The Topp–Leone odd log-logistic family of distributions

We introduce a new class of continuous distributions named the Topp–Leone odd log-logistic family, which extends the one-parameter distribution pioneered by Topp and Leone [A family of J-shaped frequency functions. J Amer Statist Assoc. 1955;50:209–219]. We study some of its mathematical properties and describe two special cases. Further, we propose a regression model based on the new Topp–Leone odd log-logistic Weibull distribution. The usefulness and flexibility of the proposed family are illustrated by means of three real data sets.     

Two-sided generalized Topp and Leone (TS-GTL) distributions

Over 50 years ago, in a 1955 issue of JASA, a paper on a bounded continuous distribution by Topp and Leone [C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50(269) (1955), pp. 209–219] appeared (the subject was dormant for over 40 years but recently the family was resurrected). Here, we shall investigate the so-called Two-Sided Generalized Topp and Leone (TS-GTL) distributions. This family of distributions is constructed by extending the Generalized Two-Sided Power (GTSP) family to a new two-sided framework of distributions, where the first (second) branch arises from the distribution of the largest (smallest) order statistic. The TS-GTL distribution is generated from this framework by sampling from a slope (reflected slope) distribution for the first (second) branch. The resulting five-parameter TS-GTL family of distributions turns out to be flexible, encompassing the uniform, triangular, GTSP and two-sided slope distributions into a single family. In addition, the probability density functions may have bimodal shapes or admitting shapes with a jump discontinuity at the ‘threshold’ parameter. We will discuss some properties of the TS-GTL family and describe a maximum likelihood estimation (MLE) procedure. A numerical example of the MLE procedure is provided by means of a bimodal Galaxy M87 data set concerning V–I color indices of 80 globular clusters. A comparison with a Gaussian mixture fit is presented.     

An absolutely continuous bivariate Topp–Leone distribution: A useful model on a bounded domain

We introduce an absolutely continuous bivariate generalization of the Topp–Leone distribution, which is a special member of the proportional reversed hazard family using a one-parameter bivariate exchangeable distribution. We show that a copula approach could also be used in defining the bivariate Topp–Leone distribution. The marginal distributions of the new bivariate distribution have also Topp–Leone distributions. We study its distributional and dependence properties. We estimate the parameters by maximum-likelihood procedure, perform a simulation study on the estimators, and apply them to a real data set. Furthermore, we give a way of generating bivariate distributions using the proposed distribution.     

Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

On selection of a suitable prior for the posterior analysis of censored mixture of Topp Leone distribution

The paper aims to select a suitable prior for the Bayesian analysis of the two-component mixture of the Topp Leone model under doubly censored samples and left censored samples for the first component and right censored samples for the second component. The posterior analysis has been carried out under the assumption of a class of informative and noninformative priors using a couple of loss functions. The comparison among the different Bayes estimators has been made under a simulation study and a real life example. The model comparison criterion has been used to select a suitable prior for the Bayesian analysis. The hazard rate of the Topp Leone mixture model has been compared for a range of parametric values.     

A monte carlo of plotting positions

A Monte Carlo study was made of the effects of using simple linear regression, on the appropriate probability paper, to estimate parameters, quantiles and cumulative probability for several distributions. These distributions were the Normal, Weibull (shape parameters 1, 2, and 4) and the Type I largest extreme-value distributions. The specific objective was to observe differences arising from choice of plotting positions. Plotting positions used were i/(n+l), (i?3)/(n+.04), (i?.5)/n, either (i?.375)/(n+.25) or (i?.4)/(n+.2), and either F[E(Y_i)] or F[E(£n Y)]. For each combination of 4 sample sizes (n=10(10)(40)), distribution, and plotting position, regression lines were found for each of N =9999 samples. Each regression line was used to estimate: (1) quantiles of 9 specific probabilities, (2) probabilities of 9 specific quantiles, and (3) return periods corresponding to 9 specific quantiles. Compa[rgrave]ison of the mean, variances, mean square error and medians of these estimates and of the regression coefficients confirm some results of Harter [Commun. Statist. A13(13), 1984] and provide further insight.     

