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.     

Some exact expressions for the mean and higher moments of functions of sample moments

Examples of exact expressions for the moments (mainly of the mean) of functions of sample moments are given. These provide checks on alternative developments such as asymptotic series for n, and simulation processes. Exact expressions are given for the mean of the square of the sample coefficient of variation, particularly in uniform sampling; Frullani integrals studied by G. H. Hardy arise. It should be kept in mind that exact results for (joint) moment generating functions (mgfs) are of interest as they produce a means of obtaining exact results for (cross) moments—including moments with negative indices. Thus an exact expression for the joint mgf of the 1st two noncentral moments can be used to obtain the mean of the (c.v.)² (but not for the mean of the c..). A general expression is given for the moment generating function of the sample variance. The limitations of Fisher's symbolic formula for the characteristic function of sample moments (or more general statistics) are noted.This research was sponsored by the Applied Mathematical Sciences Research program, Office of Energy Research, U. S. Department of Energy under contract DE-AC0584OR21400 with the Martin Marietta Energy Systems. Inc.     

A rank test based on the moments of order statistics of the modified Makeham distribution

Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.     

Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions

This paper considers multi-dimensional affine processes with continuous sample paths. By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode at a given time. In these two cases, we also compute the long-term growth rate and the explosion rate for exponential moments. These results provide a handle to study implied volatility asymptotics in models where log-returns of stock prices are described by affine processes whose exponential moments do not have an explicit formula.     

Stability of semilinear stochastic evolution equations

Stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of Liapunov function. Sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved. Three examples are given to illustrate the theory.     

On the limiting distribution of sample central moments

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

     

Simultaneous equation estimation from undersized samples

In simultaneous equation theory a sample is said to be undersized when it is smaller than the number of exogenous variables. The sample moment matrix of these variables is then singular, but equations can nevertheless be estimated when different moments (hybrid moments based on the Theil-Laitinen maximum entropy distribution) are used. Here we evaluate the adequacy of the asymptotic standard errors formulated in terms of such moments.     

Some limit theorems for the eigenvalues of a sample covariance matrix

Limit theorems are given for the eigenvalues of a sample covariance matrix when the dimension of the matrix as well as the sample size tend to infinity. The limit of the cumulative distribution function of the eigenvalues is determined by use of a method of moments. The proof is mainly combinatorial. By a variant of the method of moments it is shown that the sum of the eigenvalues, raised to k-th power, k = 1, 2,…, m is asymptotically normal. A limit theorem for the log sum of the eigenvalues is completed with estimates of expected value and variance and with bounds of Berry-Esseen type.     

协方差的非齐次估计的可容许性

本文研究协方差的非齐次二次估计的可容许性,在平方损失下,我们给出了一个非齐次二次估计在非齐次二次估计类中是协方差的容许估计的充要条件.     

协方差的二次型容许估计

本文研究方差的二次型估计的容许性,在平方损失下,我们给出了一个二次型估计在二次型估计类中是协方差的容许估计的充要条件。     

Moment Estimation for Multivariate Extreme Value Distribution in a Nested Logistic Model

This paper considers multivariate extreme value distribution in a nested logistic model. The dependence structure for this model is discussed. We find a useful transformation that transformed variables possess the mixed independence. Thus, the explicit algebraic formulae for a characteristic function and moments may be given. We use the method of moments to derive estimators of the dependence parameters and investigate the properties of these estimators in large samples via asymptotic theory and in finite samples via computer simulation. We also compare moment estimation with a maximum likelihood estimation in finite sample sizes. The results indicate that moment estimation is good for all practical purposes.     

SAMPLING INSPECTION OF RELIABILITY IN （LOG）NORMAL CASE WITH TYPE I CENSORING

This article proposes a statistical method for working out reliability sampling plans under Type I censored sample for items whose failure times have either normal or lognormal distributions. The quality statistic is a method of moments estimator of a monotonous function of the unreliability. An approach of choosing a truncation time is recommended. The sample size and acceptability constant are approximately determined by using the Cornish-Fisher expansion for quantiles of distribution. Simulation results show that the method given in this article is feasible.     

Limit distributions of multivariate kurtosis and moments under Watson rotational symmetric distributions

The limit distribution of Mardia's measure of sample multivariate kurtosis is derived for a wide class of multivariate distributions which includes both the family of elliptical and Watson rotational symmetric distributions. Explicit expressions are given for the higher-order moments of Watson rotational symmetric distributions. The problem of constructing approximate confidence intervals for the kurtosis parameter is also discussed.     

The extreme value of the generalized distances of the individual points in the multivariate normal sample

1. Summary The extreme value of the generalized distances, from the origin, ofN individual points which may be correlated each other, in thep-variate normal sample is defined and discussed. It contains, as special cases, (i) the extreme deviate from the population mean or the sample mean, (ii) the extreme deviate from the control variate and (iii) the range defined by (2.10) or (2.11) below. The exact sampling distributional theory of this statistic is extremely difficult to find, even its moments. However, the method of obtaining the approximate upper 100α percentage points for the ordinary significance levelα is given. The lower percentage points can be obtained in the similar way if necessary. In connection with the evaluation of the approximate percentage points, the two-dimensional chi-square distribution is discussed and the asymptotic formulas for the joint distribution function of the two generalized distances are given in the special forms for the present aim. The extreme deviate from the sample mean will be explained in some detail and the tables of the approximate upper 5, 2.5 and 1% points are given. For the cases (ii) and (iii) mentioned above the details are omitted and will be discussed in the case of need.     

On characterizations of continuous distributions in terms of moments of order statistics when the sample size is random

We characterize continuous distribution functions F of a population when the sample size N is a random variable. Characterizing conditions in terms of moments of the kth order statistics are given, among other things, for the uniform, exponential, Pareto, logistic, and Weibull distributions. Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló, Hungary, 1997, Part II.     

Behaviour of queueing approximations based on sample moments

We investigate moment–based queueing approximations in the presence of sampling error. Let L be the steady–state mean number in the system for a GI/M/1 queue. We focus on the estimation of L under the assumption that only sample moments of the interarrival–time distribution are known. A simulation experiment is carried out for several interarrival–time distributions. For each case, sample moments from the interarrival–time distribution are matched to an approximating phase–type distribution and the corresponding estimate L is obtained. We show that the sampling error in the moments induces bias as well as variability in L. Based on our simulation experiment, we suggest matching only two moments when the sample coefficient of variation is low or when sample size is low; otherwise, matching three moments is preferable.     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

On Moments of Bivariate Order Statistics

In the present paper, we give the exact explicit expression for the product moments (of any order) of bivariate order statistics (o.s.) from any arbitrary continuous bivariate distribution function (d.f.). Furthermore, for any arbitrary bivariate uniform d.f., universal distribution-free bounds for the differences of any two different product moments (of order (1,1) or (-1,1)) are given.     

On some properties of the “apparent length” distribution and a compound distribution

This paper extends and supplements some of the properties of the apparent length distribution introduced by Baker (J. Appl. Statist. 27(1) (2000) 2–21). Recurrence relations for moments and formulae for moments derived via difference and differential operators will be given. We also establish maximum likelihood estimator and Bayesian estimators of a parameter of this distribution. Compound distributions based on the apparent length distribution are also discussed.