Gholamhossein Hamedani

Information Measures for Pareto Distributions and Order Statistics

This paper consists of three sections. The first section gives an overview of the basic information functions, their interpretations, and dynamic information measures that have been recently developed for lifetime distributions. The second section summarizes the information features of univariate Pareto distributions, tabulates transformations of a Pareto random variable under which information measures of numerous distributions can be obtained, and gives a few characterizations of the generalized Pareto distribution. The final section summarizes information measures for order statistics and tabulates the expressions for Shannon entropies of order statistics for numerous distributions.

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.

The Gumbel-Lomax Distribution: Properties and Applications

We introduce a new four-parameter model called the Gumbel-Lomax distribution arising from the GumbelX generator recently proposed by Al-Aqtash (2013). Its density function can be right-skewed and reversed-J shaped, and can have decreasing and upside-down bathtub shaped hazard rate. Various structural properties of the new distribution are obtained including explicit expressions for the quantile function, ordinary and incomplete moments, Lorenz and Bonferroni curves, mean residual lifetime, mean waiting time, probability weighted moments, generating function and Shannon entropy. We also provide the density function for the order statistics. Some characterizations of the new distribution based on the conditional expectations of certain functions of the random variable are also proposed. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The flexibility of the new model is illustrated by means of two real lifetime data sets.

A New Weibull-G Family of Distributions

Statistical analysis of lifetime data is an important topic in reliability engineering, biomedical and social sciences and others. We introduce a new generator based on the Weibull random variable called the new Weibull-G family. We study some of its mathematical properties. Its density function can be symmetrical, left-skewed, right-skewed, bathtub and reversed-J shaped, and has increasing, decreasing, bathtub, upside-down bathtub, J, reversed-J and S shaped hazard rates. Some special models are presented. We obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Renyi entropy, order statistics and reliability. Three useful characterizations based on truncated moments are also proposed for the new family. The method of maximum likelihood is used to estimate the model parameters. We illustrate the importance of the family by means of two applications to real data sets.

The Zografos-Balakrishnan Log-Logistic Distribution: Properties and Applications

The log-logistic distribution (also known as the Fisk distribution in economics) is widely used in survival analysis when the failure rate function presents a unimodal shape. In this paper, we introduce the ZografosBalakrishnan log-logistic distribution, which contains the log-logistic distribution as a special model and has the four common shapes of the hazard hate function. We present some properties of the new distribution and estimate the model parameters by maximum likelihood. An application to a real data set shows that the new distribution can provide a better fit than other classical lifetime models such as the exponentiated Weibull distribution.

Bayes estimation of the binomial parameter n

Bayes estimation of the binomial parameter n based on a general prior distribution for n is studied. As special cases improper prior and Poisson prior (which is a natural choice) are considered, and formulae for the marginal and posterior distributions are obtained. It is shown that the assumption of improper priors in both p and n leads to implausible results.

Oscillation criteria for certain second order differential equations

Abstract Two oscillation theorems for second order equations x ″( t ) + p ( t ) f ( x ( t ), x ( h ( t ))) g ( x ′( t )) = 0 are established.

Characterizations of Continuous Distributions Based on Conditional Expectation of Generalized Order Statistics

Characterizations of probability distributions by different regression conditions on generalized order statistics have attracted the attention of many researchers, in particular Bieniek and Szynal (2003), Cramer et al. (2004), and Bieniek (2007, 2009). We present here certain characterizations of univariate continuous distributions based on the conditional expectation of adjacent generalized order statistics.

Non-linear regression with multidimensional indices

We consider a non-linear regression model when the index variable is multidimensional. Sufficient conditions on the non-linear function are given under which the least-squares estimators are strongly consistent and asymptotically normally distributed. These sufficient conditions are satisfied by harmonic type functions, which are also of interest in the one dimensional index case where Wus (Asymptotic theory of non-linear least-squares estimation, Ann. Statist. 9 (1981) 501-513) and Jennrichs (Asymptotic properties of non-linear least-squares estimators, Ann. Math. Statist. 40 (1969) 633-643) sufficient conditions are not applicable.

Empiric bayes estimation of binomial probability

An empirical Bayes estimator of a binomial parameter, based on orthogonal polynomials on (0,1), is introduced. The resulting estimator of the prior density is asymptotically optimal. The method allows one to combine Bayes and empiric Bayes methods with smoothing in a natural way.