The shape of the hazard function: Does the generalized gamma have the last word?

This paper compares the five-parameter beta generalized gamma (BGG) distribution to the three-parameter generalized gamma (GG). Both distributions include the four standard hazard shapes that we believe is an important property for any parametric family. For several BGG distributions, we select matching GGs and compute the Kullback-Liebler distance, observing remarkable agreement. We explore the beta parameters' influence on the matched GG parameters, detecting a strong connection between the distributions. Lastly, we compare the distributions using two real-data examples. We conclude from these comparisons that the BGG is not likely to be more useful for analytical purposes than the simpler GG.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Bivariate semi α-Laplace distribution and processes

The Laplace distribution is considered as a better choice for modeling whenever data exhibit high kurtosis and heavier tails than Gaussian tails. Even though this is the case, not much work has been done on bivariate Laplace distribution. In this work, we introduce and study a new class of bivariate distributions called bivariate semi α-Laplace distribution, containing bivariate Laplace distributions. Three characterizations of bivariate semi α-Laplace distribution are obtained. Relation with bivariate semi stable distribution is established. An autoregressive model with bivariate semi α-Laplace marginal distributions is developed.     

A note on the multivariate generalized asymmetric Laplace motion

Abstract

In this note, we use multivariate subordination to introduce a multivariate extension of the generalized asymmetric Laplace motion. The class introduced provides a unified framework for several multivariate extensions of the popular variance gamma process. We also show that the associated time one distribution extends the multivariate generalized asymmetric Laplace distributions proposed in the statistical literature.     

A new family of lifetime models

We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions. Some properties of the new family of distributions are derived. A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention. The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived. An application to a real data set is discussed.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT

This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.     

Tests of goodness of fit based on Phi-divergence

In this paper, we introduce a general goodness of fit test based on Phi-divergence. Consistency of the proposed test is established. We then study some special cases of tests for normal, exponential, uniform and Laplace distributions. Through Monte Carlo simulations, the power values of the proposed tests are compared with some known competing tests under various alternatives. Finally, some numerical examples are presented to illustrate the proposed procedure.     

Gamma frailty models for bivariate survival data

In this paper, we introduce the shared gamma frailty models with two different baseline distributions namely, the generalized log-logistic and the generalized Weibull. We introduce the Bayesian estimation procedure to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett related to the kidney infection data and a better model is suggested for the data.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

A NEW FAMILY OF NON-NEGATIVE DISTRIBUTIONS

We introduce a new, flexible family of distributions for non‐negative data, defined by means of a quantile function. We describe some properties of this family, and discuss several methods for estimating the parameters. The distribution is applied to an example from environmental engineering.     

Correlated gamma frailty models for bivariate survival data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data) the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this article, we introduce the gamma correlated frailty models with two different baseline distributions namely, the generalized log logistic, and the generalized Weibull. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these models to a real life bivariate survival dataset related to the kidney infection data and a better model is suggested for the data.     

Inference for the Generalized Normal Laplace Distribution

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.     

Non parametric multiple comparisons of non nested rival models

The practice for testing homogeneity of several rival models is of interest. In this article, we consider a non parametric multiple test for non nested distributions in the context of the model selection. Based on the linear sign rank test, and the known union–intersection principle, we let the magnitude of the data to give a better performance to the test statistic. We consider the sample and the non nested rival models as blocks and treatments, respectively, and introduce the extended Friedman test version to compare with the results of the test based on the linear sign rank test. A real dataset based on the waiting time to earthquake is considered to illustrate the results.     

Quasi Most Powerful Invariant Goodness-of-fit Tests

ABSTRACT.  In this paper, we develop an approximation for the most powerful invariant test of one location-scale family against another one. The approach is based on the Laplace method for integrals and yields a very accurate approximation of the density of a maximal invariant. Moreover, it can be applied to a much wider set of pairs of densities than previously possible. Many examples are worked out. The resulting test is easy to compute and its power is shown to be very close to that of the best test. By using versions of the Laplace method, the approach is extended to goodness-of-fit tests for residuals in regression and to some multivariate distributions. A small simulation study confirms the theoretical results. An example concludes the paper.     

Estimation of noncentrality parameters

We propose some estimators of noncentrality parameters which improve upon usual unbiased estimators under quadratic loss. The distributions we consider are the noncentral chi-square and the noncentral F. However, we give more general results for the family of elliptically contoured distributions and propose a robust dominating estimator.     

An absolutely continuous bivariate Topp–Leone distribution: A useful model on a bounded domain

We introduce an absolutely continuous bivariate generalization of the Topp–Leone distribution, which is a special member of the proportional reversed hazard family using a one-parameter bivariate exchangeable distribution. We show that a copula approach could also be used in defining the bivariate Topp–Leone distribution. The marginal distributions of the new bivariate distribution have also Topp–Leone distributions. We study its distributional and dependence properties. We estimate the parameters by maximum-likelihood procedure, perform a simulation study on the estimators, and apply them to a real data set. Furthermore, we give a way of generating bivariate distributions using the proposed distribution.