When conducting reliability studies, the progressive first-failure censoring (PFFC) method is useful in situations in which the units of the life testing experiment are separated into groups consisting of k units each with the intention of seeing only the first failure in each group. Using progressive first-failure censored samples, the statistical inference for the parameters, reliability, and hazard functions of the extended Rayleigh distribution (ERD) are investigated in this study. The asymptotic normality theory of maximum likelihood estimates (MLEs) is used in order to acquire the maximum likelihood estimates (MLEs) together with the asymptotic confidence intervals (Asym. CIs). Bayesian estimates (BEs) of the parameters and the reliability functions under different loss functions may be produced by using independent gamma informative priors and non-informative priors. The Markov chain Monte Carlo (MCMC) approach is used so that Bayesian computations are performed with ease. In addition, the MCMC method is used in order to create credible intervals (Cred. CIs) for the parameters, which may be used for either informative or non-informative priors. Additionally, computations for the reliability functions are carried out. A Monte Carlo simulation study is carried out in order to provide a comparison of the behaviour of the different estimations that were created for this work. At last, an actual data set is dissected for the purpose of providing an example.
In this paper, we emphasize a new one-parameter distribution with support as [1,+∞). It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as the mode, quantiles, stochastic dominance, heavy-tailed nature, moments, etc. Among the strengths of the distribution are the following: (i) the closed-form expressions and flexibility of the main functions, and in particular, the probability density function is unimodal and the hazard rate function is increasing or unimodal; (ii) the manageability of the moments; and, more importantly, (iii) it provides a real alternative to the famous Pareto distribution, also with support as [1,+∞). Indeed, the proposed distribution has different functionalities but also benefits from the heavy-right-tailed nature, which is demanded in many applied fields (finance, the actuarial field, quality control, medicine, etc.). Furthermore, it can be used quite efficiently in a statistical setting. To support this claim, the maximum likelihood, Anderson–Darling, right-tailed Anderson–Darling, left-tailed Anderson–Darling, Cramér–Von Mises, least squares, weighted least-squares, maximum product of spacing, minimum spacing absolute distance, and minimum spacing absolute-log distance estimation methods are examined to estimate the unknown unique parameter. A Monte Carlo simulation is used to compare the performance of the obtained estimates. Additionally, the Bayesian estimation method using an informative gamma prior distribution under the squared error loss function is discussed. Data on the COVID mortality rate and the timing of pain relief after receiving an analgesic are considered to illustrate the applicability of the proposed distribution. Favorable results are highlighted, supporting the importance of the findings.
In many real-world situations, systems frequently fail due to demanding operating conditions. In particular, when systems reach their lowest, highest, or both extremes operating conditions, they usually fail to accomplish their intended functions. This study considers estimating the stress–strength reliability, for a component with a strength (X) that is independent of the opposing lower bound stress (Y) and upper bound stress (Z). We assumed that the strength and stress random variables followed a generalized inverse exponential distribution with different shape parameters. Under ranked set sampling (RSS) and simple random sampling (SRS) designs, we obtained four reliability estimators using the maximum likelihood method. The first and second reliability estimators were deduced when the sample data of the strength and stress distributions used the sample design (RSS/SRS). The third reliability estimator was determined when the sample data for Y and Z were received from the RSS and the sample data for X were taken from the SRS. The fourth reliability estimator was derived when the sample data of Y and Z were selected from the SRS, while the sample data of X were taken from the RSS. The accuracy of the suggested estimators was compared using a comprehensive computer simulation. Lastly, three real data sets were used to determine the reliability.
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