In this paper we focus on two topics. Firstly, we propose $U$-statistics for the Weibull distribution parameters. The consistency and asymptotically normality of the introduced $U$-statistics are proved theoretically and by simulations. Several of methods have been proposed for estimating the parameters of Weibull distribution in the literature. These methods include: the generalized least square type 1, the generalized least square type 2, the $L$-moments, the Logarithmic moments, the maximum likelihood estimation, the method of moments, the percentile method, the weighted least square, and weighted maximum likelihood estimation. Secondly, due to lack of a comprehensive comparison between the Weibull distribution parameters estimators, a comprehensive comparison study is made between our proposed $U$-statistics and above nine estimators. In our knowledge, this work is the most comprehensive comparison study for the estimators for the Weibull distribution. Based on simulations, it turns out that different estimators may appeal for different range of the parameters. So, practitioners are allowed to chose the best estimator that is suggested by the goodness-of-fit criteria.

In this paper, a new Dirichlet process mixture model with the generalized inverse Weibull distribution as the kernel is proposed. After determining the prior distribution of the parameters in the proposed model, Markov Chain Monte Carlo methods were applied to generate a sample from the posterior distribution of the parameters. The performance of the presented model is illustrated by analyzing real and simulated data sets, in which some data are right-censored. Another potential of the proposed model demonstrated for data clustering. Obtained results indicate the acceptable performance of the introduced model.

The two-parameter discrete Weibull distribution is an important model especially in reliability studies when the data are reported on a discrete scale‎. ‎The hazard rate function of a discrete Weibull distribution is monotonically increasing and decreasing‎. ‎The present paper provides a family of parametric discrete distributions which is an infinite mixture of exponentiated discrete Weibull distributions‎, ‎and versatile in fitting increasing‎, ‎decreasing‎, ‎and bathtub-shaped failure rate models to different discrete life-test data‎. ‎Some important distributional properties of the model such as the moments‎, ‎order statistics‎, ‎and infinite divisibility are investigated and the parameters of the distribution are estimated by the maximum likelihood method‎. ‎In addition‎, ‎a real data set is analyzed to show the effectiveness of the model‎. ‎Finally we conclude the paper.

In this paper, we propose a new parametric distribution which called as the Beta Modified Weibull Power Series (BMWPS) distribution. This distribution is obtained by compounding Beta Modified Weibull (BMW) and power series distributions. BMWPS distribution contains, as special sub-models, such as Beta Modified Weibull Poisson (BMWP) distribution, Beta Modified Weibull Geometric (BMWG) distribution, Beta Modified Weibull Logarithmic (BMWL) distribution, among others. We obtain closed-form expressions for the cumulative distribution, density, survival function, failure rate function, the r-th raw moment and the moments of order statistics. A full likelihood-based approach that allows yielding maximum likelihood estimates of the BMWPS arameters is used. Finally, application to the Aarset data are given.

Tolerancing is one of the most important tools for planning, controlling, and improving quality in the industry. In order to meet the customer needs and enhance product and service quality, the design engineers use handbooks to determine the tolerance. Although the use of the statistical methods to determine the tolerance is not a new concept, the engineers for this purpose typically use the known statistical distributions such as the normal distribution. However, if the statistical distribution of the variable is unknown, a new statistical method is used. Therefore, we want to offer a flexible and proper statistical method to determine the tolerance of components of a product to enhance its performance. In this regard, Weibull distribution is proposed. To illustrate the proposed method first technical characteristics of production components were selected randomly, and thenmanufacturing parameters were determined using maximum likelihood method. Finally, the Goodness of Fit test was used to ensure the accuracy of the obtained results.

Investigation of reliability characteristics under fuzzy environments is proposed in this paper. Fuzzy Weibull distribution and lifetimes of components are using it described. Formulas of a fuzzy reliability function, fuzzy hazard function and their  a-cut set are presented. Furthermore, the fuzzy functions of series systems and parallel systems are discussed, respectively. Finally, some numerical examples are presented to illustrate how to calculate the fuzzy reliability characteristics and their  a-cut set.

A new generalized distribution called the gamma odd power generalized Weibull-G family of distributions is developed and studied. Some special models of the new family of distribution are explored. Statistical properties of the new family of distributions including the quantile function, ordinary and incomplete moments, probability weighted moments, stochastic ordering, distribution of the order statistics, and Ré, nyi entropy are presented. The maximum likelihood method is used for estimating model parameters, and Monte Carlo simulation is conducted to examine the performance of the model. The flexibility of the new family of distributions is demonstrated by means of two applications to real data sets.

Background: Rapid progression in medical and health sciences have caused survival studies, where some patients have long-term survival, especially for chronic diseases such as breast cancer. Cure models can be applicable to analyze such data. Objectives: The aim of this study was to determine the risk factors associated with breast cancer, using mixture cure fraction model. Methods: We studied data for 438 patients, who were referred to cancer research center in Shahid Beheshti University of Medical Sciences. The patients were visited and treated during 1992 to 2012 and followed-up until October 2014. The data were analyzed by mixture cure fraction model based on GMW (generalized modified Weibull) distribution and inferences were obtained with Bayesian approach, using standard MCMC (Markov Chain Monte Carlo) methods. All analyses were performed, using SPSS v20 and OpenBUGS software. The significant level was considered at 0. 05. Results: During the follow-up period, 75 (17. 12%) deaths occurred by breast cancer and the one-year overall survival rate was 98%. Covariates such as numbers of metastatic lymph nodes and histologic grade were statistically significant. Also, the cure fraction estimation was obtained 58%. Conclusions: When some patients have a long-term survival, cure models can be an interesting model to study survival and these models estimate parameters better than the traditional models such as cox model. In this paper, the mixture cure fraction model based on GMW was fitted for analysing survival times in patients with breast cancer.