Characterization of Continuous Symmetric Distributions Based on Families of Skewed Distributions

A characterization of a symmetric probability density function based on certain properties of its associated skewed family is presented. This characterization is then applied to various well-known distributions.     

A note on the Cauchy-type mixture distributions

An interesting class of continuous distributions, called Cauchy-type mixture, with potential applications in modelling erratic phenomena is introduced by Soltani and Tafakori [A class of continuous kernels and Cauchy type heavy tail distributions. Statist Probab Lett. 2013;83:1018–1027]. In this work, we provide more insights into the Cauchy-type mixture distributions, involving certain characterizations, connections with the generalized Linnik distributions and the class of discrete distributions induced by stable laws. We also prove that the Laplace transform of Cauchy-type mixture distributions when normalized by constant terms become as a density functions in terms of distributional conjugate property.     

Tables for computing shortest confidence intervals involving the f-distribution

Tables of multipliers necessary for computing the shortest confidence interval in situations involving the F-distribution are presented. Such situations are illustrated with several examples.     

Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution

This article is concerned with modifications of both maximum likelihood and moment estimators for parameters of the three-parameter gamma distribution. Modifications employed here are essentially the same as those previously considered by the authors (1980, 1981) in connection with the lognormal distribution. Sampling behavior of the estimates is indicated by a Monte Carlo simulation. For certain combinations of parameter values, these new estimators appear better than both maximum likelihood and moment estimators with respect to bias, variance and/or ease of calculation.     

Characterization of a mixture of gamma distributions via conditional finite moments

A necessary and sufficient condition that a continuous, positive random variable follow a gamma distribution is given in terms of any one of its conditional finite moments and an expression involving its failure rate. The results are then used to develop a characterization for a mixture of two gamma distributions. The general results about characterization of a mixture of gamma distributions yield several special cases that have appeared separately in recent literature, including characterization of a single exponential distribution, characterization of a single gamma distribution (in terms of either first or second moments) and a sufficient condition for a mixture of two exponential distributions (in terms of first moments). The condition in this last result is shown to be necessary also. Numerous other cases are possible, using different choices for distribution parameters along with a selection of the mixing parameter, for either individual or mixtures of distributions. Various characterizations can be expressed using higher order moments, too.     

A bayesian wls approach to generalized linear models

We use a Bayesian approach to fitting a linear regression model to transformations of the natural parameter for the exponential class of distributions. The usual Bayesian approach is to assume that a linear model exactly describes the relationship among the natural parameters. We assume only that a linear model is approximately in force. We approximate the theta-links by using a linear model obtained by minimizing the posterior expectation of a loss function.While some posterior results can be obtained analytically considerable generality follows from an exact Monte Carlo method for obtaining random samples of parameter values or functions of parameter values from their respective posterior distributions. The approach that is presented is justified for small samples, requires only one-dimensional numerical integrations, and allows for the use of regression matrices with less than full column rank. Two numerical examples are provided.     

Multiple comparisons of matched proportions

It is well known (see, e.g., Scheffé (1959)) that if confidence intervals are desired for several treatment comparisons of interest, especially after a preliminary test of significance, then the appropriate technique is to consider simultaneous confidence intervals with a certain joint confidence coefficient. Goodman (1964) derived such simultaneous confidence intervals for contrasts among several multinomial populations, each with the same number, say J, of classes. The special case involving simultaneous confidence intervals for contrasts among several binomial populations on the basis of independent samples follows simply by taking J=2. This paper now deals with the problem of construction of simultaneous confidence intervals among probabilities of ‘success’ on the basis of matched samples.     

A New Class of Non Negative Distributions Generated by Symmetric Distributions

In this article, we study a new class of non negative distributions generated by the symmetric distributions around zero. For the special case of the distribution generated using the normal distribution, properties like moments generating function, stochastic representation, reliability connections, and inference aspects using methods of moments and maximum likelihood are studied. Moreover, a real data set is analyzed, illustrating the fact that good fits can result.     

A generalization for two-sided power distributions and adjusted method of moments

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.     

