On conditional correlations defined by a class of nonlinearly truncated distributions and their applications

This article derives and studies several types of conditional correlations. The correlations are obtained by a class of two-piece scale mixture skew-normal distributions. The class is obtained by applying a set of nonlinear constraints to the bivariate scale mixture of normal distributions. The correlations of the class are invariant with respect to the choice of the scale mixing function, however, they are dependent upon the type of the nonlinear truncation. Moreover, their respective upper and lower limits are no longer 1.00 and?1.00. They are useful for the truncated data analysis, the multivariate interdependence methods (such as the principal component analysis and the factor analysis), and the random truncation modelling. Some distributional properties and the Bayesian computation of the correlations are considered when developing necessary theories and providing illustrative examples, respectively. Two applications are also given to demonstrate the usefulness of the conditional correlations in a multivariate analysis.     

Estimation of parameters of a right truncated exponential distribution

The maximum likelihood, moment and mixture of the estimators are for samples from the right truncated exponential distribution. The estimators are compared empirically when all the parameters are unknown; their bias and mean square error are investigated with the help of numerical technique. We have shown that these estimators are asymptotically unbiased. At the end, we conclude that mixture estimators are better than the maximum likelihood and moment estimators.     

Some characterizations of distributions by truncated moments

In this paper we extend GUPTA'S (1975) resLilt and show that the constant value of a truncated moment characterizes the exponential distribution. Similar results are prov¬ed in the discrete case and it is shown that two consecutive factorial moments are enough to determine the distribution. However, under mild conditions, the constancy of one fac¬torial moment is enough to guarantee that the distribution is geometric. Considering the truncation on the right, a general method of obtaining the distribution, whenever £(h(X) | Xj) is known, is fxmbitud in tnr continuous and in the discrete case. Several of the known characterization theorems toilow trivially trorn our results     

Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling

In this article, we propose mixtures of skew Laplace normal (SLN) distributions to model both skewness and heavy-tailedness in the neous data set as an alternative to mixtures of skew Student-t-normal (STN) distributions. We give the expectation–maximization (EM) algorithm to obtain the maximum likelihood (ML) estimators for the parameters of interest. We also analyze the mixture regression model based on the SLN distribution and provide the ML estimators of the parameters using the EM algorithm. The performance of the proposed mixture model is illustrated by a simulation study and two real data examples.     

Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

Skewness and mixtures of normal distributions

The effect of skewness on hypothesis tests for the existence of a mixture of univariate and bivariate normal distributions is examined through a Monte Carlo study. A likelihood ratio test based on results of the simultaneous estimation of skewness parameters, derived from power transformations, with mixture parameters is proposed. This procedure detects the difference between inherent distributional skewness and the apparent skewness which is a manifestation of the mixture of several distributions. The properties of this test are explored through a simulation study.     

Calibration model with skew scale mixtures of normal distributions

This work presents a new linear calibration model with replication by assuming that the error of the model follows a skew scale mixture of the normal distributions family, which is a class of asymmetric thick-tailed distributions that includes the skew normal distribution and symmetric distributions. In the literature, most calibration models assume that the errors are normally distributed. However, the normal distribution is not suitable when there are atypical observations and asymmetry. The estimation of the calibration model parameters are done numerically by the EM algorithm. A simulation study is carried out to verify the properties of the maximum likelihood estimators. This new approach is applied to a real dataset from a chemical analysis.     

Heteroscedastic and heavy-tailed regression with mixtures of skew Laplace normal distributions

Joint modelling skewness and heterogeneity is challenging in data analysis, particularly in regression analysis which allows a random probability distribution to change flexibly with covariates. This paper, based on a skew Laplace normal (SLN) mixture of location, scale, and skewness, introduces a new regression model which provides a flexible modelling of location, scale and skewness parameters simultaneously. The maximum likelihood (ML) estimators of all parameters of the proposed model via the expectation-maximization (EM) algorithm as well as their asymptotic properties are derived. Numerical analyses via a simulation study and a real data example are used to illustrate the performance of the proposed model.     

