Moments of scale mixtures of skew-normal distributions and their quadratic forms

We obtain the first four moments of scale mixtures of skew-normal distributions allowing for scale parameters. The first two moments of their quadratic forms are obtained using those moments. Previous studies derived the moments, but all relevant results do not allow for scale parameters. In particular, it is shown that the mean squared error becomes an unbiased estimator of σ² with skewed and heavy-tailed errors. Two measures of multivariate skewness are calculated.     

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.     

Comparison of improved regression estimators with and without moments

For estimating the coefficients in a linear regression model, the double k–class estimators are considered and the small disturbance asymptotic approximation for their density function is obtained. Then employing the criterion of concentration probability around the true parameter values, a comparison is made between the estimators possessing finite moments and the estimators having no finite moments.     

Bimodal Birnbaum–Saunders distribution with applications to non negative measurements

In this article, we introduce a new extension of the Birnbaum–Saunders (BS) distribution as a follow-up to the family of skew-flexible-normal distributions. This extension produces a family of BS distributions including densities that can be unimodal as well as bimodal. This flexibility is important in dealing with positive bimodal data, given the difficulties experienced by the use of mixtures of distributions. Some basic properties of the new distribution are studied including moments. Parameter estimation is approached by the method of moments and also by maximum likelihood, including a derivation of the Fisher information matrix. Three real data illustrations indicate satisfactory performance of the proposed model.     

A class of maximum-entropy multivariate distributions

This paper characterizes a class of multivariate distributions that includes the multinormal and is contained in the exponential family. The wide range of possible applications of these distributions is suggested by some of hte characteristics germane to them: First, they maximize Shannon's entropy among all distributions that have finite moments of given orders. As such, they constitute a class of distributions that includes the multinormal and some likely alternatives. Second, they can exhibit several modes, and, further-more, they do so with a relatively small number of parameters (compared to mixtures of multinormals). Third, they are the stationary distributions of certain diffusion processes. Fourth, they approximate, near the multinormal, the multivariate Pearson family. And fifth, the maximum likelihood estimators of their population moments are the sample moments. Two possible methods of estimating the distributions are studied in this paper: maximum likelihood estimation, and a fast procedure that can be used to find consistent estimators of the parameters via sample moments. A FORTTAN subroutine that implements the latter method is also provided.     

Moments of order statistics of the standard two-sided power distribution

The standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.     

Parameter estimates and tests of fit for infinite mixther distrirutions

Al though mixtures form a rich class of probability models, they often present difficulties for statistical inference. Likelihood functions are sometimes unbounded at certain values of the parameters, and densities often have no closed form. These features complicate hoth maximum-likelihood estimation and tests of fit based on the empirical distribution function. New inferential methods using sample characteristic functions (Cfs) and moment generating functions (MGFs) seem well-suited to mixtures. since these transforms often take simple form/ This paper reports a simulation study of the properties of estimators and tests of fit based on CFs, MGFs, and sample moments when applied to three specific families of thick tailed mixture distributios.     

An approximation of the distribution function of the LIML identifiability test statistic using the method of moments

This paper contains an application of the asymptotic expansion of a pFp() function to a problem encountered in econometrics. In particular we consider an approximation of the distribution function of the limited information maximum likelihood (LIML) identifiability test statistic using the method of moments. An expression for the Sth order asymptotic approximation of the moments of the LIML identifiability test statistic is derived and tabulated. The exact distribution function of the test statistic is approximated by a member of the class of F (variance ratio) distribution functions having the same first two integer moments. Some tabulations of the approximating distribution function are included.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.     

Parsimonious Gaussian mixture models

Parsimonious Gaussian mixture models are developed using a latent Gaussian model which is closely related to the factor analysis model. These models provide a unified modeling framework which includes the mixtures of probabilistic principal component analyzers and mixtures of factor of analyzers models as special cases. In particular, a class of eight parsimonious Gaussian mixture models which are based on the mixtures of factor analyzers model are introduced and the maximum likelihood estimates for the parameters in these models are found using an AECM algorithm. The class of models includes parsimonious models that have not previously been developed. These models are applied to the analysis of chemical and physical properties of Italian wines and the chemical properties of coffee; the models are shown to give excellent clustering performance.     

