Gamma mixture of generalized error distribution

Abstract

A new symmetric heavy-tailed distribution, namely gamma mixture of generalized error distribution is defined by scaling generalized error distribution with gamma distribution, its probability density function, k-moment, skewness and kurtosis are derived. After tedious calculation, we also give the Fisher information matrix, moment estimators and maximum likelihood estimators for the parameters of gamma mixture of generalized error distribution. In order to evaluate the effectiveness of the point estimators and the stability of Fisher information matrix, extensive simulation experiments are carried out in three groups of parameters. Additionally, the new distribution is applied to Apple Inc. stock (AAPL) data and compared with normal distribution, F-S skewed standardized t distribution and generalized error distribution. It is found that the new distribution has better fitting effect on the data under the Akaike information criterion (AIC). To a certain extent, our results enrich the probability distribution theory and develop the scale mixture distribution, which will provide help and reference for financial data analysis.     

The beta log-normal distribution

For the first time, we introduce the beta log-normal (LN) distribution for which the LN distribution is a special case. Various properties of the new distribution are discussed. Expansions for the cumulative distribution and density functions that do not involve complicated functions are derived. We obtain expressions for its moments and for the moments of order statistics. The estimation of parameters is approached by the method of maximum likelihood, and the expected information matrix is derived. The new model is quite flexible in analysing positive data as an important alternative to the gamma, Weibull, generalized exponential, beta exponential, and Birnbaum–Saunders distributions. The flexibility of the new distribution is illustrated in an application to a real data set.     

The gamma-exponentiated exponential distribution

In this paper, we introduce a new distribution generated by gamma random variables. We show that this distribution includes as a special case the distribution of the lower record value from a sequence of i.i.d. random variables from a population with the exponentiated (generalized) exponential distribution. The properties of this distribution are derived and the estimation of the model parameters is discussed. Some applications to real data sets are finally presented for illustration.     

Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.     

The heteroscedastic odd log-logistic generalized gamma regression model for censored data

We propose a four-parameter extended generalized gamma model, which includes as special cases some important distributions and it is very useful for modeling lifetime data. A advantage is that it can represent the error distribution for a new heteroscedastic log-odd log-logistic generalized gamma regression model. The proposed heteroscedastic regression model can be used more effectively in the analysis of survival data since it includes as special models several widely-known regression models. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. Overall, the new regression model is very useful to the analysis of real data.     

The log-exponentiated generalized gamma regression model for censored data

For the first time, we introduce a generalized form of the exponentiated generalized gamma distribution [Cordeiro et al. The exponentiated generalized gamma distribution with application to lifetime data, J. Statist. Comput. Simul. 81 (2011), pp. 827–842.] that is the baseline for the log-exponentiated generalized gamma regression model. The new distribution can accommodate increasing, decreasing, bathtub- and unimodal-shaped hazard functions. A second advantage is that it includes classical distributions reported in the lifetime literature as special cases. We obtain explicit expressions for the moments of the baseline distribution of the new regression model. The proposed model can be applied to censored data since it includes as sub-models several widely known regression models. It therefore can be used more effectively in the analysis of survival data. We obtain maximum likelihood estimates for the model parameters by considering censored data. We show that our extended regression model is very useful by means of two applications to real data.     

Generalized Inverse Gamma Distribution and its Application in Reliability

In this article, we introduce a new reliability model of inverse gamma distribution referred to as the generalized inverse gamma distribution (GIG). A generalization of inverse gamma distribution is defined based on the exact form of generalized gamma function of Kobayashi (1991). This function is useful in many problems of diffraction theory and corrosion problems in new machines. The new distribution has a number of lifetime special sub-models. For this model, some of its statistical properties are studied. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We also demonstrate the usefulness of this distribution on a real data set.     

Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

Comparing the mean vectors of two independent multivariate log-normal distributions

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.     

On a recent generalization of gamma distribution

In this paper, we investigate a generalized gamma distribution recentIy developed by Agarwal and Kalla (1996). Also, we show that such generalized distribution, like the ordinary gamma distribution, has a unique mode and, unlike the ordinary gamma distribution, may have a hazard rate (mean residual life) function which is upside-down bathtub (bathtub) shaped.     

