A parallel system is one of the special redundant systems that industrial systems frequently use to increase reliability and prevent unexpected failures. In this paper, a new two-parameter model called the Poisson Rayleigh distribution (PRD) is studied. Some of its statistical properties are given. Particularly, we emphasize the study of the stress–strength (SS) reliability parameter, R = p(Y < X), when X and Y have a PRD. Maximum likelihood, maximum product spacing, and Bayesian strategies are utilized to estimate the parameters. Maximum likelihood, maximum product spacing, and Bayesian techniques for R are computed. To assess how each estimation method performs, a simulation study is conducted. In order to demonstrate the adaptability of the suggested model, its goodness of fit for the PRD comparison with other models is demonstrated by application to real datasets. Finally, the SS model for the PRD was applied with two applications of real data depicting the failure times for two types of electrical insulators and pertaining to customer wait times at two banks.

This paper investigates the estimation of the stress–strength reliability parameter $R=P(Y<X)$ when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for $\mathit{R}$ is explicitly derived, enabling effective and precise calculation without numerical integration. The Garhy distribution, a flexible one-parameter lifetime model with an increasing hazard function, is confirmed by full-scale goodness-of-fit diagnostics. A Bayesian estimation model is trained on non-informative priors (normal and extended Jeffreys priors) under squared error loss. The posterior expectations are analytically intractable; we adopt two complementary methods of computation: (i) Markov Chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm and (ii) the Tierney–Kadane (TK) approximation, which provides extremely precise analytical estimates with significantly reduced computational burden. Monte Carlo simulations are large-scale and compare the proposed estimators under different censoring schemes, sample sizes, and parameter configurations in terms of bias and mean squared error (MSE). The methodology is further applied to a real medical dataset comprising kidney dialysis patient survival times, demonstrating its practical relevance in clinical reliability assessment. Results consistently indicate that Bayesian methods, particularly with the extended Jeffreys prior, outperform classical MLEs in terms of stability and accuracy, especially under heavy censoring. Moreover, the TK approximation yields estimates virtually identical to MCMC while requiring only a fraction of the computational effort. We further extend the TK framework to approximate the posterior variance of R and the expected log-likelihood, providing a fully analytical alternative to MCMC for comprehensive Bayesian inference.

This paper investigates the stress–strength reliability parameter $R=P(Y<X)$ for the Garhy distribution under upper-, lower-, and mixed-record schemes. The Garhy distribution, a flexible one-parameter lifetime model, is shown to provide an excellent fit to precipitation and medical (kidney dialysis) data, with Kolmogorov–Smirnov p-values of 0.9904 and 0.9072, respectively. Maximum likelihood estimators (MLEs) of R are developed, alongside Bayesian estimators using Jeffreys and extended Jeffreys priors under squared error loss. Markov chain Monte Carlo (MCMC) methods are employed for posterior inference. An extensive Monte Carlo simulation study reveals that: (i) MLEs converge under all record schemes but exhibit larger bias and lower efficiency compared to Bayesian estimators; (ii) Bayesian estimators, in contrast, demonstrate superior stability, lower mean squared error, and better bias control, especially for pure upper and lower records; (iii) mixed records consistently yield the most balanced and reliable estimates, capturing distributional information more effectively than single-type records; and (iv) the extended Jeffreys prior provides effective bias correction, outperforming the standard Jeffreys prior in many scenarios. Analysis of real-world datasets confirms the simulation findings, with Bayesian estimators under mixed records producing stable and accurate reliability estimates, outperforming MLE in terms of precision and stability. The results strongly advocate for Bayesian methodology with non-informative priors when assessing stress–strength reliability from record-breaking data.

