A skew factor analysis model based on the normal mean–variance mixture of Birnbaum–Saunders distribution

This paper presents a robust extension of factor analysis model by assuming the multivariate normal mean–variance mixture of Birnbaum–Saunders distribution for the unobservable factors and errors. A computationally analytical EM-based algorithm is developed to find maximum likelihood estimates of the parameters. The asymptotic standard errors of parameter estimates are derived under an information-based paradigm. Numerical merits of the proposed methodology are illustrated using both simulated and real datasets.     

Mixture inverse Gaussian distributions and its transformations,moments and applications

Skewed models are important and necessary when parametric analyses are carried out on data. Mixture distributions produce widely flexible models with good statistical and probabilistic properties, and the mixture inverse Gaussian (MIG) model is one of those. Transformations of the MIG model also create new parametric distributions, which are useful in diverse situations. The aim of this paper is to discuss several aspects of the MIG distribution useful for modelling positive data. We specifically discuss transformations, the derivation of moments, fitting of models, and a shape analysis of the transformations. Finally, real examples from engineering, environment, insurance, and toxicology are presented for illustrating some of the results developed here. Three of the four data sets, which have arisen from the consulting work of the authors, are new and have not been previously analysed. All these examples display that the empirical fit of the MIG distribution to the data is very good.     

Log-gamma-generated families of distributions

We introduce two new general families of continuous distributions, generated by a distribution F and two positive real parameters α and β which control the skewness and tail weight of the distribution. The construction is motivated by the distribution of k-record statistics and can be derived by applying the inverse probability integral transformation to the log-gamma distribution. The introduced families are suitable for modelling the data with a significantly skewed and heavy-tailed distribution. Various properties of the introduced families are studied and a number of estimations and data fitness on real data are given to illustrate the results.     

On the normality of transformed beta and unit- gamma random variables

In this paper we consider properties of the logarithmic and Tukey's lambda-type transformations of random variables that follow beta or unit-gamma distributions. Beta distributions often arise as models for random proportions, and unit-gamma distributions, although not well- known, may serve the same purpose. The latter possess many properties similar to those of beta distributions. Some transformations of random variables that follow a beta distribution are considered by Johnson (1949) and Johnson and Kotz (1970,1973). These are used to obtain a -new"random variable that potentially approximately follows a normal distribution, so that practical analyses become possible. We study normality -related properties of the above transformations. This is done for the first time for unit-gamma distributions. Under the logarithmic transformation the beta and unit-gamma distributions become, respectively, the logarithmic F and generalized logistic distributions. The distributions of the transformed beta and unit-gamma distributions after application of Tukey's lambda-type transformations cannot be derived easily; however, we obtain the first four moments and expressions for the skewness and kudos is of the transformed variables. Values of skewness and kurtosis for a variety of different parameter values are calculated, and in consequence, the near (or not near) normality of the transformed variables is evaluated. Comments on the use of the various transformations are provided..     

Comparison of estimators of a dispersion matrix under generalized pitman nearness criterion

Some equivariant estimators of the dispersion matrix of a multivariate normal population are compared using the generalized Pitman nearness criterion based on the entropy loss function. It is shown that, under the group of lower triangular transformations, a best equivariant estimator does not exist. Existence of best estimators in certain subclasses are discussed and the performances of two commonly used estimators are compared. Some properties of central chi-square distributions are obtained and used to derive the main results.     

Inferences on the coefficient of variation of an inverse gaussian distribution

The likelihood ratio test for a characteristic parameter of the inverse Gaussian distribution is derived. The parameter of interest characterizes the coefficient of variation, the skewness and the kurtosis of the distribution. The distribution of the test statistic is presented in a simplified form. Useful quanfiles of the distribution are given. Methods for constructing confidence bounds for the parameter, including Bayes highest posterior density intervals, are considered.     

An asymptotic representation of a ratio of two statistics and its applications

Some statistics in common use take a form of a ratio of two statistics.In this paper, we will discuss asymptotic properties of the ratio statistic.We obtain an asymptotic representation of the ratio with remainder term o _p(n ^-1) and a Edgeworth expansion with remainder term o(n ^-1/2) And as example, the asymptotic representation and the Edgeworth expansion of the jackknife skewness estimator for U-statistics are established and we discuss the biases of the skewness estimator theoretically.We also apply the result to an estimator of Pearson’s coefficient of variation and the sample correlation coefficient.     

Generalized Confidence Interval for the Scale Parameter of the Power-Law Process

The power-law process is widely used in the analysis of repairable system reliability. In this article, interval estimation for the scale parameter is investigated under some general conditions. A procedure to derive a generalized confidence interval for the scale parameter is presented. We also study the accuracy of the generalized confidence interval by Monte Carlo simulation. Finally, two examples are shown to illustrate the proposed procedure.     

Testing high-dimensional normality based on classical skewness and Kurtosis with a possible small sample size

Abstract

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples.     

Inference in flexible families of distributions with normal kernel

This paper addresses the inference problem for a flexible class of distributions with normal kernel known as skew-bimodal-normal family of distributions. We obtain posterior and predictive distributions assuming different prior specifications. We provide conditions for the existence of the maximum-likelihood estimators (MLE). An EM-type algorithm is built to compute them. As a by product, we obtain important results related to classical and Bayesian inferences for two special subclasses called bimodal-normal and skew-normal (SN) distribution families. We perform a Monte Carlo simulation study to analyse behaviour of the MLE and some Bayesian ones. Considering the frontier data previously studied in the literature, we use the skew-bimodal-normal (SBN) distribution for density estimation. For that data set, we conclude that the SBN model provides as good a fit as the one obtained using the location-scale SN model. Since the former is a more parsimonious model, such a result is shown to be more attractive.