Density and hazard rate estimation for right censored data using wavelet methods

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 possess pointwise and global mean square consistency, obtain the best possible asymptotic MISE convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show 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.