# Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching
## 3 Results and Discussions
### KDE, NMEM, NAPS(KDE) and NAPS(NMEM) Comparisons
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Distribution** & \(\mathbf{log_{2}(N)}\) & **NMEM** & **NAPS(KDE)** & **NAPS(NMEM)** & **RER** \\ \hline Beta(0.5, 0.5) & 10 & \(1.53\times 10^{2}\pm 1\times 10^{3}\) & \(5.22\times 10^{1}\pm 1\times 10^{2}\) & \(1.12\times 10^{1}\pm 4\) & 0.214 \\ Beta(0.5, 0.5) & 16 & \(3.64\times 10^{1}\pm 1\times 10^{2}\) & \(2.72\times 10^{1}\pm 2\times 10^{2}\) & \(3.43\pm 5\times 10^{-1}\) & 0.126 \\ Beta(0.5, 0.5) & 22 & \(3.09\times 10^{2}\pm 8\times 10^{2}\) & \(3.62\pm 7\) & \(7.22\times 10^{-1}\pm 9\times 10^{-2}\) & 0.199 \\ Beta(0.5, 1.5) & 10 & \(5.54\times 10^{1}\pm 1\times 10^{2}\) & \(1.39\times 10^{2}\pm 9\times 10^{2}\) & \(9.90\pm 3\) & 0.179 \\ Beta(0.5, 1.5) & 16 & \(3.64\times 10^{1}\pm 6\times 10^{1}\) & \(2.42\times 10^{1}\pm 1\times 10^{2}\) & \(2.96\pm 7\times 10^{-1}\) & 0.123 \\ Beta(0.5, 1.5) & 22 & \(5.80\times 10^{2}\pm 8\times 10^{2}\) & \(1.99\pm 2\) & \(6.13\times 10^{-1}\pm 8\times 10^{-2}\) & 0.307 \\ Beta(2, 0.5) & 10 & \(7.46\times 10^{1}\pm 9\times 10^{1}\) & \(3.82\times 10^{1}\pm 1\times 10^{2}\) & \(9.03\pm 2\) & 0.236 \\ Beta(2, 0.5) & 16 & \(9.88\pm 2\) & \(1.08\times 10^{1}\pm 3\times 10^{1}\) & \(2.78\pm 5\times 10^{-1}\) & 0.281 \\ Beta(2, 0.5) & 22 & \(1.42\times 10^{1}\pm 1\) & \(1.86\pm 2\) & \(5.98\times 10^{-1}\pm 7\times 10^{-2}\) & 0.317 \\ Cauchy-Beta & 10 & \(7.71\times 10^{3}\pm 4\times 10^{3}\) & \(1.00\times 10^{2}\pm 1\times 10^{2}\) & \(4.18\times 10^{1}\pm 4\times 10^{1}\) & 0.417 \\ Cauchy-Beta & 16 & \(6.29\times 10^{5}\pm 6\times 10^{4}\) & \(1.73\times 10^{3}\pm 2\times 10^{3}\) & \(5.87\times 10^{1}\pm 2\times 10^{2}\) & 0.034 \\ Cauchy-Beta & 22 & \(1.84\times 10^{7}\pm 1\times 10^{6}\) & \(7.75\times 10^{1}\pm 1\times 10^{2}\) & \(5.14\times 10^{2}\pm 9\times 10^{2}\) & **6.632** \\ GEV(0, 1, 5) & 10 & \(3.04\times 10^{2}\pm 1\times 10^{2}\) & \(5.09\times 10^{1}\pm 7\times 10^{1}\) & \(7.61\pm 1\) & 0.149 \\ GEV(0, 1, 5) & 16 & \(6.94\times 10^{2}\pm 3\times 10^{2}\) & \(3.98\times 10^{1}\pm 6\times 10^{1}\) & \(3.10\pm 5\times 10^{-1}\) & 0.078 \\ GEV(0, 1, 5) & 22 & \(3.02\times 10^{2}\pm 1\times 10^{2}\) & \(1.48\times 10^{1}\pm 2\times 10^{1}\) & \(6.66\times 10^{-1}\pm 7\times 10^{-2}\) & 