A generalization of the Marshall–Olkin family of distributions is developed using negative binomial compounding instead of geometric compounding where addition is replaced by minimum of a random number of observations X ₁,X ₂,...,X_N . Here, we consider the Rayleigh distribution and extend it to obtain a Negative Binomial Marshall–Olkin Rayleigh Distribution. Various properties of the new family are investigated. Maximum likelihood estimates are obtained. The use of the model in lifetime modeling is established by fitting it to a real data set on remission times of bladder cancer patients. Also we try to develop a reliability test plan for acceptance or rejection of a lot of products submitted for inspection with lifetimes governed by this distribution.

In this article, a generalization of the Weibull distribution called Harris extended Weibull distribution is studied, and its properties are discussed. We fit the distribution to a real-life data set to show the applicability of this distribution in reliability modeling. Also, we derive a reliability test plan for acceptance or rejection of a lot of products submitted for inspection with lifetimes following this distribution. The operating characteristic functions of the sampling plans are obtained. The producer’s risk, minimum sample sizes and associated characteristics are computed and presented in tables. The results are illustrated using two data sets on ordered failure times of products as well as failure times of ball bearings.

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C _pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.