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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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