimator based on equating population and sample quantiles are presented as alternatives. These estimators exist with probability one. The performance of these estimators is examined by making use of simulations. Asymptotic variances of the ML estimators are also given.</p> <p>Finally, an application of truncated distributions is presented. The fit of returns on common stocks to the normal, Cauchy, truncated normal, and truncated Cauchy distributions is compared via the Kolmogorov-Smirnov statistic. The results show that a truncated distribution is a better fitting model in virtually all cases.</p>","abstract_has_math":false,"creators":["Staneski, Paul G."],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Ram C. Dahiya","Bruce Rubin","N. Rao Chaganty"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1990,"date_issued":"1990-04-01T08:00:00Z","date_published":"1990-04-01T08:00:00Z","updated_at":"2026-07-24T03:35:23Z","subjects":["Cauchy distribution","Stock returns","Estimates","Applied Statistics","Finance","Partial Differential Equations"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.odu.edu/mathstat_etds/101","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ram C. Dahiya","Bruce Rubin","N. Rao Chaganty"]},{"key":"dc:creator","label":"Author","values":["Staneski, Paul G."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-10-16T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics & Statistics"]},{"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":["Cauchy distribution","Stock returns","Estimates","Applied Statistics","Finance","Partial Differential Equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.odu.edu/mathstat_etds/101"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). Alternative estimators have been proposed to deal with the problems of non-existence and \"blowing up\" of the estimates. Mittal and Dahiya (1987, 1989) employ the Bayes model estimator of Blumenthal and Marcus (1975) for the normal and Weibull cases and Hegde and Dahiya (1989) apply it to the gamma. Hegde (1986) also studies the harmonic mean estimator of Joe and Reid (1984). Here we prove a sufficient and asymptotically necessary condition for the existence of the ML estimator of the scale parameter of the truncated Cauchy distribution. A modified ML estimator and an estimator based on equating population and sample quantiles are presented as alternatives. These estimators exist with probability one. The performance of these estimators is examined by making use of simulations. Asymptotic variances of the ML estimators are also given.
Finally, an application of truncated distributions is presented. The fit of returns on common stocks to the normal, Cauchy, truncated normal, and truncated Cauchy distributions is compared via the Kolmogorov-Smirnov statistic. The results show that a truncated distribution is a better fitting model in virtually all cases.
"]},{"key":"dc:title","label":"Title","values":["The Truncated Cauchy Distribution: Estimation of Parameters and Application to Stock Returns"]}]}],"canonical_facts":{"dc:contributor":["Ram C. Dahiya","Bruce Rubin","N. Rao Chaganty"],"dc:creator":["Staneski, Paul G."],"dc:date.available":["2019-10-16T07:00:00Z"],"dc:description.abstract":["The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). Alternative estimators have been proposed to deal with the problems of non-existence and \"blowing up\" of the estimates. Mittal and Dahiya (1987, 1989) employ the Bayes model estimator of Blumenthal and Marcus (1975) for the normal and Weibull cases and Hegde and Dahiya (1989) apply it to the gamma. Hegde (1986) also studies the harmonic mean estimator of Joe and Reid (1984). Here we prove a sufficient and asymptotically necessary condition for the existence of the ML estimator of the scale parameter of the truncated Cauchy distribution. A modified ML estimator and an estimator based on equating population and sample quantiles are presented as alternatives. These estimators exist with probability one. The performance of these estimators is examined by making use of simulations. Asymptotic variances of the ML estimators are also given.
Finally, an application of truncated distributions is presented. The fit of returns on common stocks to the normal, Cauchy, truncated normal, and truncated Cauchy distributions is compared via the Kolmogorov-Smirnov statistic. The results show that a truncated distribution is a better fitting model in virtually all cases.
"],"dc:identifier":["https://digitalcommons.odu.edu/mathstat_etds/101"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Cauchy distribution","Stock returns","Estimates","Applied Statistics","Finance","Partial Differential Equations"],"dc:title":["The Truncated Cauchy Distribution: Estimation of Parameters and Application to Stock Returns"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:23Z"}