Bandwidth selection for kernel log-density estimation

Kernel estimation of the logarithm of a probability density function at a given evaluation point is studied. The properties of the kernel log-density estimator are heavily influenced by the unboundedness of the log function at zero. In particular, standard asymptotic expansions can provide a poor guide to finite sample behaviour for this estimator, with consequences for the choice of methodology for bandwidth selection. In response, a new approximate cross-validation bandwidth selector is developed. Its theoretical properties are explored and its finite sample behaviour examined in numerical experiments. The proposed methodology is then applied to estimation of log-likelihoods for a complex genetic model used in determining migration rates between village communities on the Indonesian island of Sumba.