Cramér type moderate deviations for the Grenander estimator near the boundaries of the support

The Grenander estimator (nonparametric maximum likelihood estimator) of a decreasing density function is not consistent at boundaries of support. This phenomenon has great influences on the global measures of deviation between density function and its Grenander estimator. In this article, by strong approximation technique and comparison method, we establish Cramér type moderate deviations for the Grenander estimator near the boundaries of the support. The obtained results provide a nice uniform comparison between tail probability of the estimator and limiting fluctuation distribution.