Rameshwar D. Gupta

Generalized Exponential Distributions

Summary 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.

Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

Summary In this article we study some properties of a new family of distributions, namely Exponentiated Exponentialdistribution, discussed in Gupta, Gupta, and Gupta (1998). The Exponentiated Exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed that many properties of this new family are quite similar to those of a Weibull or a gamma family, therefore this distribution can be used as a possible alternative to a Weibull or a gamma distribution. We present two reall ife data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.

Modeling failure time data by lehman alternatives

The proportional hazards model has been extensively used in the literature to model failure time data. In this paper we propose to model failure time data by F*(f) = [F(t)]θ where F(t) is the baseline distribution function and θ is a positive real number. This model gives rise to monotonic as well as non-monotonic failure rates even though the baseline failure rate is monotonic. The monotonicity of the failure rates are studied, in general, and some order relations are examined. Some examples including exponentiated Weibull, exponential, gamma and Pareto distributions are investigated in detail.

Generalized exponential distribution: different method of estimations

Recently a new distribution, named as generalized exponential distribution has been introduced and studied quite extensively by the authors. Generalized exponential distribution can be used as an alternative to gamma or Weibull distribution in many situations. In a companion paper, the authors considered the maximum likelihood estimation of the different parameters of a generalized exponential distribution and discussed some of the testing of hypothesis problems. In this paper we mainly consider five other estimation procedures and compare their performances through numerical simulations.

Estimation of P[Y<X] for Weibull distributions

This paper deals with the estimation of R=P[Y<X] when X, and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator, and the approximate maximum likelihood estimator of R are proposed. We obtain the asymptotic distribution of the maximum likelihood estimator of R. Based on the asymptotic distribution, the confidence interval of R can be obtained. We also propose two bootstrap confidence intervals. We consider the Bayesian estimate of R, and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.

Discriminating between Weibull and generalized exponential distributions

Recently the two-parameter generalized exponential (GE) distribution was introduced by the authors. It is observed that a GE distribution can be considered for situations where a skewed distribution for a non-negative random variable is needed. The ratio of the maximized likelihoods (RML) is used in discriminating between Weibull and GE distributions. Asymptotic distributions of the logarithm of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between two overlapping families of distributions for a user specified probability of correct selection and tolerance limit.

Hybrid censoring schemes with exponential failure distribution

The mixture of Type I and Type I1 censoring schemes, called the hybrid censoring, is quite important in life–testing experiments. Epstein(1954, 1960) introduced this testing scheme and proposed a two–sided confidence interval to estimate the mean lifetime, θ, when the underlying lifetime distribution is assumed to be exponential. There are some two–sided confidence intervals and credible intervals proposed by Fairbanks et al. (1982) and Draper and Guttman (1987) respectively. In this paper we obtain the exact two–sided confidence interval of θ following the approach of Chen and Bhattacharya (1988). We also obtain the asymptotic confidence intervals in the Hybrid censoring case. It is important to observe that the results for Type I and Type II censoring schemes can be obtained as particular cases of the Hybrid censoring scheme. We analyze one data set and compare different methods by Monte Carlo simulations.

A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.

Bivariate generalized exponential distribution

Recently it has been observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. The main aim of this paper is to define a bivariate generalized exponential distribution so that the marginals have generalized exponential distributions. It is observed that the joint probability density function, the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms. Several properties of this distribution have been discussed. We suggest to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters and also obtain the observed and expected Fisher information matrices. One data set has been re-analyzed and it is observed that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution.

Generalized exponential distribution: Bayesian estimations

Recently two-parameter generalized exponential distribution has been introduced by the authors. In this paper we consider the Bayes estimators of the unknown parameters under the assumptions of gamma priors on both the shape and scale parameters. The Bayes estimators cannot be obtained in explicit forms. Approximate Bayes estimators are computed using the idea of Lindley. We also propose Gibbs sampling procedure to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors, are compared with the maximum likelihood estimators using Monte Carlo simulations. One real data set has been analyzed for illustrative purposes.