本文選題：橢球等高分布 + 左球分布　； 參考：《東南大學(xué)》2016年博士論文

【摘要】：本文首先利用左球分布定義一類絕對連續(xù)型廣義橢球矩陣分布,并研究這種廣義橢球矩陣分布在非奇異變換下的有關(guān)性質(zhì)。其次,本文詳細(xì)研究基于非負(fù)連續(xù)規(guī)則變化隨機變量與橢球分布的尺度混合產(chǎn)生的多元廣義t分布族的尾相依性質(zhì)。第二章首先考慮左球分布在線性變換下的性質(zhì),然后將經(jīng)典的矩陣F分布和矩陣t分布的隨機表示式結(jié)構(gòu)中的球?qū)ΨQ分布的隨機變量擴大到左球分布類,推導(dǎo)出新的隨機矩陣F分布和矩陣t的任意Borel函數(shù)的數(shù)字特征的積分表示,利用隨機矩陣的聯(lián)合密度函數(shù)與數(shù)字特征函數(shù)之間的一一對應(yīng)關(guān)系推導(dǎo)出新條件兩個隨機矩陣的聯(lián)合概率密度函數(shù)精確表達(dá)式,就是矩陣F分布或矩陣t分布的聯(lián)合概率密度。由此將多元統(tǒng)計推斷中的重要分布F分布和矩陣t分布推廣到左球分布的范圍,我們將其稱為廣義橢球矩陣分布。最后利用這一推廣意義,研究了各種矩陣橢球分布關(guān)于非奇異合同變換下的不變性質(zhì)。第三章利用隨機結(jié)構(gòu)方法定義了幾種多元廣義t分布,這種廣義t分布看成逆伽瑪分布與多元橢球?qū)ΨQ分布的尺度混合。研究了它們的尾相依性質(zhì),通過計算概率的方法推導(dǎo)了在相關(guān)矩陣意義下的上象限尾相依系數(shù)和上極值尾相依指數(shù)的表達(dá)式,并研究了尾相依系數(shù)關(guān)于尾指數(shù)和線性相關(guān)系數(shù)之間的關(guān)系。根據(jù)我們的結(jié)論可知,這里得到的表達(dá)式比己有的多元t分布的尾相依系數(shù)的計算公式簡潔明了,統(tǒng)計意義更加清楚。關(guān)于由逆伽瑪分布和多元正態(tài)分布尺度混合產(chǎn)生的廣義多元t分布,尾相依系數(shù)與尾指數(shù)的關(guān)系比較復(fù)雜,我們給出了一個單調(diào)性的充分條件。尾相依系數(shù)與相關(guān)系數(shù)的關(guān)系:上極值相依指數(shù)關(guān)于線性相關(guān)系數(shù)一定是單調(diào)非減的關(guān)系;上尾相依系數(shù)與隨機向量的相關(guān)系數(shù)的單調(diào)關(guān)系與相關(guān)系數(shù)所對應(yīng)的隨機變量有關(guān),對此我們建立了它們之間單調(diào)非減的充要條件。通過數(shù)值模擬驗證了所有結(jié)論。關(guān)于由逆廣義Gamma分布與多元指數(shù)冪分布尺度混合產(chǎn)生的廣義多元t分布,首先通過計算概率的方法,將其尾相依系數(shù)表示成多元指數(shù)冪分布變量的數(shù)字特征的形式,其次也考慮了這種廣義多元t分布的尾相依系數(shù)的性質(zhì),并通過作了相應(yīng)的隨機模擬。用類似的方法研究了由逆廣義Gamma分布與多元Kotz型分布混合產(chǎn)生的廣義多元t分布的尾相依系數(shù)的計算公式并討論了它們的性質(zhì)。第四章通過隨機結(jié)構(gòu)的方法構(gòu)造了由規(guī)則變化隨機變量與任意球?qū)ΨQ多元分布尺度混合產(chǎn)生的一類廣義多元t分布。這一分布類可以通過靈活選取規(guī)則變化隨機變量的尾指數(shù)而得到比一般橢球分布更輕或更厚尾部的隨機向量,并且包含了第三章中定義的各種多元廣義t分布。所以,這一新的分布類是多元廣義t分布的推廣,我們稱之為規(guī)則變化尺度混合的多元廣義t分布。隨后我們利用隨機向量的copula函數(shù)推導(dǎo)了隨機向量的尾相依函數(shù)。首先將尾相依函數(shù)表示成向量的緊測度的形式,然后利用尾相依系數(shù)與尾相依函數(shù)的關(guān)系巧妙地得到這類分布族的上尾相依系數(shù)和上極值相依系數(shù)的表達(dá)式。所有的尾相依系數(shù)表示成隨機結(jié)構(gòu)式中球?qū)ΨQ分布的隨機向量的相應(yīng)分量的數(shù)字特征的函數(shù),這一結(jié)果與用概率方法推導(dǎo)出的結(jié)論完全一致,顯然用copula函數(shù)方法由于只要用到隨機向量間的結(jié)構(gòu),不用考慮邊緣分布的干擾,所以比概率方法簡單許多。上一章的所有結(jié)果可以作為本部分結(jié)論的特殊結(jié)果。最后通過數(shù)值模擬的方法驗證了所得結(jié)論。
[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.

【學(xué)位授予單位】：東南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O211.3

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