## Open Access Dissertations and Theses

#### Title

A Statistical Study of Log-Gamma Distribution

9-1993

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

#### Supervisor

Dr. N. Balakrishman

#### Abstract

In this thesis, we consider the log-gamma distribution and discuss some of its properties. We then study the order statistics from this distribution. We derive the explicit expressions for the means, variances and covariances of order statistics from the distribution for various choices of the shape parameter.

Next, we discuss the following methods of estimation of the unknown location and scale parameters (i) bext linear unbiased estimation based on Type II censored samples, (ii) best linear unbiased estimation based on optimal selected order statistics, (iii) maximum likelihood estimation based on Type II censored samples, (iv) approximate maximum likelihood estimation based on Type II censored samples. We illustrate these estimation procedures through a real-life data set by Lieblein and Zelen (1956).

We also study the construction of the confidence intervals of these parameters. Both conditional and unconditional approaches are discussed and comparison between these two approaches is made. We also discuss the determination of the tolerance limits and confidence limits for the reliability.

After discussing the estimation methods of the location and scale parameters of the log-gamma distribution, we study the estimation method of shape parameter under Type II censoring. We derive the log-likelihood function of the shape parameter and its derivative and hence Newton-Raphson algorithm can be used to obtain the maximum likelihood estimate of the parameter. Asymptotic Fisher information matrix of all three parameters is also derived and so are the asymptotic variances and covariances of the maximum likelihood estimators.

Finally, we study the relation between the log-gamma distribution and record value theory. A bivariate log-gamma model is proposed because of this relation. We also study the record values which come from populations other than log-gamma distribution. Statistical inference based on the record values is also studied.

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