Density estimation by wavelet thresholding. (English) Zbl 0860.62032

Summary: Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes \(B_{\sigma pq}\) and for a range of global \(L'_p\) error measures, \(1\leq p'<\infty\). A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when \(p'>p\), some form of nonlinearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of \(n\). A second approach, using an approximation of a Gaussian white-noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error \((p'=2)\).


62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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