Arabin Kumar Dey

Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm

In this paper we consider the Marshall-Olkin bivariate Weibull distribution. The Marshall-Olkin bivariate Weibull distribution is a singular distribution, whose both the marginals are univariate Weibull distributions. This is a generalization of the Marshall-Olkin bivariate exponential distribution. The cumulative joint distribution of the Marshall-Olkin bivariate Weibull distribution is a mixture of an absolute continuous distribution function and a singular distribution function. This distribution has four unknown parameters and it is observed that the maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. In this paper we discuss about the computation of the maximum likelihood estimators of the unknown parameters using EM algorithm. We perform some simulations to see the performances of the EM algorithm and re-analyze one data set for illustrative purpose.

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.

Discriminating Among the Log-Normal, Weibull, and Generalized Exponential Distributions

We consider model selection and discrimination among three important lifetime distributions. These three distributions have been used quite effectively to analyze lifetime data. We study the probability of correct selection using the maximized likelihood method, as it has been used in the literature. We further compute the asymptotic probability of correct selection, and compare the theoretical, and simulation results for different sample sizes, and for different model parameters. The results have been extended for Type-I censored data also. The theoretical, and simulation results match quite well. Two real data sets have been analyzed for illustrative purposes. We also suggest a method to determine the minimum sample size required to discriminate among the three distributions for a given probability of correct selection, and a user specified protection level.

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.

Bayesian analysis of three parameter absolute continuous Marshall–Olkin bivariate Pareto distribution

ABSTRACT This paper provides Bayesian analysis of absolute continuous Marshall–Olkin bivariate Pareto distribution. We consider only three parameters for this Marshall–Olkin bivariate Pareto distribution. We take two types of prior—reference prior and gamma prior for our analysis. Bayesian estimate of the parameters are calculated based on slice cum Gibbs sampler and Lindley approximation. Credible intervals are also provided for all methods and all prior distributions. A real-life data analysis is shown for illustrative purpose.