Estimating the parameters of a truncated distribution is a well known problem in statistical inference. The non-existence of the maximum likelihood estimator (m.l.e.) with positive probability in certain truncated distributions is not well known. To mention a few results in the literature:
(i) Deemer and Votaw 1955 show that the maximum likelihood estimator does not exist in a truncated negative exponential distribution on 0,T , T > 0 known, whenever the sample mean x (GREATERTHEQ) T/2.
(ii) Broeder 1955 shows that the maximum likelihood estimator of the scale parameter of a truncated gamma distribution, with the shape parameter being known, becomes infinite with positive probability whenever the sample mean x (GREATERTHEQ) α/α + 1, a > 0.
(iii) Mittal 1984 derives a sufficient condition for the non-existence of the maximum likelihood estimator in a two parameter doubly truncated normal distribution on A,B , A < B known. The m.l.e.'s become infinite whenever the sample variance exceeds (B-A)2/12. (iv) Barndoff-Neilsen 1978 (BN) gives a set of general conditions for the existence and uniqueness of a solution to the maximum likelihood equations in a minimal representation of a k-parameter exponential family which depend upon a few results from convex analysis.
Using certain results from BN 1978 , we give a unified approach to the problem of maximum likelihood estimation in the two parameter doubly truncated normal, truncated gamma, and singly truncated normal families, and obtain a set of necessary and sufficient conditions in terms of observable sample quantities. This approach basically depends upon characterizing the population and the sample moment spaces using a monotonicity property of the moments.
We also study the Bayes modal estimator introduced by Blumenthal and Marcus 1975 and the harmonic mean estimator introduced by Joe and Reid 1984 . We present certain computational results for solving the maximum likelihood equations in the above families. Simulation results for the probability of non-existence of the m.l.e., for the bias vector, and for the mean square error of the Bayes modal, the harmonic mean and the mixed estimator are presented.
","abstract_html":"<p>Estimating the parameters of a truncated distribution is a well known problem in statistical inference. The non-existence of the maximum likelihood estimator (m.l.e.) with positive probability in certain truncated distributions is not well known. To mention a few results in the literature:</p> <p>(i) Deemer and Votaw 1955 show that the maximum likelihood estimator does not exist in a truncated negative exponential distribution on 0,T , T > 0 known, whenever the sample mean x (GREATERTHEQ) T/2.</p> <p>(ii) Broeder 1955 shows that the maximum likelihood estimator of the scale parameter of a truncated gamma distribution, with the shape parameter being known, becomes infinite with positive probability whenever the sample mean x (GREATERTHEQ) α/α + 1, a > 0.</p> <p>(iii) Mittal 1984 derives a sufficient condition for the non-existence of the maximum likelihood estimator in a two parameter doubly truncated normal distribution on A,B , A < B known. The m.l.e.'s become infinite whenever the sample variance exceeds (B-A)<sup>2</sup>/12. (iv) Barndoff-Neilsen 1978 (BN) gives a set of general conditions for the existence and uniqueness of a solution to the maximum likelihood equations in a minimal representation of a k-parameter exponential family which depend upon a few results from convex analysis.</p> <p>Using certain results from BN 1978 , we give a unified approach to the problem of maximum likelihood estimation in the two parameter doubly truncated normal, truncated gamma, and singly truncated normal families, and obtain a set of necessary and sufficient conditions in terms of observable sample quantities. This approach basically depends upon characterizing the population and the sample moment spaces using a monotonicity property of the moments.</p> <p>We also study the Bayes modal estimator introduced by Blumenthal and Marcus 1975 and the harmonic mean estimator introduced by Joe and Reid 1984 . We present certain computational results for solving the maximum likelihood equations in the above families. Simulation results for the probability of non-existence of the m.l.e., for the bias vector, and for the mean square error of the Bayes modal, the harmonic mean and the mixed estimator are presented.</p>","abstract_has_math":false,"creators":["Hegde, Laxman M."],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Ram C. Dahiya","John P. Morgan","Wilkie Chaffin","N. R. Chaganty"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1986,"date_issued":"1986-01-01T08:00:00Z","date_published":"1986-01-01T08:00:00Z","updated_at":"2026-07-24T03:35:15Z","subjects":["Estimation","Truncated distribution","Parameters","Statistical inference","Mathematics","Statistics and Probability"],"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).
