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

Craig, Bradley
http://orcid.org/0009-0007-6771-8788
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1688593748026899

Abstract Details

2023, Doctor of Philosophy (Ph.D.), Bowling Green State University, Statistics.
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.
Wei Ning, Ph.D (Advisor)
Virginia Dubasik, Ph.D (Committee Member)
Craig Zirbel, Ph.D (Committee Member)
Junfeng Shang, Ph.D (Committee Member)
89 p.
Statistics
Energy Statistics; Sequential Inference; Goodness-of-Fit;

Recommended Citations

Citations

  • Craig, B. (2023). Sequential Inference and Goodness of Fit Testing using Energy Statistics for the Power Normal and Modified Power Normal Distributions [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1688593748026899

    APA Style (7th edition)

  • Craig, Bradley. Sequential Inference and Goodness of Fit Testing using Energy Statistics for the Power Normal and Modified Power Normal Distributions. 2023. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1688593748026899.

    MLA Style (8th edition)

  • Craig, Bradley. "Sequential Inference and Goodness of Fit Testing using Energy Statistics for the Power Normal and Modified Power Normal Distributions." Doctoral dissertation, Bowling Green State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1688593748026899

    Chicago Manual of Style (17th edition)

bgsu1688593748026899
169
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This open access ETD is published by Bowling Green State University and OhioLINK.