## Mathematics Ph.D. Dissertations

#### Title

Sequential Inference and Nonparametric Goodness-of-Fit Tests for Certain Types of Skewed Distributions

2019

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics/Mathematical Statistics

Amy Morgan (Other)

John Chen (Committee Member)

Craig Zirbel (Committee Member)

#### Abstract

We consider a sequence of i.i.d. random observations from a standard skew-normal distribution with skewness λ, denoted as SN(λ). We wish to test H0: λ = λ0 vs H1: λ = λ1, where λ0 and λ1 (λ0 ≠ λ1) are unknown constants, with target type-I and type-II error probabilities, denoted as α and β respectively. We first describe some interesting characteristics of the skew-normal distribution and then adopt Wald's sequential probability ratio test (SPRT) to perform the decision making and determine, on average, how many observations are needed to make such a decision. We choose numerous values of λ0 and λ1 to study how the chosen values affect the average sample number (ASN). We then compare these theoretical average sample numbers to those obtained through simulations. The approximations developed are applied to a set of BMI data.

We develop a nonparametric goodness-of-fit test for the hypothesis H0: F = SN(μ, σ, λ) vs H1: F ≠ SN(μ, σ, λ), based on the energy distance where F is the distribution of X1, ... , Xn. We first describe the energy distance and functions of energy distance, called energy statistics, along with some useful properties. We also briefly describe currently available goodness-of-fit tests for the skew-normal distribution in order to make comparisons with the proposed test. Simulations are conducted to indicate that the proposed test controls the Type-I error rate well and power studies show a higher detection rate for skew-normal than existing tests. The proposed test is applied to a set of IQ data and to the BMI data from Chapter 2.

We develop a nonparametric goodness-of-fit test for the hypothesis H0: F = SEP(μ,σ, λ,ν) vs H1: F ≠ SEP(μ, σ, λ,ν) where SEP(μ, σ, λ,ν) is the skewed exponential power distribution. This proposed test is based on the energy distance described in Chapter 3. We first describe the exponential power distribution in its symmetric version first while discussing some useful properties. Then we place the skewness parameter in the distribution. Simulations are conducted to show how the proposed test controls the Type-I error rate and power studies are performed to investigate the competitiveness of the proposed test.

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