The effect of the regularity of the error process on the performance of kernel regression estimators

This article considers estimation of regression function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f$$</EquationSource> </InlineEquation> in the fixed design model <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Y(x_i)=f(x_i)+ \epsilon (x_i), i=1,\ldots ,n$$</EquationSource> </InlineEquation>, by use of the Gasser and Müller kernel estimator. The point set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\{ x_i\}_{i=1}^{n}\subset [0,1]$$</EquationSource> </InlineEquation> constitutes the sampling design points, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\epsilon (x_i)$$</EquationSource> </InlineEquation> are correlated errors. The error process <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\epsilon $$</EquationSource> </InlineEquation> is assumed to satisfy certain regularity conditions, namely, it has exactly <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation> (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$=\!0, 1, 2, \ldots $$</EquationSource> </InlineEquation>) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$(2k+1)$$</EquationSource> </InlineEquation>th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator. Copyright Springer-Verlag Berlin Heidelberg 2013