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Investigating smooth multiple regression by the method of average derivatives. (English) Zbl 0703.62052
Let \((x_ 1,...,x_ k,y)\) be a random vector where y denotes a response on the vector x of predictor variables. We propose a technique [termed average derivative estimation (ADE)] for studying the mean response \(m(x)=E(y| x)\) through the estimation of the k vector of average derivatives \(\delta =E(m')\). The ADE procedure involves two stages: First estimate \(\delta\) using an estimator \({\hat \delta}\), and then approximate \(m(x)\) by \(\hat m(x)=\hat g(x^ T{\hat \delta})\), where \(\hat g\) is an estimator of the univariate regression of y on \(x^ T{\hat \delta}\). We argue that the ADE procedure exhibits several attractive characteristics: data summarization through interpretable coefficients, graphical depiction of the possible nonlinearity between y and \(y^ T{\hat \delta}\), and theoretical properties consistent with dimension reduction. We motivate the ADE procedure using examples of models that take the form \(m(x)=\tilde g(x^ T\beta)\). In this framework, \(\delta\) is shown to be proportional to \(\beta\) and \(\hat m(x)\) infers \(m(x)\) exactly. The focus of the procedure is on the estimator \({\hat \delta}\), which is based on a simple average of kernel smoothers and is shown to be a \(\sqrt{N}\) consistent and asymptotically normal estimator of \(\delta\). The estimator \(\hat g(\cdot)\) is a standard kernel regression estimator and is shown to have the same properties as the kernel regression of y on \(x^ T\delta\).

MSC:
62G07 Density estimation
62J02 General nonlinear regression
62J12 Generalized linear models (logistic models)
62G05 Nonparametric estimation
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