Grenander functionals and Cauchy's formula

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. Here we extend this to the proof of a general result on integrals of the Grenander estimator. We also correct Groeneboom and Pyke (1983), where the limit distribution of the sums of gamma and Poisson variables on which the conditioning was done did not have the right form. Saddle point methods and Cauchy's formula are important tools in our development.