**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 3, 441--475

**Penalized Contrast Estimation of
Density and Hazard Rate with Censored Data**

By

E. Brunel and F. Comte, Universit\'e
Ren\'e Descartes -- Paris 5 and MAP5, UMR CNRS 8145, France

SUMMARY. We consider projection estimator methods for the estimation of density and hazard rate functions based on randomly right-censored data. Two types of adaptive hazard estimators are considered. The first one is a two-step estimator defined as the ratio of a penalized contrast estimator of the subdensity and of the empirical survival function of the data. The second estimator is built by using another penalized projection contrast. Both estimators are proved to achieve automatically the standard optimal rate associated with the unknown regularity of the hazard function, but with some restriction for the ``ratio" estimator. In the examples studied here, the ratio estimator seems to be slightly better than the direct estimator.

*AMS (1991) subject classification. *62G05, 62G20.

*Key words and phrases. *Adaptive estimation, hazard rate, minimax
rate, right-censoring, nonparametric penalized contrast estimator.