On the Conditional Variance for Scale Mixtures of Normal Distributions

For a scale mixture of normal vector, X=A^1/2G, where X, G ⁿ and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X₂ X₁), is always finite a.s. for m2, where X₁ ⁿ and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function S_A, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given.     

Numerical Aspects in Estimating the Parameters of a Mixture of Normal Distributions

Abstract

Mixtures of normal distributions arise naturally in robustness analyses when distributional assumptions are not met and with the analysis of contaminated data. Typically these analyses involve substantial computational problems. We present some simulation procedures that address these difficulties.     

On a Finite Group Having a Normal Series Whose Factors Have Bicyclic Sylow Subgroups

We consider the structure of a finite group having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigate groups of odd order and A ₄-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.     

On Moments of Folded and Truncated Multivariate Normal Distributions

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.     

第二大根小于1的简单图

设G为n阶简单图,λ2(G)为G的第二大特征根．我们给出了所有使λ2(G)<1 的偶图,以及使λ2(G)<1、围长不小于4的非偶图．     

A Normal (0, 1)-Polytope None of Whose Regular Triangulations Is Unimodular

A normal (0,1) -polytope none of whose regular triangulations is unimodular is constructed. Received April 1, 1997, and in revised form June 20, 1997.     

关于正态分布的次序统计量的随机序

设$X_1,X_2,\cdots,X_n$和$X^*_1,X^*_2,\cdots,X^*_n$分别服从正态分布$N(\mu_i,\sigma^2)$和$N(\mu^*_i,\sigma^2)$,以$X_{(1)}$,$X^*_{(1)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots,X^*_n$的极小次序统计量,以$X_{(n)}$, $X^*_{(n)}$分别表示$X_1,\cdots,X_n$和$X^*_1,\cdots$,$X^*_n$的极大次序统计量. 我们得到了如下结果:(i)\,如果存在严格单调函数$f$使得$(f(\mu_{1}),\cdots,f(\mu_{n}))\succeq_{\text{m}}$ $(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\!\geq\!0$, 则$X_{(1)}\!\leq_{\text{st}}\!X^*_{(1)}$;(ii)\,如果存在严格单调函数$f$使得$(f(\mu_{1})$,$\cdots,f(\mu_{n}))\succeq_{\text{m}}(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,且$f'(x)f'(x)\leq 0$, 则$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.(iii)\,设$X_{1},X_{2},\cdots,X_{n}$和\, $X^*_{1},X^*_{2},\cdots,X^*_{n}$分别服从正态分布$N(\mu,\sigma_i^2)$和$N(\mu,\sigma_i^{*2})$,若$({1}/{\sigma_{1}},\cdots,{1}/{\sigma_{n}})\succeq_{\text{m}}({1}/{\sigma^{*}_{1}},\cdots,{1}/{\sigma^{*}_{n}})$,则有$X_{(1)}\leq_{\text{st}}X^*_{(1)}$和$X_{(n)}\geq_{\text{st}}X^*_{(n)}$同时成立.     

两个正态总体的似然比检验

对于两个正态总体N（μ1,σ1^2)和N(μ2，σ2^2)，本文讨论了统计假设H0：μ1=μ，σ1^2=σ2^2←→H1：μ1≠μ，或σ1^2≠σ2^2。给出了似然比检验统计量分布及相应的临界值表，并对其临界值表的计算给出了详细的讨论。     

Exponential Mixture Representation of Geometric Stable Distributions

We show that every strictly geometric stable (GS) random variable can be represented as a product of an exponentially distributed random variable and an independent random variable with an explicit density and distribution function. An immediate application of the representation is a straightforward simulation method of GS random variables. Our result generalizes previous representations for the special cases of Mittag-Leffler and symmetric Linnik distributions.     

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In this paper, we study the stochastic comparisons of order statistics from generalized normal distributions. We obtain some sufficient conditions for ordering results based on parameter matrix and vector majorization comparisons. These conditions are necessary in some cases.     

Stochastic Ordering of Multivariate Normal Distributions

We show an interesting identity for Ef(Y) – Ef(X), where X, Yare normally distributed random vectors and f is a function fulfilling some weak regularity condition. This identity will be used for a unified derivation of sufficient conditions for stochastic ordering results of multivariate normal distributions, some well known ones as well as some new ones. Moreover, we will show that many of these conditions are also necessary. As examples we will consider the usual stochastic order, convex order, upper orthant order, supermodular order and directionally convex order.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Dependence Properties of Multivariate Mixture Distributions and Their Applications

Consider a multivariate mixture model where the random variables X ₁, ..., X _n given (₁, ..., _n ), are conditionally independent. Conditions are obtained under which different kinds of positive dependence hold among X _i 's. The results obtained are applied to a variety of problems including the concomitants of order statistics and of record values; and to frailty models.     

Specific Features of Finite Mixtures of Normal Distributions

Properties of finite mixtures of normal distributions are considered. Their behavioral similarities and differences relative to normal distributions are studied. A practical application of finite mixtures of normal distributions for the simulating the noise of neurophysiological signals is described. It is shown that the Aitken estimate can be used for the source amplitudes in the considered model.     

底空间为对称域乘积的Hartogs域的度量等价

研究一类非齐性Hartogs域上经典度量的等价问题.首先证明了Bergman-度量和Einstein-Khler度量在这类域上等价;其次,当域的参数满足mσ+nΤ<1时,此类域上Bergman度量,Carathéodary度量,Kobalyashi度量和Einstein-Khler度量是等价的.     

奇异四元数矩阵的正态及Wishart分布

本文利用拉直算子(vec)和Kronecker积求得了四元数矩阵的实表示矩阵的一些性质,在此基础上利用实矩阵的奇异正态分布密度函数,求出了四元数矩阵的奇异正态分布的密度函数表达式.由此得到四元数矩阵奇异Wishart分布的密度函数表达式.