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.

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.

Background: : Therapies for many diseases especially cancers have been improved significantly in the recent year, so there have been an increased number of patients who do not experience mortality. In the analysis of these diseases, cure models are used instead of usual survival models. Weibull model and its generalized version, beta Weibull Poisson (BWP), are flexible models in cure models and are used in the present study to analyze breast cancer patients data. Materials and Methods: The data of the present descriptive study are from patients with breast cancer, who referred to Motamed Cancer Institute in Tehran during 1997-2006 and were followed from 2013 to 2017. A total of 270 patients were randomly selected and their individual characteristics were registered. The data were analyzed using Stata 12 and R3. 4. 1 softwares with the the significance level set at 0. 05. Results: The results showed that 43 (15. 9%) of patients deaths occurred after treatment. The rate of cure after one, three, and five years were found to be 0. 99, 0. 87, and 0. 83 respectively. The results of the current study showed that PBW Non-Mixture Cure Model (AIC=427) has better fit compared with weibull (AIC=593). Based on this model, variables of tumor size greater than 5 (P<0. 001) and tumor grade 3 (P<0. 001) are factors affecting patients cured. Also, the cure rate was estimated to be 80%. Conclusion: By studying the effects of factors affecting the occurrence of death, considering the unknown number of causes and using BWP models, the cancer trend can better be analyzed and more accurate information can be available for researchers.

As is well known, estimating parameters of the tree-parameter weibull distribution is a complicated task and sometimes contentious area with several methods vying for recognition. Weibull distribution involves in reliability studies frequently and has many applications in engineering. However estimating the parameters of Weibull distribution is crucial in classical ways. This distribution has three parameters, but for simplicity, a parameter is ridded off and as a result, the estimation of the others will be easily done. When the three-parameter distribution is of interest, the classical estimation procedures such as maximum likelihood estimation (MLE) will be quite boring. In this paper to take advantage of application of artificial neural networks (ANN) to statistics, we propose using a simple neural network to estimate three parameters of Weibull distribution simultaneously. Trained neural network similar to moment method estimates Weibull parameters based on mean, standard deviation, median, skewness and kurtosis of the sample accurately. To assess the power of the proposed method we carry out simulation study and compare the results of the proposed method with real values of the parameters.

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.