It is highly common in many real-life settings for systems to fail to perform in their harsh operating environments. When systems reach their lower, upper, or both extreme operating conditions, they frequently fail to perform their intended duties, which receives little attention from researchers. The purpose of this article is to derive inference for multi reliability where stress-strength variables follow unit Kumaraswamy distributions based on the progressive first failure. Therefore, this article deals with the problem of estimating the stress-strength function, R when $X,Y$, and Z come from three independent Kumaraswamy distributions. The classical methods namely maximum likelihood for point estimation and asymptotic, boot-p and boot-t methods are also discussed for interval estimation and Bayes methods are proposed based on progressive first-failure censored data. Lindly’s approximation form and MCMC technique are used to compute the Bayes estimate of R under symmetric and asymmetric loss functions. We derive standard Bayes estimators of reliability for multi stress–strength Kumaraswamy distribution based on progressive first-failure censored samples by using balanced and unbalanced loss functions. Different confidence intervals are obtained. The performance of the different proposed estimators is evaluated and compared by Monte Carlo simulations and application examples of real data.

This article discusses the problem of estimation with step stress partially accelerated life tests using Type-II progressively censored samples. The lifetime of items under use condition follows the two-parameters inverted Kumaraswamy distribution. The maximum likelihood estimates for the unknown parameters are computed numerically. Using the property of asymptotic distributions for maximum likelihood estimation, we constructed asymptotic interval estimates. The Bayes procedure is used to calculate estimates of the unknown parameters from symmetrical and asymmetric loss functions. The Bayes estimates cannot be obtained explicitly, therefor the Lindley's approximation and the Markov chain Monte Carlo technique are used to obtaining the Bayes estimates. Furthermore, the highest posterior density credible intervals for the unknown parameters are calculated. An example is presented to illustrate the methods of inference. Finally, a numerical example of March precipitation (in inches) in Minneapolis failure times in the real world is provided to illustrate how the approaches will perform in practice.

The current work focuses on ranked set sampling and a simple random sample as sampling approaches for determining stress–strength reliability from the inverted Topp–Leone distribution. Asymptotic confidence intervals are established, along with a maximum likelihood estimator of the parameters and stress–strength reliability. The reliability of such a system is assessed using the Bayesian approach under symmetric and asymmetric loss functions. The highest posterior density credible interval is constructed successively. The results are extracted using Monte Carlo simulation to compare the proposed estimators performance with different sample sizes. Finally, by looking at waiting time data and failure times of insulating fluid, the usefulness of the suggested technique is demonstrated.

In many real-life scenarios, systems frequently perform badly in difficult operating situations. The multiple failures that take place when systems reach their lower, higher, or extreme functioning states typically receive little attention from researchers. This study uses generalized progressive hybrid censoring to discuss the inference of $R=P(X<Y<Z)$ for a component when it is exposed to two stresses, $Y,Z$, and it has one strength X that is regarded. We assume that both the stresses and strength variables follow an exponentiated exponential distribution with a common scale parameter. We obtain R’s maximum likelihood estimator and approximate confidence intervals. In addition, the Bayesian estimators for symmetric, such as squared error, and asymmetric loss functions, such as linear exponential, are developed. Credible intervals with the highest posterior densities are established. Monte Carlo simulations are used to evaluate and compare the effectiveness of the many proposed estimators. The process is then precisely described using an analysis of real data.

Based on independent progressive type-II censored samples from two-parameter Burr-type XII distributions, various point and interval estimators of $\delta =P(Y<X)$ were proposed when the strength variable was subjected to the step–stress partially accelerated life test. The point estimators computed were maximum likelihood and Bayesian under various symmetric and asymmetric loss functions. The interval estimations constructed were approximate, bootstrap-P, and bootstrap-T confidence intervals, and a Bayesian credible interval. A Markov Chain Monte Carlo approach using Gibbs sampling was designed to derive the Bayesian estimate of $\delta$. Based on the mean square error, bias, confidence interval length, and coverage probability, the results of the numerical analysis of the performance of the maximum likelihood and Bayesian estimates using Monte Carlo simulations were quite satisfactory. To support the theoretical component, an empirical investigation based on two actual data sets was carried out.