Grenander functionals and Cauchy's formula

Understand

Let $\hat f_n$ be the nonparametric maximum likelihood estimator of a decreasing density.

• Grenander characterized this as the left-continuous slope of the least concave majorant of the empirical distribution function.
• For a sample from the uniform distribution, the asymptotic distribution of the $L_2$-distance of the Grenander estimator to the uniform density was derived in Groeneboom and Pyke (1983) by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables.
• This representation was also used in Groeneboom and Lopuhaa (1993) to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator.