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Minimax Rates for Estimating the Variance and its Derivatives in Non–Parametric Regression

In this paper the van Trees inequality is applied to obtain lower bounds for the quadratic risk of estimators for the variance function and its derivatives in non–parametric regression models. This approach yields a much simpler proof compared to previously applied methods for minimax rates. Furthermore, the informative properties of the van Trees inequality reveal why the optimal rates for estimating the variance are not affected by the smoothness of the signal g. A Fourier series estimator is constructed which achieves the optimal rates. Finally, a second–order correction is derived which suggests that the initial estimator of g must be undersmoothed for the estimation of the variance.