A new class of matrix variate elliptically contoured distributions

Summary A new class of matrix variate elliptically contoured distributions is defined. Properties of this class of distributions are studied. Examples of distributions which belong to this class are also presented. Rose-Hulman Institute of Technology Research supported by the FRC Major Grant, Bowling Grant, Bowling Green State University.     

A note on predictive inference for multivariate elliptically contoured distributions

The prediction problem is considered for the multivariate regression model with an elliptically contoured error distribution. We show that the predictive distribution under elliptical errors assumption is the same as that obtained under normally distributed error in both the Bayesian approach using an im-proper prior and the classical approach. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution.     

Influence diagnostics on the coefficient of variation of elliptically contoured distributions

In this article, we study the behavior of the coefficient of variation (CV) of a random variable that follows a symmetric distribution in the real line. Specifically, we estimate this coefficient using the maximum-likelihood (ML) method. In addition, we provide asymptotic inference for this parameter, which allows us to contrast hypothesis and construct confidence intervals. Furthermore, we produce influence diagnostics to evaluate the sensitivity of the ML estimate of this coefficient when atypical data are present. Moreover, we illustrate the obtained results by using financial real data. Finally, we carry out a simulation study to detect the potential influence of atypical observations on the ML estimator of the CV of a symmetric distribution. The illustration and simulation demonstrate the robustness of the ML estimation of this coefficient.     

Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.     

A characterization of the bivariate elliptically contoured distribution

The Cauchy distribution of the quotient of two random variables and its independence of the sum of squares of them together with some symmetry assumptions are the characteristic property of the bivariate elliptically contoured measures.     

A note on the orthant probability of the elliptically contoured distribution

When a scale matrix satisfies certain conditions, the orthant probability of the elliptically contoured distribution with the scale matrix is expressed as the same probability of the equicorrelated normal distribution.     

A note on classical Stein-type estimators in elliptically contoured models

In this paper the conditions under which a broad class of Stein-type estimators dominates the best invariant unbiased estimator of the mean of an elliptically contoured population have been established. The superiority conditions are derived for both known and unknown scale structures. Also an example is given when the general scale matrix is assumed to be known in linear regression.     

Improving on the sample covariance matrix for a complex elliptically contoured distribution

In this paper the problem of estimating the scale matrix in a complex elliptically contoured distribution (complex ECD) is addressed. An extended Haff–Stein identity for this model is derived. It is shown that the minimax estimators of the covariance matrix obtained under the complex normal model remain robust under the complex ECD model when the Stein loss function is employed.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

A new family of multivariate slash distributions

In this article, a new form of multivariate slash distribution is introduced and some statistical properties are derived. In order to illustrate the advantage of this distribution over the existing generalized multivariate slash distribution in the literature, it is applied to a real data set.     

A general family of NBU classes of life distributions

In this paper we introduce and study a family of new better than used ageing notions parameterized by a function h. The new better than used , the new better than used of a specified age and the new better than used in total time on test transform order , are special cases of this new family. We study some properties of the new family, and we give some applications of it in actuarial science, reliability theory and statistics.     

The distribution of the Liu-type estimator of the biasing parameter in elliptically contoured models

We derive the density function of the stochastic shrinkage parameters of the Liu-type estimator in elliptical models. The correctness of derivation is checked by simulations. A real data application is also provided.     

A new generalized odd log-logistic family of distributions

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the new generalized odd log-logistic family of distributions. The proposed family contains several important classes discussed in the literature as submodels such as the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, entropy measures, and order statistics, which hold for any baseline model, are presented. We also present certain characterization of the proposed distribution and derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behavior of the maximum likelihood estimator via simulation. The importance of the new family is illustrated by means of two real data sets. These applications indicate that the new family can provide better fits than other well-known classes of distributions. The beauty and importance of the new family lies in its ability to model real data.     

A new flexible generalized family for constructing many families of distributions

We propose a new flexible generalized family (NFGF) for constructing many families of distributions. The importance of the NFGF is that any baseline distribution can be chosen and it does not involve any additional parameters. Some useful statistical properties of the NFGF are determined such as a linear representation for the family density, analytical shapes of the density and hazard rate, random variable generation, moments and generating function. Further, the structural properties of a special model named the new flexible Kumaraswamy (NFKw) distribution, are investigated, and the model parameters are estimated by maximum-likelihood method. A simulation study is carried out to assess the performance of the estimates. The usefulness of the NFKw model is proved empirically by means of three real-life data sets. In fact, the two-parameter NFKw model performs better than three-parameter transmuted-Kumaraswamy, three-parameter exponentiated-Kumaraswamy and the well-known two-parameter Kumaraswamy models.     

A family of skew-symmetric-Laplace distributions

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we shall introduce another more general class of skew distributions, specially related to the Laplace distribution. This new class contains some previously known skew distributions. We shall investigate different characteristics of members of this class such as its moments, thus generalizing a result of Umbach (Stat Probab Lett 76:507?C512, 2006), limiting behavior, moment generating function, unimodality and reveal its natural occurrence as the distribution of some order statistics. In addition, we will generalize a result of Aryal and Rao (Nonlinear Anal 63:639?C646, 2005) in connection with truncated skew-Laplace distribution and study its certain stochastic orderings. Some illustrative examples are also provided.     

Two-Piece location-scale distributions based on scale mixtures of normal family

In this work, we study the maximum likelihood (ML) estimation problem for the parameters of the two-piece (TP) distribution based on the scale mixtures of normal (SMN) distributions. This is a family of skewed distributions that also includes the scales mixtures of normal class, and is flexible enough for modeling symmetric and asymmetric data. The ML estimates of the proposed model parameters are obtained via an expectation-maximization (EM)-type algorithm.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.