### Abstract

Let (Formula presented.) 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 an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä 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 the main result on the L_{2}-distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.

Original language | English |
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Number of pages | 20 |

Journal | Scandinavian Journal of Statistics |

DOIs | |

Publication status | Published - 2020 |

### Keywords

- Cauchy's formula
- Grenander estimator
- integral statistics
- saddle points