On the generalized t(GT) distribution

McDonald and Newey [J.B. McDonald and W.K. Newey, Partially adaptive estimation of regression models via the generalized t distribution, Econ. Theor. 4 (1988), pp. 428–457.] introduced the generalized t(GT) distribution. In this paper, several explicit formulas for its cumulative distribution function (cdf) are derived. These formulas will be useful for future developments in the theory and applications of the distribution. One such situation is explained and an application is provided to rainfall data from Orlando, Florida.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

Linear estimation for the extended exponential power distribution

Tiao and Lund [The use of OLUMV estimators in inference robustness studies of the location parameter of a class of symmetric distributions. J Amer Statist Assoc. 1970;65(329):370–386] tabulated the coefficients of the best linear unbiased estimators (BLUEs) of location and scale for a particular family of symmetric distributions. This family was a reparameterization of the extended exponential power distribution (EEPD) with the shape parameter restricted to be greater than or equal to one. In this work, we consider the BLU estimation of the location and scale parameters of the EEPD when the shape parameter is one-third and one-half. We obtain closed-form expressions for the single and product moments of the order statistics when the shape parameter is in general in the form of a reciprocal of an integer. These expressions are then used to determine the BLUEs and the corresponding variances for complete samples of size 20 and less. We consider some other linear estimators of the location and scale parameters and then compare them with the BLUEs. Finally, we present a numerical example to illustrate the developed results.     

Shrinkage estimation of scale in exponential distribution from censored samples

In this paper some shrunken and pretest shrunken estimators are suggested for the scale parameter of an exponential distribution, when observations become available from life test experiments. These estimators are shown to be more efficient than the usual estimator when a guessed value is nearer to the true value.     

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θ^r (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.     

An EM algorithm for the estimation of parameters of bivariate generalized exponential distribution under random left censoring

In this paper, we consider the four-parameter bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution, J. Multivariate Anal. 100 (2009), pp. 581–593] and propose an expectation–maximization algorithm to find the maximum-likelihood estimators of the four parameters under random left censoring. A numerical experiment is carried out to discuss the properties of the estimators obtained iteratively.     

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.     

Linear estimation of location and scale parameters in the generalized gamma distribution

The expressions for moments of order statistics from the generalized gamma distribution are derived. Coefficients to get the BLUEs of location and scale parameters in the generalized gamma distribution are computed. Some simple alternative linear unbiased estimates of location and scale parameters are also proposed and their relative efficiencies compared to the BLUEs are studied.     

Robust variable selection in finite mixture of regression models using the t distribution

Abstract

Variable selection in finite mixture of regression (FMR) models is frequently used in statistical modeling. The majority of applications of variable selection in FMR models use a normal distribution for regression error. Such assumptions are unsuitable for a set of data containing a group or groups of observations with heavy tails and outliers. In this paper, we introduce a robust variable selection procedure for FMR models using the t distribution. With appropriate selection of the tuning parameters, the consistency and the oracle property of the regularized estimators are established. To estimate the parameters of the model, we develop an EM algorithm for numerical computations and a method for selecting tuning parameters adaptively. The parameter estimation performance of the proposed model is evaluated through simulation studies. The application of the proposed model is illustrated by analyzing a real data set.     

Some moment properties and limit theorems of the reversed generalized logistic distribution with applications

In this paper we discuss an extended form of the logistic distribution and refer to it as the reversed generalized logistic distribution. We study some moment properties, and derive exact and explicit formulas for the mean, median, mode, variance, coefficients of skewness and kurtosis, and percentage points of this distribution. In addition, we study its limiting distributions as the shape parameter tends to zero or infinity. We also discuss some possible applications in bioassays through logistic regression approach.     

The average likelihood and a fiducial approximation: one parameter members of the generalized gamma distributions

The average likelihood, defined as the integral of the like-lihood function over the parameter space, has been used as a criterion for model selection The form of the average likelihood considered uses a uniform prior. An approximation is presented based on fiducial distributions. The sampling distributions of the average likelihood and its fiducial approximation are derived for cases of sampling from one parameter members of the general-ized gamma distributions.     

Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample

In this note explicit expressions are given for the maximum likelihood estimators of the parameters of the two-parameter exponential distribution, when a doubly censored sample is available.     

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

E-Bayesian estimation and its E-posterior risk of the exponential distribution parameter based on complete and type I censored samples

Abstract

This article studies E-Bayesian estimation and its E-posterior risk, for failure rate derived from exponential distribution, in the case of the two hyper parameters. In order to measure the estimated risk, the definition of E-posterior risk (expected posterior risk) is proposed based on the definition of E-Bayesian estimation. Moreover, under the different prior distributions of hyper parameters, the formulas of E-Bayesian estimation and formulas of E-posterior risk are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the squared error loss function. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analyzed for illustrative purposes, results are compared on the basis of E-posterior risk.     

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACT

Pareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012 Alzaatreh, A., Famoye, F., Lee, C. (2012). Gamma-Pareto distribution and its applications. J. Modern Appl. Stat. Methods. 11:78–94.[Crossref] , [Google Scholar]), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD^(IV) distribution. Various properties of the GPD^(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD^(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD^(IV) distribution.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

A Bayesian predictive approach to determining the number of components in a mixture distribution

This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.     

Mean and cell-mean imputation resulting in the use of a semicontinuous distribution and a mixture of semicontinuous distributions

Zero-inflated models are commonly used for modeling count and continuous data with extra zeros. Inflations at one point or two points apart from zero for modeling continuous data have been discussed less than that of zero inflation. In this article, inflation at an arbitrary point α as a semicontinuous distribution is presented and the mean imputation for a continuous response is discussed as a cause of having semicontinuous data. Also, inflation at two points and generally at k arbitrary points and their relation to cell-mean imputation in the mixture of continuous distributions are studied. To analyze the imputed data, a mixture of semicontinuous distributions is used. The effects of covariates on the dependent variable in a mixture of k semicontinuous distributions with inflation at k points are also investigated. In order to find the parameter estimates, the method of expectation–maximization (EM) algorithm is used. In a real data of Iranian Households Income and Expenditure Survey (IHIES), it is shown how to obtain a proper estimate of the population variance when continuous missing at random responses are mean imputed.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.