Properties of Coherent Systems with Dependent Components

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.     

A zero frequency alternative method to the moment method of estimation in finite poisson mixtures

The method of moments (MM) has been widely used for parametric estimation, as it is often computationally simple. Our interest focuses on the case of finite Poisson mixtures. The inefficiency of the method of moments relative to the Maximum Likelihood (ML) method is studied. Both the asymptotic efficiency as well as the small sample efficiency is examined. The case of samples that fail to lead to MM estimates is also considered. The results discourage the use of the MM estimators for two reasons; the first is that they are inefficient relative to the ML estimators and the second is the high probability of failing to lead to valid estimates. Another method, which considers replacing the third'moment by the zero frequency, is examined. This method turns out to be more efficient than the moment method and not very demanding computationally.     

Nonparametric bayesian inference for dichotomous response models

A Bayesian least squares approach is taken here to estimate certain parameters in generalized linear models for dichotomous response data. The method requires that only first and second moments of the probability distribution representing prior information be specified* Examples are presented to illustrate situations having direct estimates as well as those which require approximate or iterative solutions.     

The Exponentiated G Poisson Model

We study a new family of distributions defined by the minimum of the Poisson random number of independent identically distributed random variables having a general exponentiated G distribution. Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics and their moments, reliability, and Shannon entropy are derived. Maximum likelihood estimation of the model parameters is investigated. Two special models of the new family are discussed. We perform an application to a real data set to show the potentiality of the proposed family.     

GMM estimation of a realized stochastic volatility model: A Monte Carlo study

This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.     

Selection procedures based on normal and nonnormal variables with single and multiple characters

The performance of selection procedures using a single screening variable are assessed in the presence of nonnormality, in particular mixtures of bivariate normal distributions and the bivariate Edgeworth series distribution.

Screening with multiple characters in the normal situation is studied using principal components.     

The TTT transformation and a new bathtub distribution model

In this paper we use the total time on test transformation to establish a method for construction of parametric models of lifetime distributions having bathtub-shaped failure rate. We study a particular model which is simple compared to the other existing models. We derive expressions for moments and quantiles and treat estimation methods. Particularly, the maximum likelihood method is studied. Consistency proofs are given.     

On principal points for location mixtures of spherically symmetric distributions

In this paper, we investigate some properties of 2-principal points for location mixtures of spherically symmetric distributions with focus on a linear subspace in which a set of 2-principal points must lie. Our results can be viewed as an extension of those of Yamamoto and Shinozaki [2000. Two principal points for multivariate location mixtures of spherically symmetric distributions. J. Japan Statist. Soc. 30, 53–63], where a finite location mixture of spherically symmetric distributions is treated. As an extension of their paper, this paper defines a wider class of distributions, and derives a linear subspace in which a set of 2-principal points must exist. A theorem useful for comparing the mean squared distances is also established.     

Extended tables for moments of gamma distribution order statistics

Apart from having intrinsic mathematical interest, order statistics are also useful in the solution of many applied sampling and analysis problems. For a general review of the properties and uses of order statistics, see David (1981). This paper provides tabulations of means and variances of certain order statistics from the gamma distribution, for parameter values not previously available. The work was motivated by a particular quota sampling problem, for which existing tables are not adequate. The solution to this sampling problem actually requires the moments of the highest order statistic within a given set; however the calculation algorithm used involves a recurrence relation, which causes all the lower order statistics to be calculated first. Therefore we took the opportunity to develop more extensive tables for the gamma order statistic moments in general. Our tables provide values for the order statistic moments which were not available in previous tables, notably those for higher values of m, the gamma distribution shape parameter. However we have also retained the corresponding statistics for lower values of m, first to allow for checking accuracy of the computtions agtainst previous tables, and second to provide an integrated presentation of our new results with the previously known values in a consistent format     

Robust modelling for asset management

We argue that the principal way to achieve robustness is through good elicitation. This means asking the right questions, to elicit only those judgements which can be given reliably. Bayes linear methods minimise the number of prior judgements employed in the analysis. Only first- and second-order moments are specified. We present an example in which a complex problem is modelled in terms of a relatively small number of meaningful prior judgements. Sensitivity to variations in those judgements is explored here and in Goldstein and Wooff (1992).