Rate of Convergence of the Density Estimator on an Unknown Orientation

Motivated by recent work on exploratory projection pursuit, we give the kernel estimator of density on an unknown projection orientation \theta_0\left(\left\|\theta_0\right\|=1\right) by combining the estimator of θ₀ with classical kernel estimation and obtain its asymptotically normality at a fixed center. Its precise convergence rate uniformly over all canters are derived and the convergence rate in mean square is described. These results present same optimality as that of the density estimator on a know orientation. At last, we give a example of estimator \widehat\theta_n for θ₀.