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Variable bandwidth kernel hazard estimators

Variable kernel hazard estimators, where the bandwidth is allowed to depend on the exposure, are considered together with the kernel hazard analogue to Abramsons approach to bias reduction, see Abramson (1982). The theoretical analysis of the kernel hazard analogue to Abramsons approach shows that the variable bandwidth should be weighted according to the inverse square-root of the hazard. This does not seem to be as intuitively appealing as the more intuitive result known from density estimation, where the variable bandwidth is weighted according to the inverse square-root of the density. Our simulations show that the intuitive appealing exposure weighted variable bandwidth approach performs much better for small sample sizes than the theoretically excellent, but counter intuitive, variable bandwidth approach based on Abramsons approach to bias reduction. Our conclusion of this variable bandwidth study therefore is, that one should use variable bandwidth exposure weighting to stabilise basic kernel hazard smoothers. However, one should not use variable bandwidth methods to improve the rate of convergence of the bias.