**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 3, pp. 392--402

**AN INVARIANCE PRINCIPLE FOR DISCONTINUITY ESTIMATION IN SMOOTH HAZARD FUNCTIONS UNDER RANDOM CENSORING**

By

HANS-GEORG MÜLLER and JANE-LING WANG, *University of California*

SUMMARY. Consider a hazard function which is smooth with the exception of a discontinuity or a discontinuity in its $\gamma$-th derivative. Nonparametric estimators for the discontinuity location are constructed by maximizing differences between left and right sided kernel estimates using smooth kernel functions. A local derivation process around the discontinuity location is introduced and an invariance principles established. This result is applied to obtain asymptotic distributions of these estimators as well as corresponding estimators for the jump size. The methods and the results are applicable to randomly censored data.

*AMS (1991) subject classification.* 62G07, 62G20.

*Key words and phrases.* Boundary, derivatives, functional limit theorem, Gaussian process, i.i.d. representation, kernel method, Nelson estimator, survival analysis, weak convergence.