Nonparametric and semiparametric compound estimation in multiple covariates

We consider the problem of simultaneously estimating a mean response function and its partial derivatives, when the mean response function depends nonparametrically on two or more covariates. To address this problem, we propose a "compound estimation" approach, in which differentiation and estimation are interchangeable: an estimated partial derivative is exactly equal to the corresponding partial derivative of the estimated mean response function. Compound estimation yields essentially optimal convergence rates and may exhibit substantially smaller squared error in finite samples compared to local regression. We also explain how to employ compound estimation under more general circumstances, when the mean response function depends parametrically on some additional covariates and the observations are not statistically independent. In a case study, we apply compound estimation to examine how the progression of Parkinson's disease may relate to a subject's age and the signal fractal scaling exponent of the subject's recorded voice. Especially among those intermediate in age, an abnormal signal fractal scaling exponent may portend greater symptom progression.