Estimating the proportions of two populations in a mixture using linear maps

We address the problem of estimating the proportions of two statistical populations in a given mixture on the basis of an unlabeled sample of n–dimensional observations on the mixture. Assuming that the expected values of observations on the two populations are known, we show that almost any linear map from Rⁿ to R¹ yields an unbiased consistent estimate of the proportion of one population in a very easy way. We then find that linear map for which the resulting proportion estimate has minimum variance among all estimates so obtained. After deriving a simple expression for the minimum-variance estimate, we discuss practical aspects of obtaining this and related estimates.     

On beta regression residuals

We propose two new residuals for the class of beta regression models, and numerically evaluate their behaviour relative to the residuals proposed by Ferrari and Cribari-Neto. Monte Carlo simulation results and empirical applications using real and simulated data are provided. The results favour one of the residuals we propose.     

Bayesian analysis of the difference of two proportions

In this article we give the expression of the prior distribution of p₁-P₂, where P₁ and P₂ and independent proportions with a beta prior each. The expression derived for the posterior distribution of P₁-P₂ then shows the closure of the beta-difference family for independent dual Bernoulli samples. Other bayesian results are also presented.     

The beta Weibull-geometric distribution

A new five-parameter distribution called the beta Weibull-geometric (BWG) distribution is proposed. The new distribution is generated from the logit of a beta random variable and includes the Weibull-geometric distribution of Barreto-Souza et al. [The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], beta Weibull (BW), beta exponential, exponentiated Weibull, and some other lifetime distributions as special cases. A comprehensive mathematical treatment of this distribution is provided. The density function can be expressed as an infinite mixture of BW densities and then we derive some mathematical properties of the new distribution from the corresponding properties of the BW distribution. The density function of the order statistics and also estimation of the stress–strength parameter are obtained using two general expressions. To estimate the model parameters, we use the maximum likelihood method and the asymptotic distribution of the estimators is also discussed. The capacity of the new distribution are examined by various tools, using two real data sets.     

The model identification of beta distribution based on quantiles

This paper addresses the modelling of a beta distribution based on quantile pairs. The elicited beta distribution is proved to exist uniquely for any given two-quantile pairs. By combining the results of this study with the research documented in the literature, we concluded that for any given two-quantile pairs, there is one and only one beta distribution fits the two specified quantile pairs. The numerical results that verify this finding were obtained by comprehensively examining 1300 two-quantile pairs. The first four moments of the fitted beta distribution were also derived numerically using these two-quantile pairs. The variations in the first four moments, which reflect the perception of uncertainty, were plotted and were explored.     

The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.     

Libby and Novick's generalized beta exponential distribution

A new family of skewed distributions is presented. Some properties and estimation procedures for Libby and Novick's generalized beta exponential distribution, a particular member of the family, are derived. Real applications using two original data sets are described to show superior performance versus at least six known models.     

General results for the beta Weibull distribution

In this paper, we study some mathematical properties of the beta Weibull (BW) distribution, which is a quite flexible model in analysing positive data. It contains the Weibull, exponentiated exponential, exponentiated Weibull and beta exponential distributions as special sub-models. We demonstrate that the BW density can be expressed as a mixture of Weibull densities. We provide their moments and two closed-form expressions for their moment-generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and two entropies. The density of the BW-order statistics is a mixture of Weibull densities and two closed-form expressions are derived for their moments. The estimation of the parameters is approached by two methods: moments and maximum likelihood. We compare the performances of the estimates obtained from both the methods by simulation. The expected information matrix is derived. For the first time, we introduce a log-BW regression model to analyse censored data. The usefulness of the BW distribution is illustrated in the analysis of three real data sets.     

Improved point estimation for the Kumaraswamy distribution

We develop nearly unbiased estimators for the Kumaraswamy distribution proposed by Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 46 (1980), pp. 79–88], which has considerable attention in hydrology and related areas. We derive modified maximum-likelihood estimators that are bias-free to second order. As an alternative to the analytically bias-corrected estimators discussed, we consider a bias correction mechanism based on the parametric bootstrap. We conduct Monte Carlo simulations in order to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates.     

