#### Date of Award

2004

#### Level of Access Assigned by Author

Campus-Only Thesis

#### Degree Name

Master of Arts (MA)

#### Department

Mathematics

#### Advisor

Ramesh C. Gupta

#### Second Committee Member

Pushpa Gupta

#### Third Committee Member

Henrik Bresinsky

#### Abstract

The statistical analysis for comparing the means of two independent log-normal distribution is often of interest in biomedical research. For examples, in bio-availability studies, the relative potency of a new drug to that of a standard one is expressed in terms of the ration of means, and analysts often need to construct a confidence interval for the ratio or to test the null hypothesis that the ratio is one, i.e., the mean outcome of the two products are the same, (Berger and Hsu, 1996; Chow and Liu, 2000). In this connection, Zhou et al. (1997) proposed likelihood-based procedures for comparing the means of two independent log-normal populations. One is a Z test and the other is a nonparametric bootstrap approach. However, Wu et al. (2002) pointed out that the Z test does not perform well in a range of small sample settings. In this thesis, we shall study the large sample as well as small sample procedures for comparing the means of two log-normal populations. In addition to Zhou's test, we develop a Rao's score test for testing the equality of the means of two log-normal populations and compare its performance with that of Zhou's test. Assuming that the means of two log-normal distributions are the same, we shall also: 1. Develop procedures for estimating the common mean of two independent log-normal distributions. Confidence intervals for the common mean will be constructed and their performance will be examined by simulation studies in terms of the coverage probabilities and their average length. 2. Estimate *R = P(X, where X and Y come from independent log-normal distributions with equal means. The problem originated in estimating the reliability in the stress-strength model where X is the stress and Y is the strength, see Gupta et al. (1999). The performance of the confidence intervals developed will be examined by simulation studies in terms of the coverage probability and the average length of the confidence intervals.*

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#### Recommended Citation

Li, Xue, "Statistical Inference for the Common Mean of Two Independent Log-Normal Distributions and Some Applications in Reliability" (2004). *Electronic Theses and Dissertations*. 1089.

https://digitalcommons.library.umaine.edu/etd/1089

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