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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.odu.edu/mathstat_etds/83"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Estimating the parameters of a truncated distribution is a well known problem in statistical inference. The non-existence of the maximum likelihood estimator (m.l.e.) with positive probability in certain truncated distributions is not well known. To mention a few results in the literature:
(i) Deemer and Votaw 1955 show that the maximum likelihood estimator does not exist in a truncated negative exponential distribution on 0,T , T > 0 known, whenever the sample mean x (GREATERTHEQ) T/2.
(ii) Broeder 1955 shows that the maximum likelihood estimator of the scale parameter of a truncated gamma distribution, with the shape parameter being known, becomes infinite with positive probability whenever the sample mean x (GREATERTHEQ) α/α + 1, a > 0.
(iii) Mittal 1984 derives a sufficient condition for the non-existence of the maximum likelihood estimator in a two parameter doubly truncated normal distribution on A,B , A < B known. The m.l.e.'s become infinite whenever the sample variance exceeds (B-A)2/12. (iv) Barndoff-Neilsen 1978 (BN) gives a set of general conditions for the existence and uniqueness of a solution to the maximum likelihood equations in a minimal representation of a k-parameter exponential family which depend upon a few results from convex analysis.
Using certain results from BN 1978 , we give a unified approach to the problem of maximum likelihood estimation in the two parameter doubly truncated normal, truncated gamma, and singly truncated normal families, and obtain a set of necessary and sufficient conditions in terms of observable sample quantities. This approach basically depends upon characterizing the population and the sample moment spaces using a monotonicity property of the moments.
We also study the Bayes modal estimator introduced by Blumenthal and Marcus 1975 and the harmonic mean estimator introduced by Joe and Reid 1984 . We present certain computational results for solving the maximum likelihood equations in the above families. Simulation results for the probability of non-existence of the m.l.e., for the bias vector, and for the mean square error of the Bayes modal, the harmonic mean and the mixed estimator are presented.
"]},{"key":"dc:title","label":"Title","values":["Estimation in Truncated Exponential Family of Distributions"]}]}],"canonical_facts":{"dc:contributor":["Ram C. Dahiya","John P. Morgan","Wilkie Chaffin","N. R. Chaganty"],"dc:creator":["Hegde, Laxman M."],"dc:date.available":["2019-10-01T07:00:00Z"],"dc:description.abstract":["Estimating the parameters of a truncated distribution is a well known problem in statistical inference. The non-existence of the maximum likelihood estimator (m.l.e.) with positive probability in certain truncated distributions is not well known. To mention a few results in the literature:
(i) Deemer and Votaw 1955 show that the maximum likelihood estimator does not exist in a truncated negative exponential distribution on 0,T , T > 0 known, whenever the sample mean x (GREATERTHEQ) T/2.
(ii) Broeder 1955 shows that the maximum likelihood estimator of the scale parameter of a truncated gamma distribution, with the shape parameter being known, becomes infinite with positive probability whenever the sample mean x (GREATERTHEQ) α/α + 1, a > 0.
(iii) Mittal 1984 derives a sufficient condition for the non-existence of the maximum likelihood estimator in a two parameter doubly truncated normal distribution on A,B , A < B known. The m.l.e.'s become infinite whenever the sample variance exceeds (B-A)2/12. (iv) Barndoff-Neilsen 1978 (BN) gives a set of general conditions for the existence and uniqueness of a solution to the maximum likelihood equations in a minimal representation of a k-parameter exponential family which depend upon a few results from convex analysis.
Using certain results from BN 1978 , we give a unified approach to the problem of maximum likelihood estimation in the two parameter doubly truncated normal, truncated gamma, and singly truncated normal families, and obtain a set of necessary and sufficient conditions in terms of observable sample quantities. This approach basically depends upon characterizing the population and the sample moment spaces using a monotonicity property of the moments.
We also study the Bayes modal estimator introduced by Blumenthal and Marcus 1975 and the harmonic mean estimator introduced by Joe and Reid 1984 . We present certain computational results for solving the maximum likelihood equations in the above families. Simulation results for the probability of non-existence of the m.l.e., for the bias vector, and for the mean square error of the Bayes modal, the harmonic mean and the mixed estimator are presented.
"],"dc:identifier":["https://digitalcommons.odu.edu/mathstat_etds/83"],"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":["Estimation","Truncated distribution","Parameters","Statistical inference","Mathematics","Statistics and Probability"],"dc:title":["Estimation in Truncated Exponential Family of Distributions"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:15Z"}