Abhimanyu Singh Yadav

Process capability indices () are essential tools for assessing how well a process aligns with predefined quality standards. While larger values are often associated with higher process performance, they do not necessarily guarantee lower rejection rates. This work introduces a new formulated through the quantile function, particularly suited for processes that deviate from normality. The investigation centers on processes described by the power Rayleigh distribution with unknown parameters and pursues four main objectives. First, six distinct frequentist estimation techniques are proposed for the new PCI under this distributional framework. Second, the efficiency of three parametric bootstrap confidence intervals () is examined in estimating the index. Third, Bayesian estimation is carried out under two alternative loss functions ‐ the squared error loss and the linear exponential loss assuming gamma priors for the parameters. Interval estimation is addressed through approximate and bootstrap‐based confidence intervals in the frequentist setting and through highest posterior density credible intervals in the Bayesian setting. Fourth, the statistical power of the is investigated via the equivalence between confidence interval construction and two‐sided hypothesis testing. Their discriminative capacity is further explored using receiver operating characteristic curve analysis. Finally, a detailed Monte Carlo simulation is designed to evaluate the competing estimators with respect to mean squared error, coverage probability, and average interval length. To highlight the practical value of the proposed approaches, two case studies based on real data from the electronic and food industries are presented.

1. In the field of lifetime and reliability analysis, selecting an appropriate probability distribution is vital for accurately modelling the underlying failure mechanisms of components, systems, o...

This paper introduces a new family of lifetime distributions, termed the alpha power logarithmic transformed family (APLo-F), and investigates its genesis and key distributional characteristics. A particular member of this family, the alpha power logarithmic transformed exponential (APLo-E) distribution, is derived by applying the proposed transformation to the one-parameter exponential distribution. The structural behavior of the APLo-E distribution is examined in detail, and mathematical conditions governing its hazard rate function (HRF) are established. A comprehensive analytical study is carried out to derive several statistical properties, including moments, conditional moments, generating functions, entropies, mean deviation, order statistics, and inequality measures such as the Bonferroni and Lorenz curves. The parameters of the proposed model are estimated using the well-known maximum likelihood estimation (MLE) technique, and the reliability and hazard functions are further estimated through the invariance property of MLE. A Monte Carlo simulation study is performed to assess the consistency and efficiency of the estimators. Finally, the practical applicability of the proposed distribution is illustrated using reliability and survival dataset from diverse application domains. The results demonstrate the flexibility, reliability, and superior performance of the APLo-E distribution in modeling time-to-event data. Overall, this study contributes a novel class of lifetime distributions and provides robust theoretical and practical tools for analyzing reliability and survival phenomena.

The meteoric developments of cutting‐edge technologies in the modern era expedited the advancements of infrastructure in the industrial and biomedical sectors. As a result, the lifetimes of products or patients have considerably increased. In practice, lifetime data are usually modeled with a probability distribution to capture the inherent randomness in the data. When life‐testing experiments produce data containing extreme observations, resorting to heavy‐tailed distributions is often an immediate choice for the researchers. In this paper, log‐Cauchy, a super‐heavy‐tailed distribution, is explored as a lifetime model in the context of analyzing survival and reliability data, and some characterizations of this lifetime distribution are discussed. Further, the considered distribution is examined under a progressive first‐failure type‐II censoring, a technique known for reducing the test completion time when products are highly reliable. Apart from the maximum likelihood estimators, the Bayes estimators of the distribution parameters with dependent priors under a symmetric loss function are investigated under the considered censoring scheme. The estimators of the reliability and hazard functions have also been obtained in this study. Moreover, a quantile‐based measure of experiment completion time has been discussed. The performance of the estimators and the behavior of the experiment completion time have been evaluated through a Monte Carlo study. Additionally, a prediction technique has been developed to predict the potential lifetimes of the censored units. Finally, a reliability and survival data set has been used to illustrate the real‐life applicability of the proposed distribution and the associated procedures.

This article proposes the Bayes estimation of the parameter and reliability function for xgamma distribution in the presence of type-I hybrid censored observations. The Bayes estimate of the parameter has been obtained by assuming informative and non-informative priors using general entropy loss function. Obviously, censoring adds difficulties in estimation procedure; hence the Bayes estimators computed with type-I hybrid censored observation under the mentioned prior often do not assume any standard form. Therefore, Bayes estimates are computed using Tierney-Kadane approximation and Markov Chain Monte Carlo numerical technique. Further, different interval estimates namely asymptotic confidence interval, bootstrap confidence interval and highest posterior density interval along with the width of the interval and coverage probability are also discussed. The maximum likelihood estimate for the same has also been computed using non- linear maximization iterative procedure and compared with corresponding Bayes estimates using Monte Carlo simulations. The comparison of the estimators are made in terms of average loss over whole sample space and corresponding length of the interval. lastly, one medical data set has been considered for the real application of the proposed study.