Calculating nonparametric confidence intervals for quantiles using fractional order statistics

In this paper, we provide an easy-to-program algorithm for constructing the preselected 100(1 - alpha)% nonparametric confidence interval for an arbitrary quantile, such as the median or quartile, by approximating the distribution of the linear interpolation estimator of the quantile function Q L ( u ) = (1 - epsilon) X \[ n u ] + epsilon X \[ n u ] + 1 with the distribution of the fractional order statistic Q I ( u ) = Xn u , as defined by Stigler, where n = n + 1 and \[ . ] denotes the floor function. A simulation study verifies the accuracy of the coverage probabilities. An application to the extreme-value problem in flood data analysis in hydrology is illustrated.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

New results on the Ristić–Balakrishnan family of distributions

ABSTRACT

Adding new shape parameters to expand a model into a larger family of distributions to provide significantly skewed and heavy-tails plays a fundamental role in distribution theory. For any continuous baseline G distribution, Risti? and Balakrishnan (2012 Risti?, M.M., Balakrishnan, N. (2012). The gamma exponentiated exponential distribution. J. Stat. Comput. Simul. 82:1191–1206.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the gamma-generated family of distributions with an extra positive shape parameter. They presented some special models of their family but did not study its properties. This paper examines some general mathematical properties of this family which hold for any baseline model. Some distributions are studied and a number of existing results in the literature can be recovered as special cases. We estimate the model parameters by maximum likelihood and illustrate the importance of the family by means of an application to a real data set.     

Admissible minimax estimators for the shape parameter of Topp–Leone distribution

Abstract

The shape parameter of Topp–Leone distribution is estimated in this article from the Bayesian viewpoint under the assumption of known scale parameter. Bayes and empirical Bayes estimates of the unknown parameter are proposed under non informative and suitable conjugate priors. These estimates are derived under the assumption of squared and linear-exponential error loss functions. The risk functions of the proposed estimates are derived in analytical forms. It is shown that the proposed estimates are minimax and admissible. The consistency of the proposed estimates under the squared error loss function is also proved. Numerical examples are provided.     

A comparison of approximations to the distribution of average kendall tau

The average Kendall tau test for the null hypothesis of compLete randomness in n judges' rankings of r objects was introduced by Ehrenberg (1952). The superior efficiency of this test. as measured by approximate Bahadur slope has been established by Alva, Cabilio, and Feigin (1982).

The purpose of this paper is first to provide further justification for the use of this test, and second to compare the accuracy of the asymptotic distribution to other app roxtmations to the null distribution for small values of r and n. In the process we generate tables for this exact dis tribution for r=3 and n=3(1)19, r=4 and n=3(1)9, r=5 and n=3(1)5.     

On the Marshall–Olkin extended distributions

ABSTRACT

A general method of introducing a new parameter to a well-established continuous baseline cumulative function G to obtain more flexible distributions was proposed by Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]). This new family is known as Marshall–Olkin extended G family of distributions. In this article, we characterize this family as mixtures of the distributions of the minimum and maximum of random variables with cumulative function G. We demonstrate that the coefficients of the mixtures are probabilities of random variables with geometric distributions. Additionally, we present new representations for the density and cumulative functions of this class of distributions. Further, we introduce a new three-parameter continuous model for modeling rates and proportions based on the Marshall–Olkin's method. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of a real dataset.     

Moments for Some Kumaraswamy Generalized Distributions

Explicit expansions for the moments of some Kumaraswamy generalized (Kw-G) distributions (Cordeiro and de Castro, 2011 Cordeiro, G.M., de Castro, M. (2011). A new family of generalized distributions. J. Statist. Computat. Simul. 81:883–898.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are derived using special functions. We explore the Kw-normal, Kw-gamma, Kw-beta, Kw-t, and Kw-F distributions. These expressions are given as infinite weighted linear combinations of well-known special functions for which numerical routines are readily available.     

A Characterization of the Normal Family of Distributions Based on the Bias of the Maximum Likelihood Estimator

This paper characterizes the family of Normal distributions within the class of exponential families of distributions, via the structure of the bias of the maximum likelihood estimator Θ _n of the canonical parameter Θ . More specifically, when E _θ ( Θ n ) – Θ = (1/ n ) Q ( Θ ) + o (1/ n ), the equality Q ( Θ ) = 0 proves to be a property of the Normal distribution only. The same conclusion is obtained for the one-dimensional case bt assuming that Q ( Θ ) is a polynomial of Θ .     

Estimation of P(X>Y) with Topp–Leone distribution

We consider the estimation problem of the probability P=P(X>Y) for the standard Topp–Leone distribution. After discussing the maximum likelihood and uniformly minimum variance unbiased estimation procedures for the problem on both complete and left censored samples, we perform a Monte Carlo simulation to compare the estimators based on the mean square error criteria. We also consider the interval estimation of P.     

