Stress–strength reliability models involving generalized gamma and Weibull distributions

This paper deals with the estimation of the stress–strength reliability R = P(X < Y), when X and Y follow (1) independent generalized gamma (GG) distributions with only a common shape parameter and (2) independent Weibull random variables with arbitrary scale and shape parameters and generalize the proposal from Kundu and Gupta (2006), Kundu and Raqab (2009) and Ali et al. (2012).

First, a closed form expression for R is derived under the conditions (1) and (2). Next, sufficient conditions are given for the convergence of the infinite series expansions used to calculate the value of R in case (2). The models GG and Weibull are fitted by maximum likelihood using Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton method. Confidence intervals and standard errors are calculated using bootstrap. For illustration purpose, two real data sets are analyzed and the results are compared with the existing recent results available in the literature.

The proposed approaches improve the estimation of the R by not using transformations in the data and flexibilize the modeling with Weibull distributions with arbitrary scale and shape parameters.

The proposals of the paper eliminate the misestimation of R caused by subtracting a constant value from the data (Kundu and Raqab, 2009) and treat the estimation of R in a more adequate way by using the Weibull distributions without restrictions in the parameters. The two cases covered generalize a number of distributions and unify a number of stress–strength probability P(X < Y) results available in the literature.