A New Look at the Visual Performance of Nonparametric Hazard Rate Estimators 1 Olaf Gefellera and Nils Lid Hjortb a Department of Medical Statistics, University of Gottingen, Humboldtallee 32, D-37073 Gottingen, Germany b Department of Mathematics and Statistics, University of Oslo, P.B. 1053 Blindern, N-0316 Oslo, Norway Abstract: Nonparametric curve estimation by kernel methods has attracted widespread interest in theoretical and applied statistics. One area of conflict between theory and application relates to the evaluation of the performance of the estimators. Recently, Marron and Tsybakov (1995) proposed visual error criteria for addressing this issue of controversy in density estimation. Their core idea consists in using integrated alternatives to the Hausdorff distance for measuring the closeness of two sets based on the Euclidean distance. In this paper we transfer these ideas to hazard rate estimation from censored data. We are able to derive similar results that help to understand when the application of the new criteria will lead to answers that differ from those given by the conventional approach. 1 Introduction In various areas of applied and theoretical statistics nonparametric curve estimation by kernel methods has attracted widespread interest during the last two decades. Without imposing any distributional assumptions on the observed data, structural information, for example, on the underlying den- sity functions, some interesting functionals of the density (like the hazard rate) or regression curves, can be obtained by "smoothing" the empirical mass of the observations to some neighbouring environment around the ob- served data points. Starting with the pioneering work by Rosenblatt (1956) and Parzen (1962), nowadays a vast literature on the properties of kernel methods can be found (for recent textbooks see Wand and Jones (1995), Hardle (1991)). Important progress has been made recently in a variety of issues (for example, bandwidth selection, boundary behaviour, software im- plementation), mostly originating from research in density estimation, the simplest situation. However, it has been criticised repeatedly that there is an obvious gap between theory and application with respect to the evalu- ation of the performance of the nonparametric estimators. The estimated 1 A slightly modified version of this paper is to be published in I. Balderjahn, R. Mathar, M. Schader (eds.): Data Highways and Information Flooding, a Challenge for Classifi- cation and Data Analysis. Spinger Series "Studies in Classification, Data Analysis, and Knowledge Organization", Volume 8. Springer- Verlag, Berlin, to appear in 1997. 1