A New Look at the Visual Performance of Nonparametric Hazard Rate Estimators

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 {\it 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 onthe 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 approaches.

💡 Research Summary

The paper addresses a long‑standing discrepancy between theoretical and applied evaluations of non‑parametric estimators, focusing on hazard‑rate estimation from censored data. Classical performance measures rely on Lp‑norms—most commonly the L2‑norm or mean integrated squared error (MISE)—which quantify vertical deviations between an estimator and the true function. While mathematically convenient, such measures ignore the geometric shape of the estimated curve; they can penalise an estimator that captures the correct structural features (e.g., the number and location of peaks) simply because it deviates vertically in a region where the true curve is flat, and conversely reward an estimator that smooths away important features.

Marron and Tsybakov (1995) introduced “visual error criteria” for density estimation to remedy this problem. Their idea is to treat a function as a set of points in the plane (the graph) and to measure distances between two graphs using planar distances rather than vertical ones. For any point (x, y) the distance to a graph G_h is the Euclidean distance to the closest point on G_h. Integrating the squared distances over the domain yields V E₂, a visual analogue of the L2‑norm. Because the distance is asymmetric (the distance from the true graph to the estimated graph differs from the reverse), two versions V E₂(b→h) and V E₂(h→b) are defined, and a symmetrised version S E₂ is obtained by averaging them in a Pythagorean fashion. Non‑quadratic versions (V E₁, V E_∞) and the classic Hausdorff distance are also discussed.

The authors transfer this framework to hazard‑rate estimation. They consider the standard random‑censoring model: independent failure times T_i with distribution F and censoring times C_i with distribution G, observed as (X_i, δ_i) where X_i = min(T_i, C_i) and δ_i indicates an uncensored failure. The hazard rate h(x) = f(x)/(1‑F(x)) is estimated by a fixed‑bandwidth kernel estimator

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