| 摘要註: |
自從Azzalini (1985,1986) 發表偏斜常態分佈後, 一些基於常見對稱分佈的偏斜分佈之研究, 便如雨後春筍般出現。這些偏斜分佈, 不僅包含原本的對稱分佈性質, 此外還有偏斜的特性, 所以可以用來處理更廣泛的問題。 本論文分三部分, 探討三個關於偏斜-對稱分佈的主題。在第一章,先針對兩個獨立隨機變數的乘積, 給出只要乘積為對稱, 則兩個隨機變數中, 至少有一個也會是對稱的條件。接著, 對某些常見的二元隨機變數,以邊際分佈的相同與否, 分別加以討論: 不僅給出兩個隨機變數中,極大值與極小值的線性組合之機率密度函數, 並且提供線性組合之一些偏斜的性質。 在第二章, 對於兩個獨立的廣義偏斜常態隨機變數, 我們給出其比值的機率密度函數。 而何時比值會服從偏斜柯西分佈? 我們將給出充分必要條件。 在第三章, 對多元廣義偏斜常態隨機變數, 我們給一些關於二次形式及二次形式比值的動差之公式。二次形式的討論中,將用到正定矩陣和反矩陣。 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 thedistributions 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. |