Investigating Smooth Multiple Regression by the Method of Average Derivatives

Abstract Let (x 1, …, xk, y) be a random vector where y denotes a response on the vector x of predictor variables. In this article 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 δ = E(m′). The ADE procedure involves two stages: first estimate δ using an estimator , and then approximate m(x) by ), where ĝ is an estimator of the univariate regression of y on . We argue that the ADE procedure exhibits several attractive characteristics: data summarization through interpretable coefficients, graphical depiction of the possible nonlinearity between y and , and theoretical properties consistent with dimension reduction. We motivate the ADE procedure using examples of models that take the form . In this framework, δ is shown to be proportional to β and [mcirc](x) infers m(x) exactly. The focus of the procedure is on the estimator , which is based on a simple average of kernel smoother...

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