Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

Asymptotic distributions of the sphericity test in a complex multivariate normal distribution

In this paper, asymptotic expansions of the null and non-null distributions of the sphericity test criterion in the case of a complex multivariate normal distribution are obtained for the first time in terms of beta distributions. In the null case, it is found that the accuracy of the approximation by taking the first term alone in the asymptotic series is sufficient for practical purposes. In fact for p - 2. the asymptotic expansion reduces to the first term which is also the exact distribution in this case. Applications of the results to the area of inferences on multivariate time series are also given.     

Two-Term Edgeworth Expansions for the Classes of U- and V-statistics

Much effort has been devoted to deriving Edgeworth expansions for various classes of statistics that are asymptotically normally distributed, with derivations tailored to the individual structure of each class. Expansions with smaller error rates are needed for more accurate statistical inference. Two such Edgeworth expansions are derived analytically in this paper. One is a two-term expansion for the standardized U-statistic of order m, m ? 3, with an error rate o(n^{? 1}). The other is an expansion with the same error rate for the distribution of the standardized V-statistic of the same order. In deriving the Edgeworth expansion, we made use of the close connection between the V- and U-statistics, which permits to first derive the needed expansion for the related U-statistic, then extend it to the V-statistic, taking into consideration the estimation of all difference terms between the two statistics.     

On a class of distributions on the simplex

In the present paper we define and investigate a novel class of distributions on the simplex, termed normalized infinitely divisible distributions, which includes the Dirichlet distribution. Distributional properties and general moment formulae are derived. Particular attention is devoted to special cases of normalized infinitely divisible distributions which lead to explicit expressions. As a by-product new distributions over the unit interval and a generalization of the Bessel function distribution are obtained.     

Tail Behavior of the General Error Distribution

An inequality on the tail behavior of the general error distribution and an asymptotic Mills-type ratio are established. Two applications are provided. The first application considers the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the general error and normal distributions. The second application establishes the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the general error distribution.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

The effects on the distributions of sample canonical correlations under nonnormality

In this paper, we study the effects of nonnormality on the distributions of sample canonical correlations when the population canonical correlations are simple. In order to achieve the purpose, we derive asymptotic expansion formulas for the distributions of a function of the canonical correlations as well as the individual canonical correlations under nonnormal populations. We particularly discuss the distribution of sample canonical correlations under the class of elliptical population. These expansions are given by using a perturbation method. Simulation results are also given.     

Moment convergence of powered normal extremes

In this article, convergence for moments of powered normal extremes is considered under an optimal choice of normalizing constants. It is shown that the rates of convergence for normalized powered normal extremes depend on the power index. However, the dependence disappears for higher-order expansions of moments.     

Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures

An asymptotic expansion of the null distribution of the chi-square statistic based on the asymptotically distribution-free theory for general covariance structures is derived under non-normality. The added higher-order term in the approximate density is given by a weighted sum of those of the chi-square distributed variables with different degrees of freedom. A formula for the corresponding Bartlett correction is also shown without using the above asymptotic expansion. Under a fixed alternative hypothesis, the Edgeworth expansion of the distribution of the standardized chi-square statistic is given up to order O(1/n). From the intermediate results of the asymptotic expansions for the chi-square statistics, asymptotic expansions of the joint distributions of the parameter estimators both under the null and fixed alternative hypotheses are derived up to order O(1/n).     

Tail Behavior and Limit Distribution of Maximum of Logarithmic General Error Distribution

Logarithmic general error distribution, an extension of the log-normal distribution, is proposed. Some interesting properties of the log GED are derived. These properties are applied to establish the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the logarithmic general error and log-normal distributions, and to derive the asymptotic distribution of the partial maximum of an independent and identically distributed sequence obeying the log GED.     

