|| On the Convergence Rates of KNN Density Estimation
||Puning Zhao, Lifeng Lai, University of California Davis, United States|
||D6-S6-T3: Density Estimation
||Monday, 19 July, 23:40 - 00:00
||Tuesday, 20 July, 00:00 - 00:20
We analyze the L1 and L_infty convergence rates of k nearest neighbor density estimation method. Our analysis includes two different cases depending on whether the support set is bounded or not. In the first case, the probability density function has a bounded support and is bounded away from zero. We show that kNN density estimation is minimax optimal under both L1 and L_infty criteria, if the support set is known. If the support set is unknown, then the convergence rate of L1 error is not affected, while L_infty error does not converge. In the second case, the probability density function can approach zero and is smooth everywhere. Moreover, the Hessian is assumed to decay with the density values. For this case, our result shows that the L_infty error of kNN density estimation is nearly minimax optimal.