Nonparametric Mixtures Based on Skew-normal Distributions: An Application to Density Estimation

This article addresses the density estimation problem using nonparametric Bayesian approach. It is considered hierarchical mixture models where the uncertainty about the mixing measure is modeled using the Dirichlet process. The main goal is to build more flexible models for density estimation. Extensions of the normal mixture model via Dirichlet process previously introduced in the literature are twofold. First, Dirichlet mixtures of skew-normal distributions are considered, say, in the first stage of the hierarchical model, the normal distribution is replaced by the skew-normal one. We also assume a skew-normal distribution as the center measure in the Dirichlet mixture of normal distributions. Some important results related to Bayesian inference in the location-scale skew-normal family are introduced. In particular, we obtain the stochastic representations for the full conditional distributions of the location and skewness parameters. The algorithm introduced by MacEachern and Müller in 1998 MacEachern, S.N., Müller, P. (1998). Estimating mixture of Dirichlet Process models. J. Computat. Graph. Statist. 7(2):223–238.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] is used to sample from the posterior distributions. The models are compared considering simulated data sets. Finally, the well-known Old Faithful Geyser data set is analyzed using the proposed models and the Dirichlet mixture of normal distributions. The model based on Dirichlet mixture of skew-normal distributions captured the data bimodality and skewness shown in the empirical distribution.     

Inferences for Mixtures of Distributions for Centrally Censored Data with Partial Identification

In this article, several methods to make inferences about the parameters of a finite mixture of distributions in the context of centrally censored data with partial identification are revised. These methods are an adaptation of the work in Contreras-Cristán, Gutiérrez-Peña, and O'Reilly (2003 Contreras-Cristán , A. , Gutiérrez-Peña , E. , O'Reilly , F. ( 2003 ). Inferences using latent variables for mixtures of distributions for censored data with partial identification . Comm. Stat. Theor. Meth. 32 ( 4 ): 749 – 774 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the case of right censoring. The first method focuses on an asymptotic approximation to a suitably simplified likelihood using some latent quantities; the second method is based on the expectation-maximization (EM) algorithm. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to non-Bayesian methods that deal directly with the full likelihood of the mixture appealing to its asymptotic approximation. The third method, from a Bayesian perspective, uses data augmentation to work with an uncensored sample. This last method is related to a recently proposed Bayesian method in Baker, Mengersen, and Davis (2005 Baker , P. , Mengersen , K. , Davis , G. ( 2005 ). A Bayesian solution to reconstructing centrally censored distributions . J. Agr. Biol. Environ. Stat. 1 : 61 – 84 . [Google Scholar]). Our proposal of the three adapted methods is shown to provide similar inferential answers, thus offering alternative analyses.     

Tests for detecting overdispersion in poisson models

Collings and Margolin(1985) developed a locally most powerful unbiased test for detecting negative binomial departures from a Poisson model, when the variance is a quadratic function of the mean. Kim and Park(1992) developed a locally most powerful unbiased test, when the variance is a linear function of the mean. It is found that a different mean-variance structure of a negative binomial derives a different locally optimal test statistic.

In this paper Collings and Margolin's and Kim and Park's results are unified and extended by developing a test for overdispersion in Poisson model against Katz family of distributions, Our setup has two extensions: First, Katz family of distributions is employed as an extension of the negative binomial distribution. Second, the mean-variance structure of the mixed Poisson model is given by σ² = μ+cμ^r for arbitrary but fixed r. We derive a local score test for testing H₀ : c = 0. Superiority of a new test is proved by the asymtotic relative efficiency as well as the simulation study.     

A discrete distribution including the Poisson

Abstract

This article presents a new generalization of the Poisson distribution, with the parameters α > 0 and θ > 0, using the Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652.[Crossref], [Web of Science ®] , [Google Scholar]) scheme and adding a parameter to the classical Poisson distribution. The particular case of α = 1 gives the Poisson distribution. The new distribution is unimodal and has a failure rate that monotonically increases or decreases depending on the value of the parameter α. After reviewing some of the properties of this distribution, we investigated the question of parameter estimation. Expected frequencies were calculated for two data sets, one with an index of dispersion larger than one and the other with an index of dispersion smaller than one. In both cases the distribution provided a very satisfactory fit.     

Reliability characteristics of Farlie–Gumbel–Morgenstern family of bivariate distributions

Abstract

In this paper, we study the Farlie–Gumbel–Morgenstern family of bivariate distributions from a reliability point of view. The properties of this family of distributions and the association between the two variables are investigated by studying the local dependence function and the association measure defined by Clayton (1978 Clayton, D.G. (1978). A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151.[Crossref], [Web of Science ®] , [Google Scholar]). We also study the effect of the association parameter on the hazard components, the failure rate of the series system, and the regression mean residual life of a parallel system. Stochastic comparisons with respect to the association parameter are also studied. Some examples are provided to illustrate the results.