Numerical results concerning a sharp adaptive density estimator
A local adaptive kernel density estimator is described for densities \(f\) with \(\int(f^{(\beta)})^2dx\leq L^2\), where \(f^{(\beta)}\) is the derivative, may be fractional, and \(L\) is supposed to be fixed and known. A specific procedure for estimation of \(\beta\) is considered. The adaptive procedure is based on a pilot estimator \(f_n\). The adaptive bandwidth is \[ h_{n,\beta}(x)=[(\max(f_n(x),1/\ln n)\cdot\ln n)/n\beta(2\beta-1)L^2]^{1/(2\beta)}, \] and the kernel used in the final estimator is \[ K_\beta(x)=\pi^{-1}\int_0^\infty(1+|u|^{2\beta})^{-1}\cos(xu)du. \] The final locally adaptive estimator is \[ (nh_{n,\beta}(x))^{-1}\sum_{j=1}^n K_\beta\left((X_j-x)/h_{n,\beta}(x)\right), \] where \((X_1,\dots,X_n)\) is an i.i.d. sample. Simulation results are presented for some distributions.
scientific article; zbMATH DE number 1878613
scientific article