**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 3, pp. 491--506

**HOW MANY "GOOD" OBSERVATIONS DO YOU NEED FOR "FAST" DENSITY CONVOLUTION FROM SUPERSMOOTH ERRORS**

By

CHRISTIAN HESSE, *Universität Stuttgart*

SUMMARY. Estimating the density of the random variables Xj from iid observations $Y_{j} = X_{j}+\={\varepsilon}_{j}$ suffers from notoriously slow (i.e. logarithmic) rates when the errors $\={\varepsilon}_{j}$ are normal or in general belong to the class of supersmooth distributions. When $\={\varepsilon}_{j}=T_{j}\={\varepsilon}$ with $\={\varepsilon}$ supersmooth $T_{j}$ ~ Bernoulli (1-p(n)) where $p(n)~n_{-\gamma}$ we show that fast (i.e. algebric) rates in $\mathcal{L}^{\infty}$ are possible for kernel density estimators when $\gamma$ is not too large. The paper suggests that the highest quality observations, although unknown, essentially determine the performance estimators. Similar results are obtained for estimating the distribution function and derivatives of density. The theorems are proved under weak conditions which in particular do not include the existence of derivatives beyond those that are being estimated.

*AMS (1992) subject classification.* Primary 62G05, secondary 60F15.

*Key words and phrases.* Deconvolution, density estimation, kernel estimator, contamination, a.s. convergence.