R. J. Carroll

Deconvolving kernel density estimators

This paper considers estimation of a continuous bounded probability density when observations from the density are contaminated by additive measurement errors having a known distribution. Properties of the estimator obtained by deconvolving a kernel estimator of the observed data are investigated. When the kernel used is sufficiently smooth the deconvolved estimator is shown to be pointwise consistent and bounds on its integrated mean squared error are derived. Very weak assumptions are made on the measurement-error density thereby permitting a comparison of the effects of different types of measurement error on the deconvolved estimator

Nonparametric kernel and regression spline estimation in the presence of measurement error

In many regression applications both the independent and dependent variables are measured with error. When this happens, conventional parametric and nonparametric regression techniques are no longer valid. We consider two different nonparametric techniques, regression splines and kernel estimation, of which both can be used in the presence of measurement error. Within the kernel regression context, we derive the limit distribution of the SIMEX estimate. With the regression spline technique, two different methods of estimations are used. The first method is the SIMEX algorithm which attempts to estimate the bias, and remove it. The second method is a structural approach, where one hypothesizes a distribution for the independent variable which depends on estimable parameters. A series of examples and simulations illustrate the methods.

Nonparametric estimation via local estimating equations, with applications to nutrition calibration

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 formulae and plug-in an estimate of a density function. The same philosophy is used in estimating the bias of the nonparametric function, i.e., we use an empirical method 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.

Semiparametric Regression: Simple Semiparametric Models

Introduction Until now we have confined discussion to scatterplot smoothers. This setting served well to illustrate the main concepts behind smoothing. However, there is a gap between the methodology and the needs of practitioners. As exemplified by the problems described in Chapter 1, most applications of regression involve several predictors. To begin closing the gap, this chapter introduces a class of multiple regression models that have a nonparametric component involving only a single predictor and a parametric component for the other predictors. Having both parametric and nonparametric components means the models are semiparametric . This class of simple semiparametric models is important in its own right but also serves as an introduction to more complex semiparametric regression models of later chapters, where the effects of several predictors are modeled nonparametrically. Beyond Scatterplot Smoothing The end of the previous chapter closed off quite a lengthy description of how to smooth out a scatterplot and perform corresponding inference. In Chapter 3 we described three general approaches: penalized splines, local polynomial fitting, and series approximation. For penalized splines, we presented both an algorithmic approach based on ridge regression and a mixed model approach based on maximum likelihood and best prediction. There are other approaches to scatterplot smoothing that we did not describe at all.