# Some Contributions To Distribution Theory

• Rui Li

Student thesis: Phd

### Abstract

Distribution theory, which is widely applied to describe the features and properties of observations, and plays an important role in the development of statistics. However, some distributions have complicated or implicit forms of mathematical expressions, which is not able to solve practical problems in a transparent way. This thesis aims to make some contributions to distribution theory.Specifically, this thesis presents some exact distributions in a closed form, which is able to provide more accurate results and improve the computational efficiency in practice. The main contributions are listed in Chapters 2-7. Student's t distribution is one of the most famous distributions which was discovered by William Gosset in 1908. Chapter 2 gives a review of the Student's t (ST) distribution and its generalizations. This chapter is consisted of the collection of nearly 30 generalizations of the ST distribution. Also, the comparisons among some of ST generalizations are performed for fitting with stock index data in applications. The Behrens-Fisher problem is a famous problem in statistics regarding hypothesis testing and interval estimation. Commonly, the distribution of Behrens-Fisher statistic is approximated by a Student's t random variable. Chapter 3 provides the exact distribution of a modified Behrens-Fisher statistic, which is more convenient from the computational perspective. Chapter 4 manages to derive the true maximum likelihood estimators for the generalized Gaussian distribution (GGD) with the shape parameter p = 3; 4; 5. Compared with traditional numerical algorithms, theorems in this chapter allow one to compute the maximum likelihood estimator of the location parameter mu and the scale parameter sigma accurately. Since round off errors arise in many areas of signal processing, Chapter 5 aims to extend the work of Gadzhiev (2015) to estimate the mean and variance of round off errors for any continuous random variable defined on either the real line or a finite interval. The contributions are elaborated as some theorems with higher accuracy in computing mean and variance of round off errors, with no need to consider sample size. Chapter 6 provides an exact density of the sum of independent skew normal random variables. The out-performances of our method are shown in comparison with the corresponding central processing unit (CPU) operation time with an increasing number of random variables. Besides, our method appears good performance in fitting and predicting stock price over a long period of time. Estimation of population mean based on given measurements is one of the oldest problems in statistics. Chapter 7 investigates the relative performances of arithmetic mean, root mean square, geometric mean, and harmonic mean for the simplest case of two measurements from a uniform distribution.
Date of Award 1 Aug 2019 English The University of Manchester Georgi Boshnakov (Supervisor) & Nadarajah (Supervisor)

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