An Empirical Research for Likelihood-free Parameter Estimation Approach for NHPP-Based Software Reliability Models

The non-homogeneous Poisson process (NHPP) is the most widely used stochastic counting model in software reliability. The maximum likelihood estimation (MLE) is useful when the likelihood function of NHPP is available. However, it is well-known that the MLE is bound to fail in the J-shaped distributions, such as in the Weibull and gamma distribution when the shape parameter is less than 1. The maximum likelihood estimator also does not exist inside the parameter space with positive probability. The maximum likelihood equations of the NHPP-based SRM cannot be solved, if and only if the observed time to last software failure is less than twice the mean observed-time-to-failure. Furthermore, in some generalized software reliability models, it is quite hard to obtain the likelihood function in a closed form. Therefore, we apply a likelihood-free estimation approach on NHPP-based software reliability models with finite mean value function. Our method is motivated by the maximum product of spacing estimation which provides the parameter estimation without the likelihood function and intensity function. In contrast to existing likelihood-free parameter estimation methods, such as least squares estimation, our method can yield an estimator that is consistent with the underlying NHPP probability law. We have demonstrated the predictive performance of our method through real-data analysis.