Exponentiated Pareto distributions

Most Pareto distributions are defined on one side of the real line. For wider applicability, we introduce five exponentiated Pareto distributions and derive several of their properties including the moment generating function, expectation, variance, skewness, kurtosis, Shannon entropy, and the Rényi entropy.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Form-invariance under weighted sampling

In this paper, form-invariant weighted distributions are considered in an exponential family. The class of bivariate distribution with invariant property is identified under exponential weight function. The class includes some of the custom bivariate models. The form-invariant multivariate normal distributions are obtained under a quadratic weight function.     

Stability of a characterization by the identical distribution of linear forms

Hidden Markov models (HMMs) have during the last decade become a widely spread tool for modelling sequences of dependent random variables. Inference for HMMs has been considered by several authors, but so far no work has been done on estimating their order. In this paper we propose a penalized likelihood estimator for this purpose. This estimator is based on the m-dimensional distribution of HMM, and it is shown that in the limit it does not underestimate the order.     

Discrete generalized exponential distribution of a second type

In this paper, we shall attempt to introduce another discrete analogue of the generalized exponential distribution of Gupta and Kundu [Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), pp. 173–188], different to that of Nekoukhou et al. [A discrete analogue of the generalized exponential distribution, Comm. Stat. Theory Methods, to appear (2011)]. This new discrete distribution, which we shall call a discrete generalized exponential distribution of the second type (DGE₂(α, p)), can be viewed as another generalization of the geometric distribution. We shall first study some basic distributional and moment properties, as well as order statistics distributions of this family of new distributions. Certain compounded DGE₂(α, p) distributions are also discussed as the results of which some previous lifetime distributions such as that of Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] follow as corollaries. Then, we will investigate estimation of the parameters involved. Finally, we will examine the model with a real data set.     

Application domains for the Delta method

The Delta method uses truncated Lagrange expansions of statistics to obtain approximations to their distributions. In this paper, we consider statistics Y=g(μ+X), where X is any random vector. We obtain domains 𝒟 such that, when μ∈𝒟, we may apply the distribution derived from the Delta method. Namely, we will consider an application on the normal case to illustrate our approach.     

On the Sampling Distributions of the Estimated Process Loss Indices with Asymmetric Tolerances

The inverse Gaussian distribution provides a flexible model for analyzing positive, right-skewed data. The generalized variable test for equality of several inverse Gaussian means with unknown and arbitrary variances has satisfactory Type-I error rate when the number of samples (k) is small (Tian, 2006). However, the Type-I error rate tends to be inflated when k goes up. In this article, we propose a parametric bootstrap (PB) approach for this problem. Simulation results show that the proposed test performs very satisfactorily regardless of the number of samples and sample sizes. This method is illustrated by an example.     

Posterior moments of scalar functions of a covariance matrix

Given multivariate normal data and a certain spherically invariant prior distribution on the covariance matrix, it is desired to estimate the moments of the posterior marginal distributions of some scalar functions of the covariance matrix by importance sampling. To this end a family of distributions is defined on the group of orthogonal matrices and a procedure is proposed for selecting one of these distributions for use as a weighting distribution in the importance sampling process. In an example estimates are calculated for the posterior mean and variance of each element in the covariance matrix expressed in the original coordinates, for the posterior mean of each element in the correlation matrix expressed in the original coordinates, and for the posterior mean of each element in the covariance matrix expressed in the coordinates of the principal variables.     

Some Uses for Distribution-Fitting Software in Teaching Statistics

Statistics courses now make extensive use of menu-driven, interactive computer software. This article presents some insight as to how a new class of PC-based statistical software, called “distribution-fitting” software, can be used in teaching various courses in statistics.     

The Logistic-Uniform Distribution and Its Applications

In this article, we give a new family of univariate distributions generated by the Logistic random variable. A special case of this family is the Logistic-Uniform distribution. We show that the Logistic-Uniform distribution provides great flexibility in modeling for symmetric, negatively and positively skewed, bathtub-shaped, “J”-shaped, and reverse “J”-shaped distributions. We discuss simulation issues, estimation by the methods of moments, maximum likelihood, and the new method of minimum spacing distance estimator. We also derive Shannon entropy and asymptotic distribution of the extreme order statistics of this distribution. The new distribution can be used effectively in the analysis of survival data since the hazard function of the distribution can be “J,” bathtub, and concave-convex shaped. The usefulness of the new distribution is illustrated through two real datasets by showing that it is more flexible in analyzing the data than the Beta Generalized-Exponential, Beta-Exponential, Beta-Normal, Beta-Laplace, Beta Generalized half-Normal, β-Birnbaum-Saunders, Gamma-Uniform, Beta Generalized Pareto, Beta Modified Weibull, Beta-Pareto, Generalized Modified Weibull, Beta-Weibull, and Modified-Weibull distributions.