A note on Sheppard's corrections for grouping and maximum likelihood estimation

Sheppard's corrections for grouping can, in the case of an underlying normal distribution, be interpreted as a first step to the solution of the maximum likelihood equations which incorporate the grouping problem. This result of Lindley (for the univariate) and Haitovsky (for the bivariate) is generalized to the multivariate normal distribution, making use of recent results in matrix algebra. Also, formulae concerning the efficiency lost in grouping are generalized to the multivariate case.     

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.     

The disteribution of a moment estimator for a parameter of the generalized poision distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

End effects in thermoelasticity

This paper derives spatial decay bounds in a dynamical problem of thermoelasticity defined on a semi‐infinite cylindrical region. Previous results for isothermal elastodynamics and the parabolic heat equation lead us to suspect that the solution of the thermoelastic problem should tend to zero, faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance. This is reminiscent of the decay rate in the second‐order parabolic problems. Similar arguments are sketched for the case of a non‐prismatic cylinder, and a non‐linear heat conduction problem for a rigid body is considered. Some extensions and generalizations are indicated in the final section. Copyright © 2001 John Wiley & Sons, Ltd.     

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall''s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980''s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.     

贝叶斯对数线性估计

用Logits变换方法来讨论列联表的贝叶期估计，这时的估计称为贝叶斯对数线性估计，最早是由Lconard（1973，1975）提出的．Leonard讨论了二维表的对数线性贝叶斯估计；其后，Nazerat（1987）给出了三维表的贝叶斯对数线性估计．该文采用矩阵求微商方法，立足于高维表贝叶斯线性估计的较一般的表达形式；具体对四维表讨论员叶斯对数线性估计的表达和计算．     

Normal Bases and Their Dual-Bases over Finite Fields

In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.     

Maximal L 2 and Pointwise Hölder Estimates for □ b on CR Manifolds of Class C 2

In this paper we prove the optimal pointwise Hölder estimates for the Kohn's Laplacian □ _b for CR manifolds of class C ². We first derive optimal L ² theory for solutions of □ _b directly without using pseudo-differential operators. Then we use the scaling method to obtain pointwise estimates in non-istotropic Campanato spaces based on the maximal L ² theory.     

正态定限（Orthant）概率

本文采用不同以往的方法;对正态定限（Ｏｒｔｈａｎｔ）概率积分施行线性变换与极坐标变换;得到了一个新的递推公式,使积分重数至少降低了３重;并推导了四维正态定限概率的表达式,这对进行数值计算是十分有意义的．     

Equality in Distribution in a Convex Ordering Family

Let F and G be the respective distributions of nonnegative random variables X and Y satisfying the convex ordering. We investigate the class of functions h for which the equality E[h(X)] = E[h(Y)] guarantees F = G. It leads to extensions of some existing results and at the same time offers a somewhat simpler proof.     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

Fibre homogenisation

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.     

A characteristic property of the exponential distribution

The following characterization of the exponential distribution is given: Under suitable conditions on the random variables X and Y, X is exponentially distributed if and only if E[min{X, Y}]=E(X)P(X<Y).     

Correction to “On the linear combination of normal and Laplace random variables”, by Nadarajah,S., <Emphasis Type=Italic>Computational Statistics</Emphasis>, 2006, <Emphasis Type=Bold>21</Emphasis>, 63–71

In the above mentioned paper, some errors were found in the expressions given for the distribution of a linear combination of Normal and Laplace random variables, Z, given in formulae (3, Theorem 1), (6), and (7) that can lead to obtaining negative values for the mentioned distribution. The corrected versions for these expressions are presented here. In addition, the density function of Z is also provided.     

游程数的分布和矩的计算

本文较为系统地介绍了游程数的定义、分布及其矩的计算