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This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no speci®c parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The ®rst uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its¯exibility, provides a new and interesting interpretation.

We consider the problem of estimating jump points in smooth curves. Observa-.., n from a random design regression function are given. We focus essentially on the basic situation where a unique change point is present in the regression function. Based on local linear regression, a jump estimate process t Q ĉ(t) is constructed. Our main result is the convergence to a compound Poisson process with drift, of a local dilated-rescaled version of ĉ(t), under a positivity condition regarding the asymmetric kernel involved. This result enables us to prove that our estimate of the jump location converges with exact rate n −1 without any particular assumption regarding the bandwidth h n . Other consequences such as asymptotic normality are investigated and some proposals are provided for an extension of this work to more general situations. Finally we present Monte-Carlo simulations which give evidence for good numerical performance of our procedure.

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