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..     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

On the robustness of the power function of the one-sample test for the negative exponential distribution

The robustness of the power function of the standard one-sample parametric test for the mean of the negative exponential distribution is examined. The main form of departure from the exponential assumption is a mixture of negative exponential components although an alternative Gamma distribution is also examined. It is found that the test is sensitive to these departures although the effect of mixtures with short tails is less dramatic than those with long tails.     

Analysis of over-dispersed count data with extra zeros using the Poisson log-skew-normal distribution

ABSTRACT

Mixed Poisson distributions are widely used in various applications of count data mainly when extra variation is present. This paper introduces an extension in terms of a mixed strategy to jointly deal with extra-Poisson variation and zero-inflated counts. In particular, we propose the Poisson log-skew-normal distribution which utilizes the log-skew-normal as a mixing prior and present its main properties. This is directly done through additional hierarchy level to the lognormal prior and includes the Poisson lognormal distribution as its special case. Two numerical methods are developed for the evaluation of associated likelihoods based on the Gauss–Hermite quadrature and the Lambert's W function. By conducting simulation studies, we show that the proposed distribution performs better than several commonly used distributions that allow for over-dispersion or zero inflation. The usefulness of the proposed distribution in empirical work is highlighted by the analysis of a real data set taken from health economics contexts.     

Exact formula for the cumulative distribution function of the quasi-range from the logistic distribution

In this paper we obtain an exact formula for the cumulative distribution function of the rth quasi-range from the logistic distribution. For the special case r = 0, the result agrees for the rangegiven by Gupta and Shah (1965).     

Bivariate beta regression models: joint modeling of the mean,dispersion and association parameters

In this paper a bivariate beta regression model with joint modeling of the mean and dispersion parameters is proposed, defining the bivariate beta distribution from Farlie–Gumbel–Morgenstern (FGM) copulas. This model, that can be generalized using other copulas, is a good alternative to analyze non-independent pairs of proportions and can be fitted applying standard Markov chain Monte Carlo methods. Results of two applications of the proposed model in the analysis of structural and real data set are included.     

Modified maximum likelihood estimator for the mixing distribution in a finite mixture

Using the likelihood equations, a method based on the generalized inverse of matrices is proposed to derive closed-form estimates for the mixing proportions in a finite mixture, when the component distributions are known.     

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.     

A new four-parameter discrete distribution with bathtub and unimodal failure rate

In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined.     

Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity

This paper examines the nonnuU distribution of the ellipticity statistic used for testing sphericity in a bivariate normal population. The distribution function is expressed as mixtures of F distribution functions, where the mixing probabilities are binomial and mgatiire binomial. A simple approximation to the distribution is suggested and examined     

Two-stage hierarchical modeling for analysis of subpopulations in conditional distributions

In this work, we develop modeling and estimation approach for the analysis of cross-sectional clustered data with multimodal conditional distributions where the main interest is in analysis of subpopulations. It is proposed to model such data in a hierarchical model with conditional distributions viewed as finite mixtures of normal components. With a large number of observations in the lowest level clusters, a two-stage estimation approach is used. In the first stage, the normal mixture parameters in each lowest level cluster are estimated using robust methods. Robust alternatives to the maximum likelihood estimation are used to provide stable results even for data with conditional distributions such that their components may not quite meet normality assumptions. Then the lowest level cluster-specific means and standard deviations are modeled in a mixed effects model in the second stage. A small simulation study was conducted to compare performance of finite normal mixture population parameter estimates based on robust and maximum likelihood estimation in stage 1. The proposed modeling approach is illustrated through the analysis of mice tendon fibril diameters data. Analyses results address genotype differences between corresponding components in the mixtures and demonstrate advantages of robust estimation in stage 1.