Automatic smoothing and estimation in single index Poisson regression

We address the problem of smoothing parameter (h) selection when estimating the direction vector (β0) and the link function in the context of semiparametric, single index Poisson regression. The single index Poisson model (PSIM) differs from the classical nonparametric setting in two ways: first, the errors are heteroscedastic, and second, the direction parameter is unknown and has to be estimated. We propose two simple, automatic rules for simultaneously estimating β0 and h in a PSIM. The first criterion, called weighted least squares (WLS2), estimates the Kullback-Leibler risk function and has a penalty term to prevent undersmoothing in small samples. The second method, termed double smoothing (DS), is based on the estimation of an L2 approximation of the Kullback-Leibler risk and makes use of a double smoothing idea as in Wand and Gutierrez (1997). Simulations are used to investigate the behavior of various criteria in the PSIM context. Our weighted least squares and double smoothing methods out-perform both a Kullback-Leibler version of cross-validation and the weighted least squares cross-validation criterion proposed by Härdle,Hall and Ichimura (1993)