广义椭球等高分布及其性质研究

发布时间：2018-04-22 10:12

本文选题：椭球等高分布 + 左球分布　；参考：《东南大学》2016年博士论文

【摘要】：本文首先利用左球分布定义一类绝对连续型广义椭球矩阵分布,并研究这种广义椭球矩阵分布在非奇异变换下的有关性质。其次,本文详细研究基于非负连续规则变化随机变量与椭球分布的尺度混合产生的多元广义t分布族的尾相依性质。第二章首先考虑左球分布在线性变换下的性质,然后将经典的矩阵F分布和矩阵t分布的随机表示式结构中的球对称分布的随机变量扩大到左球分布类,推导出新的随机矩阵F分布和矩阵t的任意Borel函数的数字特征的积分表示,利用随机矩阵的联合密度函数与数字特征函数之间的一一对应关系推导出新条件两个随机矩阵的联合概率密度函数精确表达式,就是矩阵F分布或矩阵t分布的联合概率密度。由此将多元统计推断中的重要分布F分布和矩阵t分布推广到左球分布的范围,我们将其称为广义椭球矩阵分布。最后利用这一推广意义,研究了各种矩阵椭球分布关于非奇异合同变换下的不变性质。第三章利用随机结构方法定义了几种多元广义t分布,这种广义t分布看成逆伽玛分布与多元椭球对称分布的尺度混合。研究了它们的尾相依性质,通过计算概率的方法推导了在相关矩阵意义下的上象限尾相依系数和上极值尾相依指数的表达式,并研究了尾相依系数关于尾指数和线性相关系数之间的关系。根据我们的结论可知,这里得到的表达式比己有的多元t分布的尾相依系数的计算公式简洁明了,统计意义更加清楚。关于由逆伽玛分布和多元正态分布尺度混合产生的广义多元t分布,尾相依系数与尾指数的关系比较复杂,我们给出了一个单调性的充分条件。尾相依系数与相关系数的关系:上极值相依指数关于线性相关系数一定是单调非减的关系;上尾相依系数与随机向量的相关系数的单调关系与相关系数所对应的随机变量有关,对此我们建立了它们之间单调非减的充要条件。通过数值模拟验证了所有结论。关于由逆广义Gamma分布与多元指数幂分布尺度混合产生的广义多元t分布,首先通过计算概率的方法,将其尾相依系数表示成多元指数幂分布变量的数字特征的形式,其次也考虑了这种广义多元t分布的尾相依系数的性质,并通过作了相应的随机模拟。用类似的方法研究了由逆广义Gamma分布与多元Kotz型分布混合产生的广义多元t分布的尾相依系数的计算公式并讨论了它们的性质。第四章通过随机结构的方法构造了由规则变化随机变量与任意球对称多元分布尺度混合产生的一类广义多元t分布。这一分布类可以通过灵活选取规则变化随机变量的尾指数而得到比一般椭球分布更轻或更厚尾部的随机向量,并且包含了第三章中定义的各种多元广义t分布。所以,这一新的分布类是多元广义t分布的推广,我们称之为规则变化尺度混合的多元广义t分布。随后我们利用随机向量的copula函数推导了随机向量的尾相依函数。首先将尾相依函数表示成向量的紧测度的形式,然后利用尾相依系数与尾相依函数的关系巧妙地得到这类分布族的上尾相依系数和上极值相依系数的表达式。所有的尾相依系数表示成随机结构式中球对称分布的随机向量的相应分量的数字特征的函数,这一结果与用概率方法推导出的结论完全一致,显然用copula函数方法由于只要用到随机向量间的结构,不用考虑边缘分布的干扰,所以比概率方法简单许多。上一章的所有结果可以作为本部分结论的特殊结果。最后通过数值模拟的方法验证了所得结论。
[Abstract]:In this paper, we first define a class of absolute continuous generalized elliptic matrix distribution by using the distribution of the left sphere, and study the properties of the generalized elliptic matrix distribution under the nonsingular transformation. Secondly, in this paper, the tail dependence of the multivariate generalized t distribution family based on the scale mixing of the nonnegative continuous rule and the scale of the ellipsoid distribution is studied in detail. In the second chapter, the second chapter first considers the properties of the left spherical distribution under linear transformation. Then, the classical matrix F distribution and the random variable of the spherical symmetric distribution in the matrix t distribution are extended to the left spherical distribution class. The new random matrix F distribution and the integral representation of the digital feature of the Borel function of the matrix T are derived. By using the one-to-one correspondence between the joint density function and the digital eigenfunction of the random matrix, the exact expression of the joint probability density function of the two random matrices of the new conditions is derived, which is the joint probability density of the matrix F distribution or the matrix t distribution. Thus, the important distribution of the F distribution and the matrix t distribution in the multivariate statistical inference are extended to the distribution of the matrix. The range of the distribution of the left sphere is called the generalized ellipsoid matrix distribution. Finally, using this generalized meaning, we study the invariant properties of various matrix ellipsoid distributions with respect to the nonsingular contract transformation. In the third chapter, several generalized t distributions are defined by the stochastic structure method. This generalized t distribution is regarded as the Gama distribution and the multivariate ellipsoid. The dependent properties of the symmetric distribution are studied. By calculating the probability, the expressions of the upper quadrant tail dependence coefficient and the upper extremum dependent exponent under the correlation matrix are derived, and the relation between the tail dependence coefficient and the linear phase