Analysis of KNN Density Estimation

We analyze the convergence rates of $k$ nearest neighbor density estimation method, under notation="LaTeX">$\ell _{\alpha} $ norm with notation="LaTeX">$\alpha \in [1,\infty]$ . Our analysis includes two different cases depending on whether support set is bounded or not. In first case, probability function has a support. show that if known, then kNN estimator minimax optimal both \big[1,\infty\big)$ and =\infty If unknown, still _{1}$ , but suboptimal for >1$ not consistent _\infty second unbounded smooth everywhere. Moreover, Hessian assumed to decay values. For this our result shows error nearly optimal. The original consistent. To address issue, we design new adaptive estimator, which can select samples. Using bound comparison, popular kernel case.