一般形式下斜正态随机向量的矩

本文给出一般形式下斜正态随机向量及其平方型的矩公式. 作为应用, 计算出了斜正态随机向量的多元偏度和峰度.     

Relationships between moments of two related sets of order statistics and some extensions

Govindarajulu expressed the moments of order statistics from a symmetric distribution in terms of those from its folded form. He derived these relations analytically by dividing the range of integration suitably into parts. In this paper, we establish these relations through probabilistic arguments which readily extend to the independent and non-identically distributed case. Results for random variables having arbitrary multivariate distributions are also derived.The first author would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Multivariate Normal Slice Sampling

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time.

In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.     

A characterization of the multivariate normal distribution by using the hazard gradient

We give a general result to characterize a multivariate distribution from a relationship between the left truncated mean function and the hazard gradient function. This result allows us to obtain new characterizations of multivariate distributions. In particular, we show that, for the multivariate normal distribution, the simple relationship, obtained in standardized form by McGill (1992,Communications in Statistics. Theory Methods,21(11), 3053–3060), actually characterizes the multivariate normal distribution. Supported by Ministerio de Ciencia y Tecnologia under grant BFM2000-0362.     

A note on the moments of order statistics from doubly truncated exponential distribution

Summary In a recent paper [2], the author has obtained some recurrence relations between the moments of order statitics from the exponential and right truncated exponential distributions. In this paper, similar relations are derived for a doubly truncated exponential distribution. It is shown that one can obtain all the moments by using these recurrence relations.     

Number of Records in a Bivariate Sample with Application to Missouri River Flood Data

For a sequence of observations from a bivariate absolutely continuous distribution, two types of records are considered depending on whether a univariate record is established in both or in at least one of the components. The distributional properties of the associated univariate and bivariate record indicators are examined. Correlation between the number of component records and the first two moments of the number of bivariate records in a finite random sample are obtained. These are evaluated for the Farlie-Gumbel-Morgenstern and bivariate normal distributions. Large sample properties of these moments are explored. Our results are used to predict the number of record annual floods at two sites along the Missouri river during the next 50 years.     

The singly truncated normal distribution: A non-steep exponential family

This paper is concerned with the maximum likelihood estimation problem for the singly truncated normal family of distributions. Necessary and suficient conditions, in terms of the coefficient of variation, are provided in order to obtain a solution to the likelihood equations. Furthermore, the maximum likelihood estimator is obtained as a limit case when the likelihood equation has no solution.     

多维不可微平稳正态过程的极值

讨论了强相关多维不可微平稳正态过程的渐进性质 ,给出了其最大值的极限分布     

On discrete distributions of orderk

Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed.     

Geometric stable laws: Estimation and applications

Geometric stable laws constitute a class of limiting distributions of appropriately normalized random sums of i.i.d. random variables. We consider the problem of estimation of the parameters of univariate and multivariate geometric stable laws. Our estimation technique is based on the method of moments and yields consistent and asymptotically normal estimators. We apply our estimators to a currency exchange data and show that the geometric stable dominates Paretian stable and normal models.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

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.     

A class of weighted multivariate normal distributions and its properties

This article proposes a class of weighted multivariate normal distributions whose probability density function has the form of a product of a multivariate normal density and a weighting function. The class is obtained from marginal distributions of various doubly truncated multivariate normal distributions. The class strictly includes the multivariate normal and multivariate skew-normal. It is useful for selection modeling and inequality constrained normal mean vector analysis. We report on a study of some distributional properties and the Bayesian perspective of the class. A probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify the distribution and to implement computation, with output readily adapted for the required analysis. Necessary theories and illustrative examples are provided.     

Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

Some Truncated Distributions

Long-tailed distributions arise in many areas of the sciences. These distributions, however, suffer from the weakness of not having finite moments of all orders and this weakness has restricted their use. In this note, we introduce truncated versions of five of the most commonly known long-tailed distributions—which possess finite moments of all orders and could therefore be better models. Explicit expressions for the moments are derived for each of the truncated distributions. Several applications are illustrated using real data.     

f-Dissimilarity of Several Distributions in Testing Statistical Hypotheses

Various problems in statistics have been treated by the decision rule, based on the concept of distance between distributions. The aim of this paper is to give an approach for testing statistical hypotheses, using a general class of dissimilarity measures among k 2 distributions. The test statistics are obtained by the replacement, in the expression of the dissimilarity measure, of the unknown parameters by their maximum likelihood estimators. The asymptotic distributions of the resulting test statistics are investigated and the results are applied to multinomial and multivariate normal populations.     

A new construction for skew multivariate distributions

This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal (type A and type B) and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions.     

Sequential order statistics with an order statistics prior

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated.     

Efficient algorithms for generating truncated multivariate normal distributions

Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.