Statistics & Probability Letters 59 (2002) 61–66 A new proof of strong consistency of kernel estimation of density function and mode under random censorship Ali Gannoun ∗ ,J erˆ ome Saracco Laboratoire de Probabilit es et de Statistique, D epartement des Sciences Math ematiques, CC 051, Universit e Montpellier II, Place Eug ene Bataillon, 34095 Montpellier Cedex 5, France Received March 2001; received in revised form January 2002 Abstract In this paper, we establish a new proof of uniform consistency of kernel estimator of density function when we observe a random right censored model. This proof uses an exponential inequality established by Wang (2000). As a consequence, we obtain the almost sure convergence of the kernel estimator of the mode. c  2002 Published by Elsevier Science B.V. Keywords: Censored data; Kaplan–Meier estimator; Kernel density estimation; Mode estimation; Strong consistency 1. Introduction Consider a randomly right censored model given by two sequences X 1 ;:::;X n and Y 1 ;:::;Y n of independent and identically distributed nonnegative random variables such that X i and Y i are inde- pendent (i =1;:::;n). Let F and G be the unknown continuous distribution functions of the X ’s and the Y ’s, respectively. It is assumed that X i has a density function f with respect to Lebesgue measure on R. In the random censorship model, one cannot observe X i completely, but only Z i =min(X i ;Y i ) and  i = 5 {Xi 6Yi } for i =1;:::;n; where 5 {A} denotes the indicator function of the event A. By independence of X and Y , the distri- bution function of Z is H =1 - (1 - F )(1 - G). In survival analysis, X is the lifetime of a patient due to particular cause of interest and Y is usually the withdrawal time of the patient from the follow-up study or the time of the death due to other causes in a competing risk model. * Corresponding author. Statistical Genetics and Bioinformatics unit, Howard University, 2216 6th Street NW, Suite 206, Washington, DC 20059, USA. Fax: +1 202 265 0871. E-mail addresses: gannoun@stat.math.univ-montp2.fr, agannoun@howard.edu (A. Gannoun), saracco@stat.math.univ- montp2.fr (J. Saracco). 0167-7152/02/$-see front matter c  2002 Published by Elsevier Science B.V. PII:S0167-7152(02)00166-9