Local Estimating Equations

Abstract

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 caIibration" 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.

Semiparametric Estimation of Animal Abundance Using Capture–Recapture Data from Open Populations

“…Ruppert and Wand (1994) examined local least squares estimators for nonparametric models and these estimators are the basis of our approach, although our setting is different. Carroll, Ruppert, and Welsh (1998) developed weighted estimating methods for nonparametric estimation in a general setting, and Severini and Wong (1992) and Severini and Staniswallis (1994) have developed a generalized profile estimating equation approach to semiparametric estimation. Huggins and Yip (1999) developed nonparametric estimators for capture-recapture experiments based on martingale estimating equations by regarding the population size at capture occasions near a time t as being locally closed.…”

Estimation of the Size of an Open Population Using Local Estimating Equations II: A Partially Parametric Approach

“…where d t is the number of distinct individuals captured in t . The approach of Carroll et al (1998) then suggests the following estimating equations…”

“…Lloyd and Yip (1991) found that f (k, t, K) follows a beta-binomial distribution conditional on the past capture history and constructed a martingale estimating equation to estimate the population size in closed populations. Following the approach of Carroll et al (1998), we modify equation (5.20) in Lloyd and Yip (1991) and apply their suggested weight function to obtain the estimating equations…”

“…In particular, a variety of methods exist to incorporate heterogeneity of the capture probabilities into estimation procedures. The use of kernel smoothing and local polynomials to extend closed population models to open populations with frequent capture occasions has been examined for some population models (Huggins and Yip, 1999;Huggins et al, 2002;Yang and Huggins, 2003) and Carroll, Ruppert, and Welsh (1998) gave a procedure applicable to estimating equations in general. In capture-recapture studies Huggins and Yip (1999) initially developed a kernel estimator of the population size under model M t , where capture probabilities are only affected by capture occasions.…”

Modeling Longitudinal Data with Ordinal Response by Varying Coefficients

“…Estimation in model (2) can be done by local estimating equations (cf., Carroll, Ruppert, and Welsh, 1998). In the setting considered here, local estimation can be seen as a weighted generalized estimating equation (GEE) with working independence used in the fitting.…”