This paper addresses the estimation of the multi-component stress–strength reliability when both the strength variables and the stress variable follow the one-parameter Garhy distribution. Data are assumed to arise from a unified hybrid censoring scheme, which generalizes both Type-I and Type-II hybrid censoring. A closed-form expression for the reliability parameter $R_{m,k}=P(\mathrm{at}\mathrm{least}m\mathrm{of}(X_1,\dots ,X_k)>Y)$ is derived, enabling efficient computation. Three estimation procedures are developed: maximum likelihood estimation (MLE), Bayesian inference using Markov chain Monte Carlo (MCMC) with non-informative priors, and the Tierney–Kadane Laplace-type approximation for posterior moments. For each method, we provide complete mathematical derivations, including the likelihood function under unified hybrid censoring, the posterior conditionals, and the asymptotic distribution of the reliability via the Delta method. Furthermore, Bayesian estimation is extended to asymmetric loss functions, and posterior propriety is formally proven. To check the suitability of the proposed methods, a real data application on generator failure times in power systems is presented.

We introduce the modified exponential root distribution (MERD), a one-parameter lifetime model on $ [1, \infty) $ defined as the square of a unit-shifted exponential variate. Being a monotone transform of an exponential variable, MERD admits closed-form distribution, survival, hazard (strictly decreasing), cumulative and reversed hazard, quantile, and mean-residual-life functions, along with closed-form moments, quantile-based shape measures, order statistics, entropies, and stress-strength reliability via a single root substitution. Since the transformed data are exactly exponential, the pivot $ 2\lambda S\sim\chi^2_{2n} $ gives exact confidence intervals for the rate parameter at any sample size, unlike most comparably simple lifetime models. We derive closed-form maximum-likelihood and conjugate-gamma Bayesian estimators and evaluate their bias, mean squared error (MSE), and coverage via simulation. The MERD exhibits competitive performance with respect to one- and two-parameter models on the parsimony-penalized information criteria when evaluated on empirical datasets of biomedical survival and actuarial science, while also providing exact inference. We emphasize that these findings are dataset-specific and do not constitute a claim of universal superiority over all competing models; rather, they illustrate the practical utility of the MERD as a tractable and exactly-inferable option for bounded decreasing-hazard data. To address the fixed lower bound and scale, we develop an identifiable three-parameter location-scale extension, show its threshold is estimated by the sample minimum, and examine how far the exact-inference properties persist once threshold and scale are estimated.

We develop an explicit step–stress cumulative exposure model (CEM) formulation with closed–form equivalent times, analytic score functions, and analytic observed information for the one–parameter inverted Z–Lindley (IZL) lifetime distribution under progressive Type–Ⅱ censoring. Because the IZL quantile inverts in closed form via the Lambert $ W $ function, the equivalent start time linking the two stress stages is itself closed form, giving an explicit rather than implicit likelihood, from which we derive the exact score, prove strict concavity of the constant–stress log–likelihood for complete samples, and obtain the analytic observed information matrix, validated against finite differences. The construction extends to a multiple–step generalization; numerical change–time optimization under $ c $–, $ D $–, and $ A $–optimality for a representative complete–sample design; and a link–function extrapolation to an untested design stress with delta–method variance. Inference is triangulated across maximum likelihood, Wald, percentile bootstrap, and gamma–prior Bayesian analysis, cross–checked by the Tierney–Kadane approximation. A Monte Carlo study shows finite-sample bias and mean squared error (MSE) decreasing with sample size and near-nominal coverage for the Wald, Bayesian, and bootstrap intervals in most scenarios as well as accurate design–stress extrapolation under a correctly specified link, which degrades sharply once the link is misspecified. The principal estimation and uncertainty–quantification procedures are illustrated using a real, Type–Ⅰ–censored step–stress data set, and design–stress extrapolation is assessed separately through a four–level simulation; the real–data example thus validates the broader CEM/IZL modeling and inference approach rather than the progressive Type–Ⅱ censoring likelihood itself. On a small unmanned aerial vehicle (SUAV) reliability data set, a bootstrap–calibrated goodness–of–fit test finds no evidence against the fitted model, and IZL is among a small cluster of equally well–supported models by AICc – supporting IZL as a credible, closed-form candidate for step–stress reliability analysis.