In this paper, the estimation of the stress–strength reliability is taken into account when the stress and strength variables have unit Gompertz distributions with a similar scale parameter. The consideration of the unit Gompertz distribution in this context is because of its intriguing symmetric and asymmetric properties that can accommodate various histogram proportional-type data shapes. As the main contribution, the reliability estimate is determined via seven frequentist techniques using the ranked set sampling (RSS) and simple random sampling (SRS). The proposed methods are the maximum likelihood, least squares, weighted least squares, maximum product spacing, Cramér–von Mises, Anderson–Darling, and right tail Anderson–Darling methods. We perform a simulation work to evaluate the effectiveness of the recommended RSS-based estimates by using accuracy metrics. We draw the conclusion that the reliability estimates in the maximum product spacing approach have the lowest value compared to other approaches. In addition, we note that the RSS-based estimates are superior to those obtained by a comparable SRS approach. Additional results are obtained using two genuine data sets that reflect the survival periods of head and neck cancer patients.

The compounded Bell generalized class of distributions is proposed in this article as an alternative to the compounded Poisson generalized family of distributions. Some properties and actuarial measures are presented. The properties of a special model named Bell Weibull (BellW) are obtained such as the linear representation of density, rth moment, incomplete moment, moment generating function using Wright generalized hypergeometric function and Meijer’s G function, the pth moment of order statistics, reliability, stochastic ordering, and residual and reversed residual life. Moreover, some commonly used entropy measures, namely, Rényi, Havrda and Charvat, and Arimoto and Tsallis entropy are obtained for the special model. From the inferential side, parameters are estimated using maximum likelihood estimation. The simulation study is performed to highlight the behavior of estimates. Some actuarial measures including expected shortfall, value at risk, tail value at risk, tail variance, and tail variance premium for the BellW model are presented with the numerical illustration. The usefulness of the proposed family is evaluated using insurance claims and COVID-19 datasets. Convincing results are obtained.

In this paper, we emphasize a new one-parameter distribution with support as $[1,+\infty )$. 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 $[1,+\infty )$. 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.

This paper presents a new probability distribution called the DUS Lindley distribution, created by applying the DUS transformation to the traditional Lindley distribution. The study provides an in-depth analysis of the distribution's statistical properties. These properties include a variety of statistical measures such as the probability density function, cumulative distribution function, failure rate, survival function, reverse hazard function, Mills ratio, mean residual life, mean past life, moments, conditional moments, characteristic function, order statistics, entropy measures, likelihood ratio test and Lorenz and Bonferroni curves. Parameter estimation is performed using several methods including weighted least squares, maximum likelihood estimation, Cramer-Von Mises estimation, least squares and Anderson-Darling estimation. The paper also explores the estimation of system reliability and evaluates the performance of maximum likelihood estimators through simulation studies across different sample sizes. Finally, the DUS Lindley distribution is applied to two real-world datasets, demonstrating a better fit than other well-known distributions.

This article examines the new inverse unit exponential distribution, utilizing both classical and Bayesian methodologies; it begins by presenting the general properties of the proposed model, highlighting characteristic features, such as the presence of a reverse-J or increasing and inverted bathtub-shaped hazard rate function. Furthermore, it explores various statistical properties of the suggested distribution. It employs 12 methods to estimate the associated parameters. A Monte Carlo simulation is conducted to evaluate the accuracy of the estimation procedure. Even for small samples, the results indicate that biases and mean square errors decrease as the sample size increases, demonstrating the robustness of the estimation method. The application of the suggested distribution to real datasets is then discussed. Empirical evidence, using model selection criteria and goodness-of-fit test statistics, supports the assertion that the suggested model outperforms several existing models considered in the study. This article extends its analysis to the Bayesian framework. Using the Hamiltonian Monte Carlo with the no U-turn sampling algorithm, 8000 real samples are generated. The convergence assessment reveals that the chains are convergent and the samples are independent. Subsequently, using the posterior samples, the parameters of the proposed model are estimated, and credible intervals and highest posterior density intervals are computed to quantify uncertainty. The applicability of the suggested model to real data under both classical and Bayesian methodologies provides insights into its statistical properties and performance compared to existing models.

The terrible operating constraints of many real-world events cause systems to malfunction regularly. The failure of systems to perform their intended duties when they reach their lowest, highest, or both extreme operating conditions is a phenomenon that researchers rarely focus on. The multi-stress strength reliability $$R = P(W<X<Z)$$ R = P ( W < X < Z ) is deemed in this study for a component whose strength X falls between two stresses, W, and Z, where X, W, and Z are independently inverted Kumaraswamy distributed. Both maximum likelihood and maximum product spacing procedures are employed to obtain the reliability estimator under simple random sampling (SRS) and ranked set sampling (RSS) methodologies. Four scenarios for reliability estimators are considered. The reliability estimator in the first and second cases can be determined by applying the same sample design (RSS/SRS) to the strength and stress distributions. When the sample data for W and Z originate from RSS while those for X are acquired from SRS, the third reliability estimator is calculated. The drawn data of the strength and stress random variables, which are obtained from SRS and RSS, respectively, are taken into consideration in the final scenario. The effectiveness of the suggested estimators is compared using a comprehensive computer simulation. Lastly, three real data sets have been used to determine reliability estimators.