Bayesian Inference in Dependent Right Censoring

In this article, we consider dependent right censoring when the lifetime and censoring variables have a Marshall–Olkin bivariate exponential distribution and obtain MLEs, MMEs and UMVUEs of the unknown parameters. The Bayes estimators as well as the Posterior Regret Gamma Minimax (PRGM) estimators of the parameters of interest under the SEL function are also obtained and a Monte Carlo simulation study is carried out to compare these estimators.     

Analysis of incomplete data in the presence of dependent competing risks from Marshall–Olkin bivariate Weibull distribution under progressive hybrid censoring

This article considers the statistical analysis of dependent competing risks model with incomplete data under Type-I progressive hybrid censored condition using a Marshall–Olkin bivariate Weibull distribution. Based on the expectation maximum algorithm, maximum likelihood estimators for the unknown parameters are obtained, and the missing information principle is used to obtain the observed information matrix. As the maximum likelihood approach may fail when the available information is insufficient, Bayesian approach incorporated with auxiliary variables is developed for estimating the parameters of the model, and Monte Carlo method is employed to construct the highest posterior density credible intervals. The proposed method is illustrated through a numerical example under different progressive censoring schemes and masking probabilities. Finally, a real data set is analyzed for illustrative purposes.     

On an Extension of the Exponentiated Weibull Distribution

In this article, the exponentiated Weibull distribution is extended by the Marshall-Olkin family. Our new four-parameter family has a hazard rate function with various desired shapes depending on the choice of its parameters and, thus, it is very flexible in data modeling. It also contains two mixed distributions with applications to series and parallel systems in reliability and also contains several previously known lifetime distributions. We shall study some basic distributional properties of the new distribution. Some closed forms are derived for its moment generating function and moments as well as moments of its order statistics. The model parameters are estimated by the maximum likelihood method. The stress–strength parameter and its estimation are also investigated. Finally, an application of the new model is illustrated using two real datasets.     

Comparison of estimation methods for the Marshall–Olkin extended Lindley distribution

The aim of this paper is to compare the parameters' estimations of the Marshall–Olkin extended Lindley distribution obtained by six estimation methods: maximum likelihood, ordinary least-squares, weighted least-squares, maximum product of spacings, Cramér–von Mises and Anderson–Darling. The bias, root mean-squared error, average absolute difference between the true and estimate distributions' functions and the maximum absolute difference between the true and estimate distributions' functions are used as comparison criteria. Although the maximum product of spacings method is not widely used, the simulation study concludes that it is highly competitive with the maximum likelihood method.     

The Weibull Marshall–Olkin family: Regression model and application to censored data

We introduce a new class of distributions called the Weibull Marshall–Olkin-G family. We obtain some of its mathematical properties. The special models of this family provide bathtub-shaped, decreasing-increasing, increasing-decreasing-increasing, decreasing-increasing-decreasing, monotone, unimodal and bimodal hazard functions. The maximum likelihood method is adopted for estimating the model parameters. We assess the performance of the maximum likelihood estimators by means of two simulation studies. We also propose a new family of linear regression models for censored and uncensored data. The flexibility and importance of the proposed models are illustrated by means of three real data sets.     

A new useful three-parameter extension of the exponential distribution

A three-parameter extension of the exponential distribution is introduced and studied in this paper. The new distribution is quite flexible and can be used effectively in modelling survival data, reliability problems, fatigue life studies and hydrological data. It can have constant, decreasing, increasing, upside-down bathtub (unimodal), bathtub-shaped and decreasing–increasing–decreasing hazard rate functions. We provide a comprehensive account of the mathematical properties of the new distribution and various structural quantities are derived. We discuss maximum likelihood estimation of the model parameters for complete sample and for censored sample. An empirical application of the new model to real data is presented for illustrative purposes. We hope that the new distribution will serve as an alternative model to other models available in the literature for modelling real data in many areas.     

The Marshall–Olkin extended generalized Rayleigh distribution: Properties and applications

In this paper, a new extension for the generalized Rayleigh distribution is introduced. The proposed model, called Marshall–Olkin extended generalized Rayleigh distribution, arises based on the scheme introduced by Marshall and Olkin (1997) Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]. A comprehensive account of the mathematical properties of the new distribution is provided. We discuss about the estimation of the model parameters based on two estimation methods. Empirical applications of the new model to real data are presented for illustrative purposes.     

On the Marshall–Olkin extended distributions

ABSTRACT

A general method of introducing a new parameter to a well-established continuous baseline cumulative function G to obtain more flexible distributions was proposed by Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]). This new family is known as Marshall–Olkin extended G family of distributions. In this article, we characterize this family as mixtures of the distributions of the minimum and maximum of random variables with cumulative function G. We demonstrate that the coefficients of the mixtures are probabilities of random variables with geometric distributions. Additionally, we present new representations for the density and cumulative functions of this class of distributions. Further, we introduce a new three-parameter continuous model for modeling rates and proportions based on the Marshall–Olkin's method. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of a real dataset.     

Statistical Inference Based on Progressively Type II Censored Data from Weibull Model

In this article, we consider the problem of estimating the shape and scale parameters and predicting the unobserved removed data based on a progressive type II censored sample from the Weibull distribution. Maximum likelihood and Bayesian approaches are used to estimate the scale and shape parameters. The sampling-based method is used to draw Monte Carlo (MC) samples and it has been used to estimate the model parameters and also to predict the removed units in multiple stages of the censored sample. Two real datasets are presented and analyzed for illustrative purposes and Monte carlo simulations are performed to study the behavior of the proposed methods.     

