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.

In order to produce more flexible models in the reliability theory field, the Bayesian inference of 𝑚-component reliability model with the non-identical-component strengths for MODIFIED WEIBULL DISTRIBUTION under the progressive censoring scheme is considered. One of the key benefits is the generality of this model, so it includes some cases studied previously, such as multi-component stress-strength model with one and two non-identical-component and stress-strength models. In addition, the study of progressive censored data discussed in this paper is critical in many practical situations. The problem is considered in three cases: when the two common parameters for strengths and stress variables are unknown, known, and general. In each case, the approximation methods, such as the MCMC and Lindley’s approximation, are used to consider the -component stress-strength parameter. The Monte Carlo simulation study compares the performance of different methods—finally, a demonstration of how the proposed model may be utilized to analyse real data sets.

Recently, reliability has become a critical criterion for product quality and decision-making that covers a wide range of subjects, including failure analysis systems. Performing a reliability analysis is essential for the study of operating safety in industrial systems. In this study, we list evaluation methods and perform real-time reliability analyses. The real-time reliability modeling of a Reverse Osmosis system is addressed in this paper. The model will help create effective maintenance while extending the subsystems' lifespan. To achieve our goal, we suggested the 2-parameter MODIFIED WEIBULL DISTRIBUTION. The simulation was performed using Maple software. The evaluation for each subsystem was displayed in the result and analyses section. The conclusion, however, draws a broad conclusion about the study.

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.

This paper describes the point and interval estimation of the unknown parameters of MODIFIED WEIBULL DISTRIBUTION under the adaptive Type-II progressive censored samples‎. ‎First‎, ‎we obtain the maximum likelihood estimation of parameters‎. ‎Because maximum likelihood estimations should be solved in numerical methods and cannot be derived in a closed form‎, ‎the approximate maximum likelihood estimations of the parameters are achieved‎. ‎Also‎, ‎asymptotic confidence intervals are obtained by earning the asymptotic DISTRIBUTION of the parameters‎. ‎Moreover‎, ‎two bootstrap confidence intervals are derived‎. ‎Second‎, ‎the Bayesian estimation of parameters is approximated using the Markov chain Monte Carlo algorithm and Lindley's method‎. ‎Furthermore‎, ‎the highest posterior density credible intervals of the parameters are derived‎. ‎Finally‎, ‎the different proposed estimations have been compared by the simulation studies and one data set is analyzed to illustrative aims.

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.

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.

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.