Fiducial and Confidence Distributions for Real Exponential Families

We develop an easy and direct way to define and compute the fiducial distribution of a real parameter for both continuous and discrete exponential families. Furthermore, such a distribution satisfies the requirements to be considered a confidence distribution. Many examples are provided for models, which, although very simple, are widely used in applications. A characterization of the families for which the fiducial distribution coincides with a Bayesian posterior is given, and the strict connection with Jeffreys prior is shown. Asymptotic expansions of fiducial distributions are obtained without any further assumptions, and again, the relationship with the objective Bayesian analysis is pointed out. Finally, using the Edgeworth expansions, we compare the coverage of the fiducial intervals with that of other common intervals, proving the good behaviour of the former.     

Expansions of the coverage probabilities of prediction region based on a shrinkage estimator

An expansion formula for the coverage probability of prediction region based on a shrinkage estimator proposed by Joshi [Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist., 38, 1868–1875.] is obtained. Its error bound is evaluated in terms of a function of an unknown parameter. Applying this result, three types of asymptotic expansions are derived. These expansions show inadmissibility of the usual prediction region.     

Second-order and resampling approximation of finite population U-statistics based on stratified samples

We construct one-term Edgeworth expansions to distributions of U statistics and Studentized U-statistics, based on stratified samples drawn without replacement. Replacing the cumulants defining the expansions by consistent jackknife estimators, we obtain empirical Edgeworth expansions. The expansions provide second-order approximations that improve upon the normal approximation. Theoretical results are illustrated by a simulation study where we compare various approximations to the distribution of the commonly used Gini's mean difference estimator.     

Stein's two sample test of a general linear hypothesis and a noncentral f–type distribution

Stein's two–sample procedure for a general linear model is studied and derived in terms of matrices in which the error tems are distributed as multivatriate student t–error terms. Tests and confidence regions are constructed in a similar way to classical linear models which involves percentage points of student t and F distributions. The advantages of taking two samples are: the variance of the error terms is known, and the power of tests are size of confidence regions are controllable. A new distribution called noncentral F–type distribution different from the nencentral F is found when considerinf the power of the test of general linear hypothesis.     

A random weighting approach for posterior distributions

ABSTRACT

In Bayesian theory, calculating a posterior probability distribution is highly important but typically difficult. Therefore, some methods have been proposed to deal with such problem, among which, the most popular one is the asymptotic expansions of posterior distributions. In this paper, we propose an alternative approach, named a random weighting method, for scaled posterior distributions, and give an ideal convergence rate, o(n^{( ? 1/2)}), which serves as the theoretical guarantee for methods of numerical simulations.     

Modified Edgeworth and Cornish-Fisher Expansions with Unknown Cumulants and no Singularities

In this paper modified Edgeworth and Cornish-Fisher expansions are introduced. These expansions do not require knowing the cumulants of the distributions involved. More importantly, the new expansions have no singularities.     

Information theoretic results for circular distributions

The broad class of generalized von Mises (GvM) circular distributions has certain optimal properties with respect to information theoretic quantities. It is shown that, under constraints on the trigonometric moments, and using the Kullback–Leibler information as the measure, the closest circular distribution to any other is of the GvM form. The lower bounds for the Kullback–Leibler information in this situation are also provided. The same problem is also considered using a modified version of the Kullback–Leibler information. Finally, series expansions are given for the entropy and the normalizing constants of the GvM distribution.     

Multivariate semi-α-Laplace distributions

A multivariate semi-α-Laplace distribution (denoted by Ms-αLaplace) is introduced and studied in this paper. It is more general than the multivariate Linnik and Laplace distributions proposed by Sabu and Pillai (1991) or Anderson (1992). The Ms-αLaplace distribution has univariate semi-α-Laplace (Pillai, 1985) as marginal distribution. Various characterization theorems of the Ms-αLaplace distribution based on the closure property of the normalized geometric sum are proved.     

Third-Order Asymptotic Distributions of Classic Test Statistics for One-Parameter Exponential Family Models

ABSTRACT

In this article we derive third-order asymptotic expansions for the non null distribution functions of four classic statistics under a sequence of local alternatives in one-parameter exponential family models. Our results are quite general and cover a wide range of important distributions.     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.