Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

A likelihood ratio test to discriminate exponential–Poisson and gamma distributions

The exponential–Poisson (EP) distribution with scale and shape parameters β>0 and λ∈?, respectively, is a lifetime distribution obtained by mixing exponential and zero-truncated Poisson models. The EP distribution has been a good alternative to the gamma distribution for modelling lifetime, reliability and time intervals of successive natural disasters. Both EP and gamma distributions have some similarities and properties in common, for example, their densities may be strictly decreasing or unimodal, and their hazard rate functions may be decreasing, increasing or constant depending on their shape parameters. On the other hand, the EP distribution has several interesting applications based on stochastic representations involving maximum and minimum of iid exponential variables (with random sample size) which make it of distinguishable scientific importance from the gamma distribution. Given the similarities and different scientific relevance between these models, one question of interest is how to discriminate them. With this in mind, we propose a likelihood ratio test based on Cox's statistic to discriminate the EP and gamma distributions. The asymptotic distribution of the normalized logarithm of the ratio of the maximized likelihoods under two null hypotheses – data come from EP or gamma distributions – is provided. With this, we obtain the probabilities of correct selection. Hence, we propose to choose the model that maximizes the probability of correct selection (PCS). We also determinate the minimum sample size required to discriminate the EP and gamma distributions when the PCS and a given tolerance level based on some distance are before stated. A simulation study to evaluate the accuracy of the asymptotic probabilities of correct selection is also presented. The paper is motivated by two applications to real data sets.     

Discriminating Between the Log-Normal and Log-Logistic Distributions

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed.     

A Comparison of Bayes Factors for Separated Models: Some Simulation Results

Some alternative Bayes Factors: Intrinsic, Posterior, and Fractional have been proposed to overcome the difficulties presented when prior information is weak and improper prior are used. Additional difficulties also appear when the models are separated or non nested. This article presents both simulation results and some illustrative examples analysis comparing these alternative Bayes factors to discriminate among the Lognormal, the Weibull, the Gamma, and the Exponential distributions. Simulation results are obtained for different sample sizes generated from the distributions. Results from simulations indicates that these alternative Bayes factors are useful for comparing non nested models. The simulations also show some similar behavior and that when both models are true they choose the simplest model. Some illustrative example are also presented.     

Discriminating between the log-normal and generalized exponential distributions

The two-parameter generalized exponential distribution was recently introduced by Gupta and Kundu (Austral. New Zealand J. Statist. 40 (1999) 173). It is observed that the Generalized Exponential distribution can be used quite effectively to analyze skewed data set as an alternative to the more popular log-normal distribution. In this paper, we use the ratio of the maximized likelihoods in choosing between the log-normal and generalized exponential distributions. We obtain asymptotic distributions of the logarithm of the ratio of the maximized likelihoods and use them to determine the required sample size to discriminate between the two distributions for a user specified probability of correct selection and tolerance limit.     

Absolute continuous bivariate generalized exponential distribution

Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.     

Discriminating between the Weibull and log-normal distributions for Type-II censored data

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.     

Discriminating between the generalized Rayleigh and Weibull distributions

Generalized Rayleigh (GR) and Weibull (WE) distributions are used quite effectively for analysing skewed lifetime data. In this paper, we consider the problem of selecting either GR or WE distribution as a more appropriate fitting model for a given data set. We use the ratio of maximized likelihoods (RML) for discriminating between the two distributions. The asymptotic and simulated distributions of the logarithm of the RML are applied to determine the probability of correctly selecting between these two families of distributions. It is examined numerically that the asymptotic results work quite well even for small sample sizes. A real data set involving the annual rainfall recorded at Los Angeles Civic Center during 25 years is analysed to illustrate the procedures developed here.     

Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies

The generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.     

A unified view on skewed distributions arising from selections

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew‐normal and skew‐elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.     

Performance of information criteria for spatial models

Model choice is one of the most crucial aspect in any statistical data analysis. It is well known that most models are just an approximation to the true data-generating process but among such model approximations, it is our goal to select the ‘best’ one. Researchers typically consider a finite number of plausible models in statistical applications, and the related statistical inference depends on the chosen model. Hence, model comparison is required to identify the ‘best’ model among several such candidate models. This article considers the problem of model selection for spatial data. The issue of model selection for spatial models has been addressed in the literature by the use of traditional information criteria-based methods, even though such criteria have been developed based on the assumption of independent observations. We evaluate the performance of some of the popular model selection critera via Monte Carlo simulation experiments using small to moderate samples. In particular, we compare the performance of some of the most popular information criteria such as Akaike information criterion (AIC), Bayesian information criterion, and corrected AIC in selecting the true model. The ability of these criteria to select the correct model is evaluated under several scenarios. This comparison is made using various spatial covariance models ranging from stationary isotropic to nonstationary models.     

