Characterization of a mixture of gamma distributions via conditional finite moments

A necessary and sufficient condition that a continuous, positive random variable follow a gamma distribution is given in terms of any one of its conditional finite moments and an expression involving its failure rate. The results are then used to develop a characterization for a mixture of two gamma distributions. The general results about characterization of a mixture of gamma distributions yield several special cases that have appeared separately in recent literature, including characterization of a single exponential distribution, characterization of a single gamma distribution (in terms of either first or second moments) and a sufficient condition for a mixture of two exponential distributions (in terms of first moments). The condition in this last result is shown to be necessary also. Numerous other cases are possible, using different choices for distribution parameters along with a selection of the mixing parameter, for either individual or mixtures of distributions. Various characterizations can be expressed using higher order moments, too.     

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.     

Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values

Let X _{U (1)} < X _{U (2)} < … < X _{U ( n )} < … be the sequence of the upper record values from a population with common distribution function F. In this paper, we first give a theorem to characterize the generalized mixtures of geometric distribution by the relation between E[(X _{U ( n +1)}–X _{U ( n )})²|X _{U ( n )} = x] and the function of the failure rate of the distribution, for any positive integer n. Secondly, we also use the same relation to characterize the generalized mixtures of exponential distribution. The characterizing relations were motivated by the work of Balakrishnan and Balasubramanian (1995). Received: March 31, 1999; revised version: November 22, 1999     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

Consistent estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data

In this paper, we propose a method of estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs well compared with another prominent method of estimation in terms of bias and root mean-squared error in small-sample situations. Finally, two real data sets are used for illustrating the proposed method.     

Extremal analysis of short series with outliers: sea-levels and athletics records

The analysis of extreme values is often required from short series which are biasedly sampled or contain outliers. Data for sea-levels at two UK east coast sites and data on athletics records for women's 3000 m track races are shown to exhibit such characteristics. Univariate extreme value methods provide a poor quantification of the extreme values for these data. By using bivariate extreme value methods we analyse jointly these data with related observations, from neighbouring coastal sites and 1500 m races respectively. We show that using bivariate methods provides substantial benefits, both in these applications and more generally with the amount of information gained being determined by the degree of dependence, the lengths and the amount of overlap of the two series, the homogeneity of the marginal characteristics of the variables and the presence and type of the outlier.     

Inverse Lindley power series distributions: a new compounding family and regression model with censored data

This paper introduces a new class of distributions by compounding the inverse Lindley distribution and power series distributions which is called compound inverse Lindley power series (CILPS) distributions. An important feature of this distribution is that the lifetime of the component associated with a particular risk is not observable, rather only the minimum lifetime value among all risks is observable. Further, these distributions exhibit an unimodal failure rate. Various properties of the distribution are derived. Besides, two special models of the new family are investigated. The model parameters of the two sub-models of the new family are obtained by the methods of maximum likelihood, least square, weighted least square and maximum product of spacing and compared them using the Monte Carlo simulation study. Besides, the log compound inverse Lindley regression model for censored data is proposed. Three real data sets are analyzed to illustrate the flexibility and importance of the proposed models.     

Confidence intervals based on L-moments for quantiles of the GP and GEV distributions with application to market-opening asset prices data

In a ground-breaking paper published in 1990 by the Journal of the Royal Statistical Society, J.R.M. Hosking defined the L-moment of a random variable as an expectation of certain linear combinations of order statistics. L-moments are an alternative to conventional moments and recently they have been used often in inferential statistics. L-moments have several advantages over the conventional moments, including robustness to the the presence of outliers, which may lead to more accurate estimates in some cases as the characteristics of distributions. In this contribution, asymptotic theory and L-moments are used to derive confidence intervals of the population parameters and quantiles of the three-parametric generalized Pareto and extreme-value distributions. Computer simulations are performed to determine the performance of confidence intervals for the population quantiles based on L-moments and to compare them to those obtained by traditional estimation techniques. The results obtained show that they perform well in comparison to the moments and maximum likelihood methods when the interest is in higher quantiles, or even best. L-moments are especially recommended when the tail of the distribution is rather heavier and the sample size is small. The derived intervals are applied to real economic data, and specifically to market-opening asset prices.     

Stochastic volatility in mean models with scale mixtures of normal distributions and correlated errors: A Bayesian approach

A stochastic volatility in mean model with correlated errors using the symmetrical class of scale mixtures of normal distributions is introduced in this article. The scale mixture of normal distributions is an attractive class of symmetric distributions that includes the normal, Student-t, slash and contaminated normal distributions as special cases, providing a robust alternative to estimation in stochastic volatility in mean models in the absence of normality. Using a Bayesian paradigm, an efficient method based on Markov chain Monte Carlo (MCMC) is developed for parameter estimation. The methods developed are applied to analyze daily stock return data from the São Paulo Stock, Mercantile & Futures Exchange index (IBOVESPA). The Bayesian predictive information criteria (BPIC) and the logarithm of the marginal likelihood are used as model selection criteria. The results reveal that the stochastic volatility in mean model with correlated errors and slash distribution provides a significant improvement in model fit for the IBOVESPA data over the usual normal model.     

Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data

In this paper, we establish the existence and uniqueness of the maximum-likelihood estimates of the parameters of a general class of inverse exponentiated distributions based on complete as well as progressively Type-I and Type-II censored data.     

Model selection in finite mixture of regression models: a Bayesian approach with innovative weighted g priors and reversible jump Markov chain Monte Carlo implementation

Finite mixture of regression (FMR) models are aimed at characterizing subpopulation heterogeneity stemming from different sets of covariates that impact different groups in a population. We address the contemporary problem of simultaneously conducting covariate selection and determining the number of mixture components from a Bayesian perspective that can incorporate prior information. We propose a Gibbs sampling algorithm with reversible jump Markov chain Monte Carlo implementation to accomplish concurrent covariate selection and mixture component determination in FMR models. Our Bayesian approach contains innovative features compared to previously developed reversible jump algorithms. In addition, we introduce component-adaptive weighted g priors for regression coefficients, and illustrate their improved performance in covariate selection. Numerical studies show that the Gibbs sampler with reversible jump implementation performs well, and that the proposed weighted priors can be superior to non-adaptive unweighted priors.     

Reliability of a soccer player based on the bivariate Rayleigh distribution with right censored and ignorable missing data

In this paper, we study the performance of a soccer player based on analysing an incomplete data set. To achieve this aim, we fit the bivariate Rayleigh distribution to the soccer dataset by the maximum likelihood method. In this way, the missing data and right censoring problems, that usually happen in such studies, are considered. Our aim is to inference about the performance of a soccer player by considering the stress and strength components. The first goal of the player of interest in a match is assumed as the stress component and the second goal of the match is assumed as the strength component. We propose some methods to overcome incomplete data problem and we use these methods to inference about the performance of a soccer player.     

Joint analysis of occurrence and time to stability after entrance into the Italian labour market: an approach based on a Bayesian cure model with structured stochastic search variable selection

Precarious employment is a serious social problem, especially in those countries, such as Italy, where there are limited benefits from social security. We investigate this phenomenon by analysing the initial part of the career of employees starting with unstable contracts for a panel of Italian workers. Our aim is to estimate the probability of getting a stable job and to detect factors influencing both this probability and the duration of precariousness. To answer these questions, we use an ad hoc mixture cure rate model in a Bayesian framework.