Density and Hazard Rate Estimation for Right-Censored Data by Using Wavelet Methods

Abstract

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.

Adaptive Estimation of Hazard Rate with Censored Data

“…The splines projection space leads to the same kind of results, but we choose to describe it separately because its generating basis is not orthonormal, even if it has also some very similar properties as compared to the bases already described. More specifically, we consider dyadic B-splines on the unit interval [0,1]. Let N r be the B-spline of order r which has knots at the points 0, 1, .…”

“…Patil (1997) considers a wavelet estimator for uncensored data but his approach can not be easily extended to censored data. Dyadic linear wavelets for incomplete data are used by Antoniadis et al (1999). They propose a two-step procedure to estimate the density function of the lifetime and then define the hazard estimator by taking the ratio with some estimator of the survival function.…”

Estimation of Hazard Function and its Associated Factors in Gastric Cancer Patients using Wavelet and Kernel Smoothing Methods

“…The common methods for estimation of hazard function (Antoniadis, 1989;O'Sullivan, 1988;Kooperberg and stone, 1992) have time-consuming and complex calculations or programming. Recently Wavelet estimation of hazard function is considered that fast speed calculations is its excellent advantage (Antoniadis et al, 1999).…”

Estimação de efeitos variantes no tempo em modelos tipo Cox via bases de Fourier e ondaletas Haar

“…Por exemplo, as ondaletas são utilizadas em estimação de densidades (Chesneau et al, 2013;Cutillo et al , 2013;Donoho, 1993;Hall e Patil, 1995a,b), na estimação de densidade e taxa de falha para dados censurados (Antoniadis et al, 1994(Antoniadis et al, , 1999, em análise de regressão não paramétrica (Donoho e Johnstone, 1995;Donoho et al, 1995Donoho et al, , 1998Hall e Patil, 1996;Nason e Silverman, 1994).…”

“…Na literatura, diversos trabalhos estatísticos relacionados com as ondaletas vêm sendo produzidos, nas mais diversas áreas. Por exemplo, elas são utilizadas na estimação de densidades (Chesneau et al , 2013;Cutillo et al, 2013;Donoho, 1993;Hall e Patil, 1995a,b), na estimação de densidade e taxa de falha para dados censurados (Antoniadis et al, 1994(Antoniadis et al, , 1999) e na análise de regressão não paramétrica (Donoho e Johnstone, 1995;Donoho et al, 1995Donoho et al, , 1998Hall e Patil, 1996;Nason e Silverman, 1994), entre outras áreas. Devido à sua analogia com as séries de Fourier, considere o espaço L 2 (R) de todas as funções mensuráveis quadrado integrável sobre R. Assim, se β 0 ∈ L 2 (R), então…”