A penalized maximum likelihood estimate of $$f(0+)$$ when $$f$$ is non- increasing.(English)Zbl 0822.62020

Summary: The problem of estimating the value at $$0 +$$ of a non-increasing density $$f$$ (on $$(0, \infty))$$ is considered. It is shown, by example, that the problem is interesting, and it is noted that the nonparametric maximum likelihood estimator is inconsistent. A penalized maximum likelihood estimator is derived as an alternative, and its properties studied through simulations and asymptotic analysis. In particular, the penalized maximum likelihood estimator is shown to be consistent.

MSC:

 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference