Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions

Abstract

In this paper, we derive Bayesian estimators of the parameters of modified power series distributions inflated at any of a support point under linex and general entropy loss function. We assume that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution inflated at a given point.     

On k-distorted generalized discrete family of distributions

We propose a new generalized discrete family of distributions which permits inflation/deflation at any single point in the support of distribution. Using the same, a zero-distorted generalized discrete family of distributions is introduced and some of its properties are studied. As an illustration, we study in detail, zero-distorted generalized Poisson distribution. Real-life applications of this distribution using well-known data sets are reported, which include an actuarial application of the proposed model.     

On the Bayes estimators of the parameters of size-biased generalized power series distributions

ABSTRACT

In this paper, we derive the Bayes estimators of functions of parameters of the size-biased generalized power series distribution under squared error loss function and weighted square error loss function. The results of size-biased GPSD are then used to obtain particular cases of the size-biased negative binomial, size-biased logarithmic series, and size-biased Poisson distributions. These estimators are better than the classical minimum variance unbiased estimators in the sense that they increase the range of the estimation. Finally, an example is provided to illustrate the results and a goodness of fit test is done using the maximum likelihood and Bayes estimators.     

Structural changes in multivariate regression models

This study generalizes the work of chin choy and Broemeling (1980) who investigated the change in the regression parameters of univariate linear models.

The marginal posterior distributions of the change point, the regression matrices,and the precision matrix are found with the use of a proper multivariate normal-Wishart distribution for the parameters of the model.

A numerical study is undertaken in order to gain some insight into the effect that changes in sample size and certain parameter values have on these marginal posterior distributions.     

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.     

Change Point Detection in The Skew-Normal Model Parameters

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.     

Baysian estimation of the parameters in two modified poission distributions

Posterior distributions and moment are derived for the generalized Poisson and the excess zeroes Poisson distributions.Three examples are presented where both maximum likelihood and posterior estimates are given.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Quasi Bayesian likelihood

If the form of the distribution of data is unknown, the Bayesian method fails in the parametric inference because there is no posterior distribution of the parameter. In this paper, a theoretical framework of Bayesian likelihood is introduced via the Hilbert space method, which is free of the distributions of data and the parameter. The posterior distribution and posterior score function based on given inner products are defined and, consequently, the quasi posterior distribution and quasi posterior score function are derived, respectively, as the projections of the posterior distribution and posterior score function onto the space spanned by given estimating functions. In the space spanned by data, particularly, an explicit representation for the quasi posterior score function is obtained, which can be derived as a projection of the true posterior score function onto this space. The methods of constructing conservative quasi posterior score and quasi posterior log-likelihood are proposed. Some examples are given to illustrate the theoretical results. As an application, the quasi posterior distribution functions are used to select variables for generalized linear models. It is proved that, for linear models, the variable selections via quasi posterior distribution functions are equivalent to the variable selections via the penalized residual sum of squares or regression sum of squares.     

Non–zero inflated modified power series distributions

In this paper, we study the class of inflated modified power series distributions (IMPSD) where inflation occurs at any of the support points. This class include among other the generalized Poisson, the generalized negative binomial, the generalized logarithmic series and the lost games distributions. We give expressions for the moments, factorial moments and central moments of the IMPSD. The maximum likelihood estimation of the parameters of the IMPSD and the variance – covariance matrix of the estimators is obtained. We derive these estimators and their information matrices for mentioned above particular members of IMPSD class. The second part of this paper deals with the distribution of sum of independent and identically distributed random variables taking values s, s+1. s + 2, …, s ≥ 0, with modified power series distributions inflated at the point s.     

A new four-parameter lifetime distribution

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley geometric (EPLG) distribution, obtained by compounding EPL and geometric distributions. The new distribution arises in a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The distribution exhibits decreasing, increasing, unimodal and bathtub-shaped hazard rate functions, depending on its parameters. It contains several lifetime distributions as particular cases: EPL, new generalized Lindley, generalized Lindley, power Lindley and Lindley geometric distributions. We derive several properties of the new distribution such as closed-form expressions for the density, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Simulation studies are also provided. Finally, two real data applications are given for showing the flexibility and potentiality of the new distribution.     

On asymptotic properties of Bayesian partially linear models

In this paper, we present large sample properties of a partially linear model from the Bayesian perspective, in which responses are explained by the semiparametric regression model with the additive form of the linear component and the nonparametric component. For this purpose, we investigate asymptotic behaviors of posterior distributions in terms of consistency. Specifically, we deal with a specific Bayesian partially linear regression model with additive noises in which the nonparametric component is modeled using Gaussian process priors. Under the Bayesian partially linear model using Gaussian process priors, we focus on consistency of posterior distribution and consistency of the Bayes factor, and extend these results to generalized additive regression models and study their asymptotic properties. In addition we illustrate the asymptotic properties based on empirical analysis through simulation studies.     

Bayes Estimation of Intraclass Correlation Coefficients Under Unequal Family Sizes

Bayesian inference for the intraclass correlation ρ is considered under unequal family sizes. We obtain the posterior distribution of ρ and then compare the performance of the Bayes estimator (posterior mean of ρ) with that of Srivastava's (1984) estimator through simulation. Simulation study shows that the Bayes estimator performs better than the Srivastava's estimator in terms of lower mean square error. We also obtain large sample posteriors of ρ based on the asymptotic posterior distribution and based on the Laplace approximation.     

Robust likelihood functions in Bayesian inference

In order to deal with mild deviations from the assumed parametric model, we propose a procedure for accounting for model uncertainty in the Bayesian framework. In particular, in the derivation of posterior distributions, we discuss the use of robust pseudo-likelihoods, which offer the advantage of preventing the effects caused by model misspecifications, i.e. when the underlying distribution lies in a neighborhood of the assumed model. The influence functions of posterior summaries, such as the posterior mean, are investigated as well as the asymptotic properties of robust posterior distributions. Although the use of a pseudo-likelihood cannot be considered orthodox in the Bayesian perspective, it is shown that, also through some illustrative examples, how a robust pseudo-likelihood, with the same asymptotic properties of a genuine likelihood, can be useful in the inferential process in order to prevent the effects caused by model misspecifications.     

Bayesian Point Estimation Using the Predictive Distribution

There is considerable question about how a Bayesian might provide a point estimate for a parameter when no loss function is specified. The mean, median, and mode of the posterior distribution have all been suggested. This article considers a natural Bayesian estimator based on the predictive distribution of future observations. It is shown that for the set of parameters that admit an unbiased estimate, this predictive estimate coincides with the posterior mean of the parameter. It is argued that this result provides some justification for use of the posterior mean as a Bayesian point estimate when there is no loss structure.     

A random weighting approach for posterior distributions

ABSTRACT

In Bayesian theory, calculating a posterior probability distribution is highly important but typically difficult. Therefore, some methods have been proposed to deal with such problem, among which, the most popular one is the asymptotic expansions of posterior distributions. In this paper, we propose an alternative approach, named a random weighting method, for scaled posterior distributions, and give an ideal convergence rate, o(n^{( ? 1/2)}), which serves as the theoretical guarantee for methods of numerical simulations.     

Bayesian analysis of linear models exhibiting changes in mean and precisionat an unknown time point

This paper analyses a linear model in which both the mean and the precision change exactly once at an unknown point in time. Posterior distributions are found for the unknown time point at which the changes occurred and for the ratio of the precisions. The Bayesian predictive distribution of k future observations is also derived. It is shown that the unconditional posterior distribution of the ratio of precisions is a mixture of F-type distributions and the predictive distribution is a mixture of multivariate t distributions.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.