## Mathematics Ph.D. Dissertations

# Sequential Inference and Goodness of Fit Testing using Energy Statistics for the Power Normal and Modified Power Normal Distributions

## Date of Award

2023

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Statistics

## First Advisor

Wei Ning (Advisor)

## Second Advisor

Virginia Dubasik (Committee Member)

## Third Advisor

Craig Zirbel (Committee Member)

## Fourth Advisor

Junfeng Shang (Committee Member)

## Abstract

We consider a sequence of i.i.d random variables from two different distributions, the power normal distribution and the modified power normal distribution. We denote these throughout the dissertation as PN(λ) and MPN(λ). For the sequential analysis, consider the hypothesis test H0: λ=λ0 vs. Ha: λ=λ1, where λ0≠λ1 and are both unknown constants. Choose type-I and type-II error probabilities to be constants denote them as α and β. In Chapter 1 we begin our analysis with the power normal distribution. Wald's sequential probability ratio test is used to determine the number of observations needed to make a decision regarding the hypothesis testing. We then develop the Operating Characteristic (OC) function which determines the probability of rejecting the null hypothesis based on the true value of our parameter. This function is used to find the Average Sample Number (ASN), which is the average number of observations needed to make a decision regarding the hypothesis test. These are used to determine how different values of λ affect average sample number. Finally we apply our method to a real dataset and illustrate its performance. In Chapter 2 we develop a non-parametric goodness of fit test for the hypothesis: H0: F=PN(λ) vs. Ha: F ≠ PN(λ) based on the energy statistic where F is the distribution of X1, ...,Xn. First, energy statistics and their function are described. After deriving the test, Type-I error levels are checked and the power of the test is compared to other well known non-parametric tests. In Chapter 3 the modified power normal distribution is introduced. As in Chapter 1, Wald's sequential probability ratio test is performed to find the operating characteristic function as well as the average sample number. Simulations are performed to see how the theoretical calculations compare to simulated data for the average sample number. Finally we use a real dataset to see how well it works in practice. In Chapter 4 we develop a non-parametric goodness of fit test for the hypothesis: H0: F=MPN(λ) vs. Ha: F ≠ MPN(λ) based on the energy distance. After deriving the test Type-I error levels are checked. The test is then compared to other well known non-parametric goodness of fit tests.

## Recommended Citation

Craig, Bradley, "Sequential Inference and Goodness of Fit Testing using Energy Statistics for the Power Normal and Modified Power Normal Distributions" (2023). *Mathematics Ph.D. Dissertations*. 92.

https://scholarworks.bgsu.edu/math_diss/92