Hermite polynomials and truncated trivariate normal distributions

The moments of a trivariate and in general of a multivariate normal distribution, which is truncated with respect to a single variable, are obtained by using properties of Hermite polynomials. An expression for the truncated correlation coefficient is derived in terms of the true population correlation coefficient and the truncation point. The values of this truncated correlation coefficient are tabulated for given values of the true correlation coefficient and a few selected values of the truncation point. A listing of the computer program for this purpose is also given.     

On finite mixtures of geometric and negative binomial distributions

Finite mixtures of distributions have been getting increasing use in the applied literature. In the continuous case, linear combinations of exponentials and gammas have been shown to be well suited for modeling purposes. In the discrete case, the focus has primarily been on continuous mixing, usually of Poisson distributions and typically using gammas to describe the random parameter, But many of these applications are forced, especially when a continuous mixing distribution is used. Instead, it is often prefe-rable to try finite mixtures of geometries or negative binomials, since these are the fundamental building blocks of all discrete random variables. To date, a major stumbling block to their use has been the lack of easy routines for estimating the parameters of such models. This problem has now been alleviated by the adaptation to the discrete case of numerical procedures recently developed for exponential, Weibull, and gamma mixtures. The new methods have been applied to four previously studied data sets, and significant improvements reported in goodness-of-fit, with resultant implications for each affected study.     

On symmetry of finite mixtures of normal distributions

In this note we derive a necessary and sufficient condition for a distribution obtained by taking a finite mixture of multivariate normal distributions to be symmetric about zero. The result derived also holds for mixtures of symmetric stable distributions, including the Cauchy distribution.     

The conditional variance for gamma mixtures of normal distributions

This paper is concerned with obtaining an expression for the conditional variance-covariance matrix when the random vector is gamma scaled of a multivariate normal distribution. We show that the conditional variance is not degenerate as in the multivariate normal distribution, but depends upon a positive function for which various asymptotic properties are derived. A discussion section is included commenting on the usefulness of these results     

Median regression model with left truncated and right censored data

We study the problem of fitting a heteroscedastic median regression model from left-truncated and right-censored data. It is demonstrated that the adapted Efron's self-consistency equation of McKeague et al. (2001) can be extended to analyze left-truncated and right-censored data. We evaluate the finite sample performance of the proposed estimators through simulation studies.     

Estimation of the parameters of a truncated gamma distribution

This paper deals with the estimation of the parameters of a truncated gamma distribution over (0,τ), where τ is assumed to be a real number. We obtain a necessary and sufficient condition for the existence of the maximum likelihood estimator(MLE). The probability of nonexistence of MLE is observed to be positive. A simulation study indicates that the modified maximum likelihood estimator and the mixed estimator, which exist with probability one,are to be preferred over MLE. The bias, the mean square error, and the probability of nearness form a basis of our simulation study.     

On characterizing mixtures of some life distributions

Received: October 15, 1998; revised version: January 10, 2000     

Compounded inverse Weibull distributions: Properties,inference and applications

ABSTRACT

In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Fitting insurance and economic data with outliers: a flexible approach based on finite mixtures of contaminated gamma distributions

Insurance and economic data are frequently characterized by positivity, skewness, leptokurtosis, and multi-modality; although many parametric models have been used in the literature, often these peculiarities call for more flexible approaches. Here, we propose a finite mixture of contaminated gamma distributions that provides a better characterization of data. It is placed in between parametric and non-parametric density estimation and strikes a balance between these alternatives, as a large class of densities can be implemented. We adopt a maximum likelihood approach to estimate the model parameters, providing the likelihood and the expected-maximization algorithm implemented to estimate all unknown parameters. We apply our approach to an artificial dataset and to two well-known datasets as the workers compensation data and the healthcare expenditure data taken from the medical expenditure panel survey. The Value-at-Risk is evaluated and comparisons with other benchmark models are provided.