Model misspecification of Log-Normal and Birnbaum-Saunders distributions

Model misspecification can be a serious issue in any lifetime data analysis. Assumption of the correct model is very important particularly in prediction for future observations and for estimating the tail probabilities of any lifetime distribution. In this paper we have considered the model misspecification of the log-normal and Birnbaum-Saunders distributions. These two distributions have striking similarities both in terms of the probability density functions and hazard functions, in certain range of parameter values, which makes it extremely difficult to detect the correct model. Hence, to identify the correct model, first the conventional method of ratio of maximized likelihood approach has been tried. The necessary theoretical results have been derived. Based on extensive simulations it has been observed that for certain range of parameter values the model misspecification can be quite high even for very large sample sizes. Some of the other methods like the ratio of maximized product of spacings and minimized Kolmogorov-Smirnov distance, have also been explored. But none of the method performs uniformly better than the other over the entire range of the parameter space. Some counter intuitive results have also been obtained. Hence, we finally conclude that it is extremely difficult to choose the correct model in case of log-normal and Birnbaum-Saunders distribution for certain range of parameter values. Finally, we have suggested a modified ratio of maximized likelihood approach, and the performances are quite satisfactory.