Parameterization and inference for nonparametric regression problems

We consider local likelihood or local estimating equations, in which a multivariate function Theta(.) is estimated but a derived function lambda(.) of Theta(.) is of interest. In many applications, when most naturally formulated 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 or linear, at least to second order. In such circumstances, the simplest standard 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 models, nonparametric generalized linear models, etc. We show that, in local linear kernel regression, inference about the derived function lambda(.) is facilitated without a loss of power by reparameterization so that lambda(.) is itself a component of Theta(.). Our approach is in contrast with the standard practice of choosing Theta(.) for convenience and allowing lambda(.) to be a non-linear function of Theta(.). The methods are applied to an important data set in nutritional epidemiology.