Non-parametric graduation using kernel methods

Abstract

LetEbe an event whose probability of occurrence depends on some continuous variablex, P(E|x) =qxFor example,Emay be death andxage,Emay be incidence of lung cancer andxamount of smoking, orEmay be reconviction of a parolee withxprevious criminal convictions (with suitable definitions of the underlying time interval for the occurrence ofE). Given observations on n individuals with characteristicxand the incidence ofE, it is desired to estimate the functionqx.The simplest case is when the data are grouped—supposexoccurs innxcases of whichEoccurssxtimes. Then the elementary crude estimate isThis paper describes a simple, non-parametric method of graduating observed rates or probabilities of the formThe technique has been used for smoothing data sets arising in medicine and criminology (Copas(1)(2)) and is extended here to an actuarial example and the results compared with more traditional approaches.