Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

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.     

Comparison of Interval Estimators of Pr(X < Y) in the Two-parameter Exponential Distribution

We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.     

Generalized elliptical distributions

A family of distributions generated by an operator acting on generalized normal density is introduced. This family contains as particular cases many known distributions, including the generalized normal, generalized t, and generalized gamma distributions. Several mathematical properties of the family (including expansions, characteristic function, moments, cumulants, and order statistics properties) are derived. Estimation procedures are derived too by the method of moments, method of maximum likelihood, and the method of empirical characteristic function. A real data application is presented. Finally, extensions to the multivariate case are outlined.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Generalized normal ARMA model applied to the areas of economy,hydrology, and public policy

The main purpose of this article is the presentation of a new class of time series models which is the merge output of the generalized normal distribution with ideas from the GARMA model. Symmetrically, tails that may be lighter or heavier than the Gaussian distribution, and Gaussian and Laplace distributions as special cases, are the main advantages of the use of generalized normal distribution. The proposed model is called generalized normal autoregressive moving average (GN-ARMA). We exemplify the application of the proposed model adjusting it to the three time series, which are from the areas of economy, hydrology, and public policy.     

Poisson-Driven Stationary Markov Models

ABSTRACT

We propose a simple yet powerful method to construct strictly stationary Markovian models with given but arbitrary invariant distributions. The idea is based on a Poisson-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.     

Two-Sample Bayesian Prediction Intervals of Generalized Order Statistics Based on Multiply Type II Censored Data

In this article, two-sample Bayesian prediction intervals of generalized order statistics (GOS) based on multiply Type II censored data are derived. To illustrate these results, the Pareto, Weibull, and Burr-Type XII distributions are used as examples. Finally, a numerical illustration of the sequential order statistics from the Pareto distribution is presented.     

Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data

Highly skewed and non-negative data can often be modeled by the delta-lognormal distribution in fisheries research. However, the coverage probabilities of extant interval estimation procedures are less satisfactory in small sample sizes and highly skewed data. We propose a heuristic method of estimating confidence intervals for the mean of the delta-lognormal distribution. This heuristic method is an estimation based on asymptotic generalized pivotal quantity to construct generalized confidence interval for the mean of the delta-lognormal distribution. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities, expected interval lengths and reasonable relative biases. Finally, the proposed method is employed in red cod densities data for a demonstration.     

Testing for Generalized Lorenz Dominance

The Generalized Lorenz dominance can be used to take account of differences in mean income as well as income inequality in case of two income distributions possessing unequal means. Asymptotically distribution-free and consistent tests have been proposed for comparing two generalized Lorenz curves in the whole interval [p ₁, p ₂] where 0 < p ₁ < p ₂ < 1. Size and power of the test has been derived.     

Generalized Confidence Interval Estimation for the Difference in Paired Areas Under the ROC Curves in the Absence of a Gold Standard

Receiver operating characteristic (ROC) curves can be used to assess the accuracy of tests measured on ordinal or continuous scales. The most commonly used measure for the overall diagnostic accuracy of diagnostic tests is the area under the ROC curve (AUC). A gold standard (GS) test on the true disease status is required to estimate the AUC. However, a GS test may be too expensive or infeasible. In many medical researches, the true disease status of the subjects may remain unknown. Under the normality assumption on test results from each disease group of subjects, we propose a heuristic method of estimating confidence intervals for the difference in paired AUCs of two diagnostic tests in the absence of a GS reference. This heuristic method is a three-stage method by combining the expectation-maximization (EM) algorithm, bootstrap method, and an estimation based on asymptotic generalized pivotal quantities (GPQs) to construct generalized confidence intervals for the difference in paired AUCs in the absence of a GS. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities and expected interval lengths. The numerical example using a published dataset illustrates the proposed method.     

Confidence intervals for coefficients of variation in two-parameter exponential distributions

This article examines confidence intervals for the single coefficient of variation and the difference of coefficients of variation in the two-parameter exponential distributions, using the method of variance of estimates recovery (MOVER), the generalized confidence interval (GCI), and the asymptotic confidence interval (ACI). In simulation, the results indicate that coverage probabilities of the GCI maintain the nominal level in general. The MOVER performs well in terms of coverage probability when data only consist of positive values, but it has wider expected length. The coverage probabilities of the ACI satisfy the target for large sample sizes. We also illustrate our confidence intervals using a real-world example in the area of medical science.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Generalized Empirical Likelihood Inference in Generalized Linear Models for Longitudinal Data

In this article, the generalized linear model for longitudinal data is studied. A generalized empirical likelihood method is proposed by combining generalized estimating equations and quadratic inference functions based on the working correlation matrix. It is proved that the proposed generalized empirical likelihood ratios are asymptotically chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. In addition, the maximum empirical likelihood estimates of parameters are obtained, and their asymptotic normalities are proved. Some simulations are undertaken to compare the generalized empirical likelihood and normal approximation-based method in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. An example of a real data is used for illustrating our methods.     

Generalized inferences on the common mean vector of several multivariate normal populations

The hypothesis testing and confidence region are considered for the common mean vector of several multivariate normal populations when the covariance matrices are unknown and possibly unequal. A generalized confidence region is derived using the concepts of generalized method based on the generalized p$p$-value. The generalized confidence region is illustrated with two numerical examples. The merits of the proposed method are numerically compared with those of existing methods with respect to their expected area or expected d-dimensional volumes and coverage probabilities under different scenarios.     

Generalized Gumbel distribution

A generalization of the Gumbel distribution is presented to deal with general situations in modeling univariate data with broad range of skewness in the density function. This generalization is derived by considering a logarithmic transformation of an odd Weibull random variable. As a result, the generalized Gumbel distribution is not only useful for testing goodness-of-fit of Gumbel and reverse-Gumbel distributions as submodels, but it is also convenient for modeling and fitting a wide variety of data sets that are not possible to be modeled by well-known distributions. Skewness and kurtosis shapes of the generalized Gumbel distribution are illustrated by constructing the Galton’s skewness and Moor’s kurtosis plane. Parameters are estimated by using maximum likelihood method in two different ways due to the fact that the reverse transformation of the proposed distribution does not change its density function. In order to illustrate the flexibility of this generalization, wave and surge height data set is analyzed, and the fitness is compared with Gumbel and generalized extreme value distributions.     

Generalized confidence intervals for the process capability indices in general random effect model with balanced data

In this paper, we consider the interval estimation problem on the process capability indices in general random effect model with balanced data. The confidence intervals for three commonly used process capability indices are developed by using the concept of generalized confidence interval. Furthermore, some simulation results on the coverage probability and expected value of the generalized lower confidence limits are reported. The simulation results indicate that the proposed confidence intervals do provide quite satisfactory coverage probabilities.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

Testing Homogeneity of Inverse Gaussian Scale Parameters Based on Generalized Likelihood Ratio

This article presents a new procedure for testing homogeneity of scale parameters from k independent inverse Gaussian populations. Based on the idea of generalized likelihood ratio method, a new generalized p-value is derived. Some simulation results are presented to compare the performance of the proposed method and existing methods. Numerical results show that the proposed test has good size and power performance.     

Tests for generalized exponential laws based on the empirical Mellin transform

Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.