Extended Poisson process modelling of dilution series data

Summary.  Data comprising colony counts, or a binary variable representing fertile (or sterile) samples, as a dilution series of the containing medium are analysed by using extended Poisson process modelling. These models form a class of flexible probability distributions that are widely applicable to count and grouped binary data. Standard distributions such as Poisson and binomial, and those representing overdispersion and underdispersion relative to these distributions can be expressed within this class. For all the models in the class, likelihoods can be obtained. These models have not been widely used because of the perceived difficulty of performing the calculations and the lack of associated software. Exact calculation of the probabilities that are involved can be time consuming although accurate approximations that use considerably less computational time are available. Although dilution series data are the focus here, the models are applicable to any count or binary data. A benefit of the approach is the ability to draw likelihood-based inferences from the data.     

Some nonparametric methods for changepoint problems

A general model for changepoint problems is discussed from a nonparametric viewpoint. The test statistics introduced are based on Cramér-von Mises functionals of certain processes and are shown to converge in distribution to corresponding Gaussian functionals (under the assumption of no change in distribution, H₀). We also demonstrate how the distribution of the limiting Gaussian functionals may be tabulated. Finally, properties of the tests under the alternative hypothesis of exactly one changepoint occurring are studied, and some examples are given.     

On some generalized farlie-gumbel-morgenstern distributions-II regression,correlation and further generalizations

Regression and correlation properties of the generalized Farlie-Gumbel-Morgenstem distributions introduced in Johnson and Kotz (1975) are studied. Further generalizations of these distributions are considered.     

A Pitman measure of similarity in k-means for clustering heavy-tailed data

One of the most popular methods and algorithms to partition data to k clusters is k-means clustering algorithm. Since this method relies on some basic conditions such as, the existence of mean and finite variance, it is unsuitable for data that their variances are infinite such as data with heavy tailed distribution. Pitman Measure of Closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. In this article using PMC, based on k-means clustering, a new distance and clustering algorithm is developed for heavy tailed data.     

A wider class of lagrange distributions of the second kind

Starting with the second Lagrange expansion, with f(z) and g(z) as two probability generating functions defined on non-negative integers, Janardan and Rao( 1983) introduced a new class of discrete distributions called the Lagrange Distributions (LD₂) of the second kind. In this note, this class of LD₂ distributions is widened by removing the restriction that the functions f(z) and g(z) be probability generation functions. It is also shown that the class of modified power series distributions is a subclass of LD₂.     

On the Bayesian analysis of categorical data: the problem of nonresponse

It is demonstrated how a suitably chosen prior for the frequency parameters can streamline the Bayesian analysis of categorical data with missing entries due to nonresponse or other causes. The two cases where the data follow the Multinomial or the Hypergeometric model are treated separately. In the first case it is adequate to restrict the prior (for the cell probabilities) to the class of Dirichlet distributions. In the case of the Hypergeometric model it is convenient to select a prior from the class of Dirichlet-Multinomial (DM) distributions. The DM distributions are studied in some details.     

An analysis of quantile measures of kurtosis: center and tails

The consequences of substituting the denominator Q ₃(p)  −  Q ₁(p) by Q ₂  −  Q ₁(p) in Groeneveld’s class of quantile measures of kurtosis (γ ₂(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ ₂(p) and the alternative class of kurtosis measures κ₂(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions. The authors thank the referee for the careful review.     

A note on finding critical values for a locally optimal multivariate test statistic

A previous paper provided a locally optimal test statistic for combining s independent test statistics of a common multivariate hypothesis. Difficulty in calculating the critical values limited the applicability of the test. In this paper, it is shown that approximate critical values can be easily calculated, with negligible loss of accuracy for most situations.     

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.     

ASYMPTOTIC DISTRIBUTION OF THE LARGEST EIGENVALUE

ABSTRACT

This paper studies the asymptotic distribution of the largest eigenvalue of the sample covariance matrix. The multivariate distribution for the population is assumed to be elliptical with finite kurtosis 3κ. An expression as an expectation is obtained for the distribution function of the largest eigenvalue regardless of the multiplicity, m, of the population's largest eigenvalue. The asymptotic distribution function and density function are evaluated numerically for m = 2,3,4,5. The bootstrap of the average of the m largest eigenvalues is shown to be consistent for any underlying distribution with finite fourth-order cumulants.