Parameterization and inference for nonparametric regression problems

We consider local likelihood or local estimating equations, in which a mult ivariate function Theta(.) is estimated but a derived function lambda(.) of Theta(.) is of interest. In many applications, when most naturally formula ted the derived function is a non-linear function of Theta(.). In trying to understand whether the derived non-linear function is constant or linear, a problem arises with this approach: when the function is actually constant or linear, the expectation of the function estimate need not be constant o r linear, at least to second order. In such circumstances, the simplest sta ndard methods in nonparametric regression for testing whether a function is constant or linear cannot be applied. We develop a simple general solution which is applicable to nonparametric regression, varying-coefficient model s, nonparametric generalized linear models, etc. We show that, in local lin ear kernel regression, inference about the derived function lambda(.) is fa cilitated without a loss of power by reparameterization so that lambda(.) i s itself a component of Theta(.). Our approach is in contrast with the stan dard practice of choosing Theta(.) for convenience and allowing lambda(.) t o be a non-linear function of Theta(.). The methods are applied to an impor tant data set in nutritional epidemiology.