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.     

The Conway–Maxwell–Poisson-generalized gamma regression model with long-term survivors

In this paper, we proposed a flexible cure rate survival model by assuming the number of competing causes of the event of interest following the Conway–Maxwell distribution and the time for the event to follow the generalized gamma distribution. This distribution can be used to model survival data when the hazard rate function is increasing, decreasing, bathtub and unimodal-shaped including some distributions commonly used in lifetime analysis as particular cases. Some appropriate matrices are derived in order to evaluate local influence on the estimates of the parameters by considering different perturbations, and some global influence measurements are also investigated. Finally, data set from the medical area is analysed.     

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.     

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.     

Generalized gamma parameter estimation and moment evaluation

The generalized gamma distribution includes the exponential distribution, the gamma distribution, and the Weibull distribution as special cases. It also includes the log-normal distribution in the limit as one of its parameters goes to infinity. Prentice (1974) developed an estimation method that is effective even when the underlying distribution is nearly log-normal. He reparameterized the density function so that it achieved the limiting case in a smooth fashion relative to the new parameters. He also gave formulas for the second partial derivatives of the log-density function to be used in the nearly log-normal case. His formulas included infinite summations, and he did not estimate the error in approximating these summations.

We derive approximations for the log-density function and moments of the generalized gamma distribution that are smooth in the nearly log-normal case and involve only finite summations. Absolute error bounds for these approximations are included. The approximation for the first moment is applied to the problem of estimating the parameters of a generalized gamma distribution under the constraint that the distribution have mean one. This enables the development of a correspondence between the parameters in a mean one generalized gamma distribution and certain parameters in acoustic scattering theory.     

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 generalized gamma distribution

For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied. It contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), Weibull and gamma distributions and thus could be a better model for analysing positive skewed data. The new density function can be expressed as a linear combination of GG densities. We derive explicit expressions for moments, generating function and other statistical measures. The elements of the expected information matrix are provided. The usefulness of the new model is illustrated by means of a real data set.     

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.     

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 McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

A note on gamma difference distributions

It is the aim of this note to point out that the double gamma difference distribution recently introduced by [Augustyniak M, and Doray, LG. Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variables. J Statist Comput Simul. 2012;82:1621–1634] is well known in financial econometrics: it is the symmetric variance gamma family of distributions. We trace back to the various origins of this distribution. In addition, we consider in some detail the difference of two independent gamma distributed random variables with different shape parameters.     

A new generalization of the Weibull-geometric distribution with bathtub failure rate

In this paper, the researchers attempt to introduce a new generalization of the Weibull-geometric distribution. The failure rate function of the new model is found to be increasing, decreasing, upside-down bathtub, and bathtub-shaped. The researchers obtained the new model by compounding Weibull distribution and discrete generalized exponential distribution of a second type, which is a generalization of the geometric distribution. The new introduced model contains some previously known lifetime distributions as well as a new one. Some basic distributional properties and moments of the new model are discussed. Estimation of the parameters is illustrated and the model with two known real data sets is examined.     

Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.     

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.     

Mixture formulations of a bivariate negative binomial distribution with applications

This paper considers further mixture formulations of the bivariate negative binomial (BNB) distribution of Edwards and Gurland (1961) and Subrahmaniam (1966). These formulations and some known ones are applied (1) to obtain a bivariate generalized negative binomial (BGNB) distribution of Bhattacharya (1966), (2) to establish a connection between the accident-proneness models given by the BNB, BGNB and Bhattacharya's bivariate distributions, and (3) to compute the grade correlation and distribution function of the Wicksell-Kibble bivariate gamma distribution.     

An extension of the generalized linear failure rate distribution

In this article, we introduce a new extension of the generalized linear failure rate (GLFR) distributions. It includes some well-known lifetime distributions such as extension of generalized exponential and GLFR distributions as special sub-models. In addition, it can have a constant, decreasing, increasing, upside-down bathtub (unimodal), and bathtub-shaped hazard rate function (hrf) depending on its parameters. We provide some of its statistical properties such as moments, quantiles, skewness, kurtosis, hrf, and reversible hrf. The maximum likelihood estimation of the parameters is also discussed. At the end, a real dataset is given to illustrate the usefulness of this new distribution in analyzing lifetime data.     

Statistical model and inference for pharmacokinetic data

In this paper, we consider a statistical model for the drug concentration–time profiles that are obtained in a pharmacokinetic (PK) study when the drug is orally administered. In the proposed statistical PK model, the subject-specific concentration–time curve is described by the one-compartment PK model with first-order absorption and elimination. Moreover, a multivariate generalized gamma distribution is developed for the joint distribution of the drug concentrations that are repeatedly measured from the same subject. We then construct confidence intervals for the subject–exposure parameters which provide a further insight into the individual exposure of the drug under study. The proposed statistical PK model and the associated inference are then applied to illustrate a real data set. A simulation study is also implemented to investigate the performances of the coverage probability and expected length of the proposed confidence intervals. Finally, we give conclusions and discussions on the application of the proposed procedures.     

A new four-parameter discrete distribution with bathtub and unimodal failure rate

In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined.     

A New Discrete Distribution Related to Generalized Gamma Distribution and Its Properties

In this article, a new discrete distribution related to the generalized gamma distribution (Stacy, 1962) is derived from a statistical mechanical setup. This new distribution can be seen as generalization of two-parameter discrete gamma distribution (Chakraborty and Chakravarty, 2012) and encompasses discrete version of many important continuous distributions. Some basic distributional and reliability properties, parameter estimation by different methods, and their comparative performances using simulation are investigated. Two-real life data sets are considered for data modeling and likelihood ratio test for illustrating the advantages of the proposed distribution over two-parameter discrete gamma distribution.