Lower bounds on bandwidth selection in hazard estimation

In the setting of nonparametric hazard estimation under right random censorship by the kernel method, asymptotic lower bounds for bandwidth selection are provided. If the error criterion is the Integrated Squared Error (ISE), and if the distribution function of the underlying lifetime is sufficiently regular, then it is shown that the relative error of any data-driven bandwidth selector cannot be reduced below order n^-1/10 asymptotically. On the other hand, if the error criterion is the Mean Integrated Squared Error (MISE), the relative error of bandwidth selection can be reduced to order n^-1/2, when the hazard function is sufficiently smooth. Possible exensions to the multivariate setting are pointed out. These results are similar with those obtained by Hall and Marron (1991) in univariate density estimation without censoring.