General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.     

Orthogonality of the mean and error distribution in generalized linear models

We show that the mean-model parameter is always orthogonal to the error distribution in generalized linear models. Thus, the maximum likelihood estimator of the mean-model parameter will be asymptotically efficient regardless of whether the error distribution is known completely, known up to a finite vector of parameters, or left completely unspecified, in which case the likelihood is taken to be an appropriate semiparametric likelihood. Moreover, the maximum likelihood estimator of the mean-model parameter will be asymptotically independent of the maximum likelihood estimator of the error distribution. This generalizes some well-known results for the special cases of normal, gamma, and multinomial regression models, and, perhaps more interestingly, suggests that asymptotically efficient estimation and inferences can always be obtained if the error distribution is non parametrically estimated along with the mean. In contrast, estimation and inferences using misspecified error distributions or variance functions are generally not efficient.     

E-Bayesian estimation and its E-MSE under the scaled squared error loss function,for exponential distribution as example

This paper is concerned with using the E-Bayesian method for computing estimates of exponential distribution. In order to measure the estimated error, based on the E-Bayesian estimation, we proposed the definition of E-MSE(expected mean square error). Moreover, the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively, these estimations are derived based on a conjugate prior distribution for the unknown parameter under the scaled squared error loss function. The properties of E-MSE under different scaled parameters are also provided. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes. Results are compared on the basis of E-MSE.     

Small Sample Tests for Shape Parameters of Gamma Distributions

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.     

A precise estimator for the log-normal mean

The log-normal distribution is frequently encountered in applications. The uniformly minimum variance unbiased (UMVU) estimator for the log-normal mean is given explicitly by a formula found by Finney in 1941. In contrast to this the most commonly used estimator for a log-normal mean is the sample mean. This is possibly due to the complexity of the formula given by Finney. A modified maximum likelihood estimator which approximates the UMVU estimator is derived here. It is sufficiently simple to be implemented in elementary spreadsheet applications. An elementary approximate formula for the root-mean-square error of the suggested estimator and the UMVU estimator is presented. The suggested estimator is compared with the sample mean, the maximum likelihood, and the UMVU estimators by Monte Carlo simulation in terms of root-mean-square error.     

Approximating hidden Gaussian Markov random fields

Summary.  Gaussian Markov random-field (GMRF) models are frequently used in a wide variety of applications. In most cases parts of the GMRF are observed through mutually independent data; hence the full conditional of the GMRF, a hidden GMRF (HGMRF), is of interest. We are concerned with the case where the likelihood is non-Gaussian, leading to non-Gaussian HGMRF models. Several researchers have constructed block sampling Markov chain Monte Carlo schemes based on approximations of the HGMRF by a GMRF, using a second-order expansion of the log-density at or near the mode. This is possible as the GMRF approximation can be sampled exactly with a known normalizing constant. The Markov property of the GMRF approximation yields computational efficiency.The main contribution in the paper is to go beyond the GMRF approximation and to construct a class of non-Gaussian approximations which adapt automatically to the particular HGMRF that is under study. The accuracy can be tuned by intuitive parameters to nearly any precision. These non-Gaussian approximations share the same computational complexity as those which are based on GMRFs and can be sampled exactly with computable normalizing constants. We apply our approximations in spatial disease mapping and model-based geostatistical models with different likelihoods, obtain procedures for block updating and construct Metropolized independence samplers.     

Modified method on the means for several log-normal distributions

Among statistical inferences, one of the main interests is drawing the inferences about the log-normal means since the log-normal distribution is a well-known candidate model for analyzing positive and right-skewed data. In the past, the researchers only focused on one or two log-normal populations or used the large sample theory or quadratic procedure to deal with several log-normal distributions. In this article, we focus on making inferences on several log-normal means based on the modification of the quadratic method, in which the researchers often used the vector of the generalized variables to deal with the means of the symmetric distributions. Simulation studies show that the quadratic method performs well only for symmetric distributions. However, the modified procedure fits both symmetric and skew distribution. The numerical results show that the proposed modified procedure can provide the confidence interval with coverage probabilities close to the nominal level and the hypothesis testing performed with satisfactory results.     

The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.     

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.