#### Date of Award

Winter 1986

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

Ram C. Dahiya

#### Committee Member

John P. Morgan

#### Committee Member

Wilkie Chaffin

#### Committee Member

N. R. Chaganty

#### 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.

#### DOI

10.25777/zbz7-sh18

#### Recommended Citation

Hegde, Laxman M..
"Estimation in Truncated Exponential Family of Distributions"
(1986). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/zbz7-sh18

https://digitalcommons.odu.edu/mathstat_etds/83