Kernel density estimations for maximum of two independent random variables

The distribution of the maximum of two independent random variables is important in reliability, investment, management science, insurance business, medical science, etc. For most practical applications, the density function provides important information about the distribution. The maximum of two independent random variables has a density function that can be estimated by two different kernel type estimators. In this article, we discuss these two kernel estimators, called direct and indirect kernel estimators. We find their asymptotic mean square errors and central limit theorems from which comparisons are made through examples and simulations.