relation is studied. The formula obtained in this paper is simpler and clearer than the formula for the dependence coefficient of the multivariate t distribution that we have, and the statistical significance is clearer. The relation between the tail dependence coefficient and the tail exponent is more complex about the generalized multivariate t distribution produced by the mixture of Gama distribution and the multidimensional normal distribution scale, and we give a sufficient condition for the monotonicity. The relation between the dependence coefficient of the tail dependence and the correlation coefficient: the dependence coefficient of the upper extremum on the linear correlation must be a monotone non subtraction relation; the monotonicity relation between the correlation coefficient of the dependence coefficient of the upper and the tail and the random vector is related to the random variable corresponding to the correlation coefficient, so we establish the necessary and sufficient conditions for the monotone non subtraction between them. The simulation verifies all conclusions. With regard to the generalized multivariate t distribution generated by the mixture of the inverse generalized Gamma distribution and the multivariable exponentiation power distribution, the tail dependence coefficient is expressed as the digital feature of the multivariate exponential power distribution variable by the method of calculating the probability. Secondly, the tail dependence coefficient of the generalized multivariate t distribution is also considered. The properties of the generalized multivariate t distribution produced by the inverse generalized Gamma distribution and the multiple Kotz type distribution are studied by a similar method, and their properties are discussed. In the fourth chapter, the random variables and the free sphere are constructed by the method of random structure. A class of generalized multivariate t distribution generated by a mixture of symmetric multivariate distribution scales. This distribution class can obtain a random vector which is lighter or thicker than the general ellipsoid distribution by selecting the tail exponents of random variables, and contains all kinds of generalized t distributions defined in the third chapter. So, this new distribution class It is the generalization of the multivariate generalized t distribution. We call it the multivariate generalized t distribution of the rule change scale. Then we use the copula function of the random vector to derive the tail dependent function of the random vector. First, the tail dependent function is expressed as the tight measure of the vector, and then the relation between the tail dependence coefficient and the tail dependence function is skillful The expression of the dependent coefficient of the upper and tail dependence and the dependence coefficient of the upper extremum is obtained. All the tail dependence coefficients represent the function of the digital feature of the corresponding component of the random vector of the spherically symmetric distribution in the random structural formula. This result is exactly the same as that derived from the probability method. It is obvious that the copula function method is obviously used. As long as the structure between the random vectors is used, the interference of the edge distribution is not considered, so it is much simpler than the probability method. All the results in the last chapter can be used as a special result of the conclusion of this part. Finally, the results of the numerical simulation are verified.

【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O211.3

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