Inequalities involving expectations of selected functions in reliability theory to characterize distributions

Recently, authors have studied inequalities involving expectations of selected functions, viz. failure rate, mean residual life, aging intensity function, and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time (X ? x, instead of X > x) and their applications. In the present work we consider reversed hazard rate, expected inactivity time, and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.     

A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

A characterization of invariant power-series abundance distributions

Consider a population the individuals in which can be classified into groups. Let y, the number of individuals in a group, be distributed according to a probability function f(y;ø_o) where the functional form f is known. The random variable y cannot be observed directly, and hence a random sample of groups cannot be obtained. Consider a random sample of N individuals from the population. Suppose the N individuals are distributed into S groups with x₁, x₂, …, x_S representatives respectively. The random variable x, the number of individuals in a group in the sample, will be a fraction of its population counterpart y, and the distributions of x and y need not have the same functional form. If the two random variables x and y have the same functional form for their distributions, then the particular common distribution is called an invariant abundance distribution. The paper provides a characterization of invariant abundance distributions in the class of power-series distributions.     

On a characteristic property of generalized Pareto distributions, extreme value distributions and their max domains of attraction

Using the independence of an arbitrary random variable Y and the weighted minima of independent, identically distributed random variables with weights depending on Y, we characterize extreme value distributions and generalized Pareto distributions. A discussion is made about an analogous characterization for distributions in the max domains of attraction of extreme value limit laws.     

The linear combination,product and ratio of Laplace random variables

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this paper, the exact distributions of the linear combination α X+β Y, the product |X Y| and the ratio |X/Y| are derived when X and Y are independent Laplace random variables. The Laplace distribution, being the oldest model for continuous data, has been one of the most popular models for measurement errors in engineering.     

Estimation in nonlinear mixed-effects models using heavy-tailed distributions

Nonlinear mixed-effects models are very useful to analyze repeated measures data and are used in a variety of applications. Normal distributions for random effects and residual errors are usually assumed, but such assumptions make inferences vulnerable to the presence of outliers. In this work, we introduce an extension of a normal nonlinear mixed-effects model considering a subclass of elliptical contoured distributions for both random effects and residual errors. This elliptical subclass, the scale mixtures of normal (SMN) distributions, includes heavy-tailed multivariate distributions, such as Student-t, the contaminated normal and slash, among others, and represents an interesting alternative to outliers accommodation maintaining the elegance and simplicity of the maximum likelihood theory. We propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed-effects and variance components, using a stochastic approximation of the EM algorithm. We compare the performance of the normal and the SMN models with two real data sets.     

Random Convex Combinations of m-Generalized Order Statistics

The situation of identical distributions of a single m-generalized order statistic and a random convex combination of two neighboring m-generalized order statistics with beta-distributed weights is studied, and related characterizations of generalized Pareto distributions are shown.     

Non-ergodic martingale estimating functions and related asymptotics

Well-known estimation methods such as conditional least squares, quasilikelihood and maximum likelihood (ML) can be unified via a single framework of martingale estimating functions (MEFs). Asymptotic distributions of estimates for ergodic processes use constant norm (e.g. square root of the sample size) for asymptotic normality. For certain non-ergodic-type applications, however, such as explosive autoregression and super-critical branching processes, one needs a random norm in order to get normal limit distributions. In this paper, we are concerned with non-ergodic processes and investigate limit distributions for a broad class of MEFs. Asymptotic optimality (within a certain class of non-ergodic MEFs) of the ML estimate is deduced via establishing a convolution theorem using a random norm. Applications to non-ergodic autoregressive processes, generalized autoregressive conditional heteroscedastic-type processes, and super-critical branching processes are discussed. Asymptotic optimality in terms of the maximum random limiting power regarding large sample tests is briefly discussed.     

Phase Type Distributions with Finite Support

This research is motivated by the fact that many random variables of practical interest have a finite support. For fixed a < b, we consider the distribution of a random variable X = (a + Ymod(b ? a)), where Y is a phase type (PH) random variable. We demonstrate that as we traverse for Y the entire set of PH distributions (or even any subset thereof like Coxian that is dense in the class of distributions on [0, ∞)), we obtain a class of matrix exponential distributions dense in (a, b). We call these Finite Support Phase Type Distributions (FSPH) of the first kind. A simple example shows that though dense, this class by itself is not very efficient for modeling; therefore, we introduce (and derive the EM algorithms for) two other classes of finite support phase type distributions (FSPH). The properties of denseness, connection to Markov chains, the EM algorithm, and ability to exploit matrix-based computations should all make these classes of distributions attractive not only for applied probability but also for a much wider variety of fields using statistical methodologies.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

Simulations and computations of nonparametric density estimates for the deconvolution problem

The nonparametric density function estimation using sample observations which are contaminated with random noise is studied. The particular form of contamination under consideration is Y = X + Z, where Y is an observable random variableZ is a random noise variable with known distribution, and X is an absolutely continuous random variable which cannot be observed directly. The finite sample size performance of a strongly consistent estimator for the density function of the random variable X is illustrated for different distributions. The estimator uses Fourier and kernel function estimation techniques and allows the user to choose constants which relate to bandwidth windows and limits on integration and which greatly affect the appearance and properties of the estimates. Numerical techniques for computation of the estimated densities and for optimal selection of the constant are given.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.     

A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

SERIES REPRESENTATION OF DOUBLY NONCENTRAL t DISTRIBUTION BY USE OF THE MELLIN INTEGRAL TRANSFORM

ABSTRACT

The Mellin integral transform is widely used to find the distributions of products and quotients of independent random variables defined over the positive domain. But it is hardly used to derive the distributions defined over both positive and negative values of the random variables. In this paper, the Mellin integral transform is applied to obtain the doubly noncentral t density and its distribution function in convergent series forms.     

On some study of skew-t distributions

Abstract

In this note, through ratio of independent random variables, new families of univariate and bivariate skew-t distributions are introduced. Probability density function for each skew-t distribution will be given. We also derive explicit forms of moments of the univariate skew-t distribution and recurrence relations for its cumulative distribution function. Finally we illustrate the flexibility of this class of distributions with applications to a simulated data and the volcanos heights data.     

Power kurtosis transformations: Definition, properties and ordering

Summary Heavy tail distributions can be generated by applying specific non-linear transformations to a Gaussian random variable. Within this work we introduce power kurtosis transformations which are essentially determined by their generator function. Examples are theH-transformation of Tukey (1960), theK-transformation of MacGillivray and Cannon (1997) and theJ-transformation of Fischer and Klein (2004).Furthermore, we derive a general condition on the generator function which guarantees that the corresponding transformation is actually tail-increasing. In this case the exponent of the power kurtosis transformation can be interpreted as a kurtosis parameter. We also prove that the transformed distributions can be ordered with respect to the partial ordering of van Zwet (1964) for symmetric distributions.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

SINGULAR ELLIPTICAL DISTRIBUTION: DENSITY AND APPLICATIONS

ABSTRACT

The paper present an explicit expression for the density of a n-dimensional random vector with a singular Elliptical distribution. Based on this, the densities of the generalized Chi-squared and generalized t distributions are derived, examining the Pearson Type VII distribution and Kotz Type distribution (as specific Elliptical distributions). Finally, the results are applied to the study of the distribution of the residuals of an Elliptical linear model and the distribution of the t-statistic, based on a sample from an Elliptical population.