Sensitivity and other properties of wavelet regression and density estimators

Abstract: We give a unified, non-iterative formulation for wavelet estimators that can be applied in density estimation, regression on a regular grid and regression with a random design. This formulation allows us to better understand the bias due to a given method of coefficients estimation at high resolution. We also introduce functional representations for estimators of interest. The proposed formulation is well suited for the study of estimation bias and sensitivity analysis and, in the second part, we compute the influence function of various wavelet estimators. This tool allows us to see how the influence of observations can differ strongly depend-ing on their locations. The lack of shift-invariance can be investigated and the influence function can be used to compare different approximation schemes for the wavelet estimator. We show that a local linear regression-type approximation for the higher resolution coefficients induces more extreme and variable influence of the observations on the final estimator than does the standard approximation. New approximation schemes are proposed. Key words and phrases: Approximation kernel, influence function, irregular design, functional, sensitivity to the design, shift invariance. 1