Inferences of the Multicomponent Stress–Strength Reliability for Burr XII Distributions

Abstract

Multicomponent stress–strength reliability (MSR) is explored for the system with Burr XII distributed components under Type-II censoring. When the distributions of strength and stress variables have Burr XII distributions with common or unequal inner shape parameters, the existence and uniqueness of the maximum likelihood estimators are investigated and established. The associated approximate confidence intervals are obtained by using the asymptotic normal distribution theory along with the delta method and parametric bootstrap procedure, respectively. Moreover, alternative generalized pivotal quantities-based point and confidence interval estimators are developed. Additionally, a likelihood ratio test is presented to diagnose the equivalence of both inner shape parameters or not. Conclusively, Monte Carlo simulations and real data analysis are conducted for illustration.

The Reliability Inference for Multicomponent Stress–Strength Model under the Burr X Distribution

“…And the strength and stress distributions are suggested to have Burr X distributions with different scale parameters for the current monthly capacity applied. It is worth mentioning that the point estimates of R s,k under different parameters for both Burr X distributions were consistent with the point estimate results of R s,k under the Burr XII distribution modeling studied by Lio et al [18], where both Burr XII distributions had one common parameter, while the Burr X modeling had different parameters for the same data sets considered.…”

“…The monthly water capacities in the months of August, September, and December from 1980 to 2015, which were accessed on 19 September 2021, were utilized for the demonstration of the processes presented. The data set was also studied under the Rayleigh distribution and Burr XII one, respectively, by Wang et al [20] and Lio et al [18].…”

“…The work by Shawky and Khan [17] was based on random samples from inverse Weibull distributions. The work by Lio et al [18] was based on type II sample of strength and complete random sample of stress Burr XII distributions. The work by Sauer et al [19] was based on progressively type II censored samples from generalized Pareto distributions.…”

“…The Burr XII distribution has two shape parameters. For more information about Burr XII, readers may refer to Lio et al [18]. The Burr X distribution considered in the current study has two parameters and the probability density function (PDF) and CDF, respectively, are defined as…”

Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch

“…The Burr XII distribution has been the basis of generating many lifetime distributions, such as the beta Burr XII distribution [89] and the Sine Burr distribution [90]. The Burr XII distribution has been applied to a number of applications, such as the multicomponent stress-strength reliability [91] and disease data analysis with competing risks [92,93].…”