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.     

On a test for exponentiality against Laplace order dominance

Surles and Padgett recently introduced two-parameter Burr Type X distribution, which can also be described as the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both the distributions can be used quite effectively to analyze skewed data set. For a given data set the problem of selecting either generalized Rayleigh or log-normal distribution is discussed in this paper. The ratio of maximized likelihood (RML) is used in discriminating between the two distributing functions. Asymptotic distributions of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between these two families of distributions for a used specified probability of correct selection and the tolerance limit.     

On the comparison of the Fisher information of the log-normal and generalized Rayleigh distributions

Surles and Padgett recently considered two-parameter Burr Type X distribution by introducing a scale parameter and called it the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both distributions can be used quite effectively to analyze skewed data set. In this paper, we mainly compare the Fisher information matrices of the two distributions for complete and censored observations. Although, both distributions may provide similar data fit and are quite similar in nature in many aspects, the corresponding Fisher information matrices can be quite different. We compute the total information measures of the two distributions for different parameter ranges and also compare the loss of information due to censoring. Real data analysis has been performed for illustrative purposes.     

The new unified representation of multivariate skewed distributions

There are so many proposals in construction skewed distributions, and it is worth finding an overall class which covers all of these proposals. We introduce a new unified representation of multivariate skewed distributions. We will show that this new unified multivariate form of skewed distributions includes all of the continuous multivariate skewed distributions in the literature. This new unified representation is based on the multivariate probability integral transformation and can be decomposed into one factor that is original multivariate symmetric probability density function (pdf) f on ? ^k and skewed factor defined by a pdf p on [0, 1] ^k . This decomposition leads us to prove some useful properties of this new unified form. Stochastic representations and basic properties of this new form are also investigated in this article. Our work is motivated by considering the different skewing mechanisms which lead to different skewed distributions and show that all of these common-used distributions can be viewed as a new unified form.     

An extension of the generalized exponential distribution

The two-parameter generalized exponential distribution has been used recently quite extensively to analyze lifetime data. In this paper the two-parameter generalized exponential distribution has been embedded in a larger class of distributions obtained by introducing another shape parameter. Because of the additional shape parameter, more flexibility has been introduced in the family. It is observed that the new family is positively skewed, and has increasing, decreasing, unimodal and bathtub shaped hazard functions. It can be observed as a proportional reversed hazard family of distributions. This new family of distributions is analytically quite tractable and it can be used quite effectively to analyze censored data also. Analyses of two data sets are performed and the results are quite satisfactory.     

A Class of Multivariate Bilateral Selection t Distributions and Its Properties

This article proposes a class of multivariate bilateral selection t distributions useful for analyzing non-normal (skewed and/or bimodal) multivariate data. The class is associated with a bilateral selection mechanism, and it is obtained from a marginal distribution of the centrally truncated multivariate t. It is flexible enough to include the multivariate t and multivariate skew-t distributions and mathematically tractable enough to account for central truncation of a hidden t variable. The class, closed under linear transformation, marginal, and conditional operations, is studied from several aspects such as shape of the probability density function, conditioning of a distribution, scale mixtures of multivariate normal, and a probabilistic representation. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

On A method for selecting a distributional model

A method for selecting a distributional model for a random variable, given a random sample of observations of it, is studied for various cases. The problems considered include those of choosing between the Weibull and lognormal distributions, between the lognormal and gamma distributions, and between the gamma and Weibull distributions, as well as choosing one of the three. Simulation studies were performed to estimate probabilities of correct selection for the method when it is applied to these problems     

A comparison of model selection criteria

There has been significant new work published recently on the subject of model selection. Notably Rissanen (1986, 1987, 1988) has introduced new criteria based on the notion of stochastic complexity and Hurvich and Tsai(1989) have introduced a bias corrected version of Akaike's information criterion. In this paper, a Monte Carlo study is conducted to evaluate the relative performance of these new model selection criteria against the commonly used alternatives. In addition, we compare the performance of all the criteria in a number of situations not considered in earlier studies: robustness to distributional assumptions, collinearity among regressors, and non-stationarity in a time series. The evaluation is based on the number of times the correct model is chosen and the out of sample prediction error. The results of this study suggest that Rissanen's criteria are sensitive to the assumptions and choices that need to made in their application, and so are sometimes unreliable. While many of the criteria often perform satisfactorily, across experiments the Schwartz Bayesian Information Criterion (and the related Bayesian Estimation Criterion of Geweke-Meese) seem to consistently outperfom the other alternatives considered.