| Summary: | 碩士 === 國立高雄大學 === 統計學研究所 === 97 === Since Azzalini (1985,1986) introduced the univariate skew-normal distribution, there are many investigations about the skew distributions based on certain symmetric probability density functions. Because these classes of the skew distributions include the original symmetric distribution and have some properties like the original one and yet is skew, hence it is more useful to handle related problems.
In this thesis, we consider three topics of the symmetric and skew distributions. In Chapter 1, we will discuss the case Z = UV first, where U and V are assumed to be independent. Under some conditions, we will show that if Z is symmetric, then at least one of U and V is symmetrically distributed. Next for certain bivariate symmetric random variables X and Y, we will find the
distributions of M = aU + bV, where a and b are constants, U=max{X, Y} and V = min{X,Y}. When X and Y are assumed or not assumed to be identically distributed, we will present the distributions and skew properties of M, respectively.
In Chapter 2, we will present the probability density function of the ratio of two generalized skew-normal distributed random variables. We also give necessary and sufficient conditions when the ratio is skew-Cauchy distributed.
In Chapter 3, some formulas for the central inverse moments of a quadratic form and of the ratio of two quadratic forms are established for multivariate skew normal random variables. They relate the quadratic forms which are determined by positive definite matrices to that defined by the inverse matrices.
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