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Let $\hat f_n$ be the nonparametric maximum likelihood estimator of adecreasing density. Grenander characterized this as the left-continuous slopeof the least concave majorant of the empirical distribution function. For asample from the uniform distribution, the asymptotic distribution of the$L_2$-distance of the Grenander estimator to the uniform density was derived inGroeneboom and Pyke (1983) by using a representation of the Grenander estimatorin terms of conditioned Poisson and gamma random variables. This representationwas also used in Groeneboom and Lopuhaa (1993) to prove a central limit resultof Sparre Andersen on the number of jumps of the Grenander estimator. Here weextend this to the proof of a general result on integrals of the Grenanderestimator. We also correct Groeneboom and Pyke (1983), where the limitdistribution of the sums of gamma and Poisson variables on which theconditioning was done did not have the right form. Saddle point methods andCauchy's formula are important tools in our development.

Grenander functionals and Cauchy's formula