The Marshall–Olkin extended generalized Lindley distribution: Properties and applications

In this article, we define and study a new three-parameter model called the Marshall–Olkin extended generalized Lindley distribution. We derive various structural properties of the proposed model including expansions for the density function, ordinary moments, moment generating function, quantile function, mean deviations, Bonferroni and Lorenz curves, order statistics and their moments, Rényi entropy and reliability. We estimate the model parameters using the maximum likelihood technique of estimation. We assess the performance of the maximum likelihood estimators in a simulation study. Finally, by means of two real datasets, we illustrate the usefulness of the new model.     

Estimation on dependent right censoring scheme in an ordinary bivariate geometric distribution

Discrete lifetime data are very common in engineering and medical researches. In many cases the lifetime is censored at a random or predetermined time and we do not know the complete survival time. There are many situations that the lifetime variable could be dependent on the time of censoring. In this paper we propose the dependent right censoring scheme in discrete setup when the lifetime and censoring variables have a bivariate geometric distribution. We obtain the maximum likelihood estimators of the unknown parameters with their risks in closed forms. The Bayes estimators as well as the constrained Bayes estimates of the unknown parameters under the squared error loss function are also obtained. We considered an extension to the case where covariates are present along with the data. Finally we provided a simulation study and an illustrative example with a real data.     

An Estimator of the Survival Function Basedon the Semi-Markov Model Under Dependent Censorship

Lee and Wolfe (Biometrics vol. 54 pp. 1176–1178, 1998) proposed the two-stage sampling design for testing the assumption of independent censoring, which involves further follow-up of a subset of lost-to-follow-up censored subjects. They also proposed an adjusted estimator for the survivor function for a proportional hazards model under the dependent censoring model. In this paper, a new estimator for the survivor function is proposed for the semi-Markov model under the dependent censorship on the basis of the two-stage sampling data. The consistency and the asymptotic distribution of the proposed estimator are derived. The estimation procedure is illustrated with an example of lung cancer clinical trial and simulation results are reported of the mean squared errors of estimators under a proportional hazards and two different nonproportional hazards models.     

On the dependent competing risks using Marshall–Olkin bivariate Weibull model: Parameter estimation with different methods

Competing risks models are of great importance in reliability and survival analysis. They are often assumed to have independent causes of failure in literature, which may be unreasonable. In this article, dependent causes of failure are considered by using the Marshall–Olkin bivariate Weibull distribution. After deriving some useful results for the model, we use ML, fiducial inference, and Bayesian methods to estimate the unknown model parameters with a parameter transformation. Simulation studies are carried out to assess the performances of the three methods. Compared with the maximum likelihood method, the fiducial and Bayesian methods could provide better parameter estimation.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.     

Bayes Estimation Based on Joint Progressive Type II Censored Data Under LINEX Loss Function

In the lifetime experiments, the joint censoring scheme is useful for planning comparative purposes of two identical products manufactured coming from different lines. In this article, we will confine ourselves to the data obtained by conducting a joint progressive Type II censoring scheme on the basis of the two combined samples selected from the two lines. Moreover, it is supposed that the distributions of lifetimes of the two products satisfy in a proportional hazard model. A general form for the distributions is considered, and we tackle the problem of obtaining Bayes estimates under the squared error and linear-exponential (LINEX) loss functions. As a special case, the Weibull distribution is discussed in more detail. Finally, the estimated risks of the various estimators obtained are compared using the Monte Carlo method.     

Bayesian Inference in Marshall–Olkin Bivariate Exponential Shared Gamma Frailty Regression Model under Random Censoring

Many analyses in the epidemiological and the prognostic studies and in the studies of event history data require methods that allow for unobserved covariates or “frailties”. We consider the shared frailty model in the framework of parametric proportional hazard model. There are certain assumptions about the distribution of frailty and baseline distribution. The exponential distribution is the commonly used distribution for analyzing lifetime data. In this paper, we consider shared gamma frailty model with bivariate exponential of Marshall and Olkin (1967 Marshall, A.W., Olkin, I. (1967). A multivariate exponential distribution. J. Am. Stat. Assoc. 62:30–44.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) distribution as baseline hazard for bivariate survival times. We solve the inferential problem in a Bayesian framework with the help of a comprehensive simulation study and real data example. We fit the model to the real-life bivariate survival data set of diabetic retinopathy data. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model and then compare the true values of the parameters with the estimated values for different sample sizes.     

On the Additive Weibull Distribution

We derive explicit expressions for the moments, incomplete moments, quantile function and generating function of the additive Weibull model pioneered by Xie and Lai (1995 Xie, M., Lai, C.D. (1995). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safety 52:87–93.[Crossref], [Web of Science ®] , [Google Scholar]), which is a quite flexible distribution for fitting lifetime data with bathtub-shaped failure rate function. In addition, we estimate the model parameters by maximum likelihood and determine the observed information matrix. The flexibility of the additive Weibull distribution is illustrated by means of one application to real data.     

Inference Under Right Censoring in a Discrete Setup

In this article, we consider the right random censoring scheme in a discrete setup when the lifetime and censoring variables are independent and have geometric distributions with means 1/θ₁ and 1/θ₂, respectively. We first obtain the Maximum Likelihood and Method of Moment estimators of the unknown parameters. We also find the Bayes and Posterior Regret Gamma Minimax estimators of the parameters for the two cases when the prior distributions are dependent and independent, assuming a squared error loss function. We then discuss the Proportional Hazard model, and obtain Maximum Likelihood estimators of the unknown parameters and derive the Bayes estimators assuming squared error loss using Markov Chain Monte Carlo methods.