0.045 \\ GEV(1, 1, 5) & 10 & \(4.03\times 10^{2}\pm 7\times 10^{2}\) & \(1.23\times 10^{1}\pm 3\) & \(8.69\pm 2\) & 0.705 \\ GEV(1, 1, 5) & 16 & \(2.78\times 10^{1}\pm 3\times 10^{1}\) & \(1.25\times 10^{1}\pm 5\times 10^{1}\) & \(3.29\pm 5\times 10^{-1}\) & 0.262 \\ GEV(1, 1, 5) & 22 & \(2.14\times 10^{1}\pm 3\times 10^{1}\) & \(2.52\times 10^{1}\pm 9\times 10^{1}\) & \(6.76\times 10^{-1}\pm 6\times 10^{-2}\) & 0.032 \\ GEV(2, 2, 2) & 10 & \(2.75\times 10^{1}\pm 1\times 10^{1}\) & \(1.38\times 10^{1}\pm 5\times 10^{1}\) & \(7.68\pm 1\) & 0.556 \\ GEV(2, 2, 2) & 16 & \(2.03\times 10^{1}\pm 1\times 10^{1}\) & \(2.80\pm 2\) & \(2.31\pm 3\times 10^{-1}\) & 0.824 \\ GEV(2, 2, 2) & 22 & \(2.01\times 10^{1}\pm 2\times 10^{1}\) & \(1.94\times 10^{1}\pm 2\times 10^{2}\) & \(5.45\times 10^{-1}\pm 5\times 10^{-2}\) & 0.028 \\ Generalized-Pareto & 10 & \(1.97\times 10^{1}\pm 6\) & \(1.17\times 10^{1}\pm 2\times 10^{1}\) & \(6.96\pm 1\) & 0.593 \\ Generalized-Pareto & 16 & \(1.18\times 10^{1}\pm 1\) & \(2.83\pm 5\times 10^{-1}\) & \(2.25\pm 4\times 10^{-1}\) & 0.798 \\ Generalized-Pareto & 22 & \(1.04\times 10^{1}\pm 5\times 10^{-1}\) & \(8.41\times 10^{-1}\pm 7\times 10^{-2}\) & \(4.28\times 10^{-1}\pm 3\times 10^{-2}\) & 0.509 \\ Normal & 10 & \(7.33\pm 4\) & \(5.33\times 10^{1}\pm 8\times 10^{1}\) & \(7.67\pm 1\) & **1.047** \\ Normal & 16 & \(1.40\pm 1\) & \(3.74\times 10^{1}\pm 6\times 10^{1}\) & \(3.15\pm 5\times 10^{-1}\) & **2.239** \\ Normal & 22 & \(2.90\times 10^{-1}\pm 3\times 10^{-1}\) & \(9.68\pm 1\times 10^{1}\) & \(6.71\times 10^{-1}\pm 7\times 10^{-2}\) & **2.311** \\ Stable & 10 & \(7.82\times 10^{3}\pm 1\times 10^{3}\) & \(3.87\times 10^{3}\pm 3\times 10^{4}\) & \(1.37\times 10^{1}\pm 4\times 10^{1}\) & 0.004 \\ Stable & 16 & \(4.08\times 10^{5}\pm 5\times 10^{4}\) & \(7.39\pm 4\times 10^{1}\) & \(2.99\pm 3\times 10^{-1}\) & 0.405 \\ Stable & 22 & \(7.31\times 10^{6}\pm 1\times 10^{6}\) & \(9.07\times 10^{-1}\pm 8\times 10^{-2}\) & \(5.14\times 10^{-1}\pm 3\times 10^{-2}\) & 0.567 \\ Trimodal-Normal & 10 & \(2.06\times 10^{1}\pm 3\) & \(4.59\times 10^{1}\pm 9\times 10^{1}\) & \(However, for a robust estimator that is fully automatic and will not require prior knowledge of the sample distribution, NAPS(NMEM) provides the most consistent high accuracy estimates across diverse distributions, and an extensive range of sample sizes.