Doctoral Dissertations


Renren Zhao


"In the first section, we consider small sample equivalence tests for exponentiality. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require a much larger sample size, in comparison to classical "difference tests", to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. We perform simulation studies to assess the bias of various tests and show that all of the Bayes' posteriors we consider are integrable. Our simulation studies show that the saddlepoint-approximated UMPU method performs remarkably well for small sample sizes and is the only method which consistently exhibits an empirical significance level close to the nominal five percent rate.

In the second section, we consider small sample equivalence tests for mean-to-variance ratio from two normal populations. In general, optimal equivalence tests for the means of two homoskedastic normal populations do not exist unless the common population variance is known. However, we show that if one considers the mean-to-variance ratio then there does exist a uniformly most powerful unbiased (UMPU) equivalence testing procedure. Furthermore, our procedure involves an intractable conditional distribution which we reproduce to a high degree of accuracy using saddlepoint approximations. We also develop six competing equivalence testing procedures for the mean-to-variance ratio. Four of these procedures are Bayesian and the remaining two are based upon the principle of confidence interval inclusion. Small sample simulation studies show that our UMPU method outperforms all competing methods by exhibiting an empirical significance level which is not statistically significantly different from the nominal five percent rate, for all simulation settings"--Abstract, page iii.


Paige, Robert

Committee Member(s)

Samaranayake, V. A.
Wen, Xuerong Meggie
Olbricht, Gayla R.
Gregory, Gelles


Mathematics and Statistics

Degree Name

Ph. D. in Mathematics and Statistics


Missouri University of Science and Technology

Publication Date

Spring 2015


vii, 53 pages

Note about bibliography

Includes bibliographic references (page 52).


© 2015 Renren Zhao, All rights reserved.

Document Type

Dissertation - Open Access

File Type




Thesis Number

T 11365

Electronic OCLC #