Date of Completion

6-29-2018

Embargo Period

6-23-2018

Keywords

Statistical Information, Confidence Interval Estimation, Sampling Strategies, Two Sample Comparison, Two-Stage Methodologies, Purely Sequential Procedures

Major Advisor

Nitis Mukhopadhyay

Associate Advisor

Joseph Glaz

Associate Advisor

Vladimir Pozdnyakov

Field of Study

Statistics

Degree

Doctor of Philosophy

Open Access

Campus Access

Abstract

Fisher's (1934,1956) "Nile" example is a classic. He revisited this example a number of times and it was reproduced in a 1973 edition (Fisher 1973, pp. 122). The "Nile" example involves a bivariate random variable (X; Y ) having a joint probability density function (p.d.f.) given by f(x; y; theta) = exp(-theta*x-theta^(-1)*y); where x > 0; y > 0 and theta(> 0) is an unknown parameter. This dissertation is started by working on the problems involving MLE, information, sufficiency and ancillary complement, where we provide general evaluations for useful information entities, including unconditional information in the MLE T for theta, and conditional information in T about theta given S with S being an ancillary complement of T. Estimation of theta is directly linked to chances of potential flooding of the banks of a river. Thus, we construct a fxed-width confidence interval for theta using an appropriate purely sequential methodology. Due to the fact that the lower confidence limit of that confidence interval can be negative with a positive probability and no quick-fix modification seems appropriate, we further construct a fixed-accuracy confidence interval. We find it very appealing that the optimal fixed sample size can be determined exactly to make such fixed-accuracy confidence interval have a preassigned coverage probability without requiring a sequential or multistage sampling strategy. It is important to estimate P(X > a); a > 0. We develop bounded-length confidence interval estimations for P(X > a) with a preassigned confidence coefficient using both purely sequential and two-stage methodologies. We show that both methodologies enjoy asymptotic first-order efficiency, asymptotic consistency and second-order efficiency properties. After presenting substantial theoretical investigations, we have also implemented extensive sets of computer simulations to empirically validate the theoretical properties. The fourth fold of this dissertation involves the two sample comparisons for normal distribution, when the variances are unknown and unequal. We develop general methodologies and procedures to test hypotheses regarding the difference of the mean values of normal populations. For preassigned levels of both type-I and type-II error probabilities, two-stage and purely sequential procedures are developed to determine the sample sizes for performing the test, along with the decision rules for making conclusions. We construct these methodologies under both unequal and equal sample size scenarios. We show that both two-stage methodologies and purely sequential procedures enjoy certain appealing properties. For comparing the mean values of two dependent normal populations, we provide two-stage and purely sequential procedures to determine the sample sizes for performing the test, along with corresponding decision rules by requiring certain levels of type-I and type-II error probabilities. Extensive sets of computer simulations empirically validate the theoretical properties.

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