Local estimating equations

Estimating equations have found wide popularity recently in parametric problems, yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula. Motivated by a problem in nutritional epidemiology, we use estimating equations to derive nonparametric estimators of a “parameter” depending on a predictor. The nonparametric component is estimated via local polynomials with loess or kernel weighting; asymptotic theory is derived for the latter. In keeping with the estimating equation paradigm, variances of the nonparametric function estimate are estimated using the sandwich method, in an automatic fashion, without the need (typical in the literature) to derive asymptotic formulas and plug-in an estimate of a density function. The same philosophy is used in estimating the bias of the nonparametric function; that is, an empirical method is used without deriving asymptotic theory on a case-by-case basis. The methods are applied to a series of examples. The application to nutrition is called “nonparametric calibration” after the term used for studies in that field. Other applications include local polynomial regression for generalized linear models, robust local regression, and local transformations in a latent variable model. Extensions to partially parametric models are discussed.