β-divergence loss for the kernel density estimation with bias reduced

Figures & data

Table 1. $RE\left(\hat{h}\right)$ for normal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$
		$\mu =0$	$\sigma =1$
50	0.934	0.953	0.853	0.723	0.703
200	0.945	0.925	0.955	0.931	0.903
700	0.990	0.945	0.982	0.987	0.952
		$\mu =0$	$\sigma =0.5$
50	0.870	0.837	0.867	0.890	0.905
200	0.937	0.880	0.897	0.954	0.932
700	0.964	0.930	0.929	0.842	0.858
		$\mu =0$	$\sigma =0.1$
50	0.584	0.767	0.634	0.631	0.623
200	0.553	0.892	0.625	0.879	0.721
700	0.529	0.946	0.612	0.877	0.813
		$\mu =1$	$\sigma =1$
50	0.864	0.899	0.904	0.853	0.876
200	0.938	0.928	0.914	0.962	0.987
700	0.973	0.952	0.974	0.927	0.932
		$\mu =1$	$\sigma =0.5$
50	0.882	0.852	0.823	0.734	0.872
200	0.963	0.880	0.780	0.925	0.967
700	0.836	0.925	0.743	0.943	0.780
		$\mu =1$	$\sigma =0.1$
50	0.230	0.770	0.611	0.587	0.554
200	0.101	0.912	0.686	0.769	0.687
700	0.051	0.949	0.727	0.880	0.721
		$\mu =5$	$\sigma =1$
50	0.400	0.810	0.723	0.889	0.457
200	0.285	0.945	0.852	0.934	0.579
700	0.222	0.963	0.967	0.978	0.789
		$\mu =5$	$\sigma =0.5$
50	0.2390	0.852	0.712	0.845	0.831
200	0.1390	0.926	0.897	0.915	0.805
700	0.0817	0.956	0.945	0.921	0.878
		$\mu =5$	$\sigma =0.1$
50	0.1360	0.588	0.702	0.645	0.613
200	0.0523	0.458	0.764	0.758	0.802
700	0.0205	0.341	0.861	0.655	0.841

Table 2. $E\left(\hat{h}\right)$ for normal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$	$h_{MISE}$
			$\mu =0$	$\sigma =1$
50	0.464	0.528	0.530	0.520	0.323	0.347
200	0.362	0.393	0.399	0.383	0.321	0.328
700	0.287	0.302	0.310	0.293	0.308	0.309
			$\mu =0$	$\sigma =0.5$
50	0.330	0.397	0.425	0.343	0.223	0.286
200	0.248	0.267	0.312	0.248	0.193	0.280
700	0.196	0.197	0.242	0.200	0.186	0.244
			$\mu =0$	$\sigma =0.1$
50	0.134	0.104	0.358	0.098	0.510	0.041
200	0.087	0.060	0.027	0.087	0.485	0.038
700	0.068	0.043	0.219	0.057	0.421	0.0370
			$\mu =1$	$\sigma =1$
50	0.520	0.590	0.592	0.588	0.429	0.426
200	0.404	0.437	0.444	0.434	0.395	0.423
700	0.316	0.336	0.344	0.333	0.354	0.345
			$\mu =1$	$\sigma =0.5$
50	0.401	0.430	0.479	0.373	0.326	0.342
200	0.320	0.298	0.373	0.265	0.280	0.282
700	0.254	0.214	0.287	0.212	0.233	0.239
			$\mu =1$	$\sigma =0.1$
50	0.366	0.103	0.464	0.203	0.0422	0.0451
200	0.276	0.061	0.342	0.053	0.0380	0.0380
700	0.221	0.0428	0.267	0.0426	0.0343	0.0314
			$\mu =5$	$\sigma =1$
50	1.290	0.770	1.400	0.608	0.420	0.475
200	0.989	0.477	0.1.070	0.441	0.330	0.470
700	0.768	0.353	0.829	0.336	0.442	0.272
			$\mu =5$	$\sigma =0.5$
50	1.270	0.468	1.370	0.369	0.310	0.295
200	0.961	0.297	1.040	0.262	0.210	0.286
700	0.750	0.208	0.810	0.197	0.209	0.270
			$\mu =5$	$\sigma =0.1$
50	1.270	0.0982	1.370	0.0745	0.045	0.0415
200	0.955	0.061	1.030	0.053	0.040	0.0385
700	0.745	0.0424	0.804	0.040	0.039	0.0339

Table 3. $E|\hat{h}/h_{MISE}-1|$ for normal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$
		$\mu =0$	$\sigma =1$
50	0.124	0.072	0.077	0.0874	0.379
200	0.0785	0.0829	0.1050	0.0578	0.1620
700	0.0396	0.0717	0.0572	0.0436	0.0509
		$\mu =0$	$\sigma =0.5$
50	0.1370	0.1510	0.1670	0.1510	0.4560
200	0.0655	0.1360	0.0882	0.1010	0.2490
700	0.0559	0.0729	0.0537	0.0818	0.0104
		$\mu =0$	$\sigma =0.1$
50	0.772	0.2530	0.3000	0.3990	0.4410
200	0.674	0.1210	0.1250	0.1520	0.2070
700	0.679	0.0772	0.0506	0.0726	0.185
		$\mu =1$	$\sigma =1$
50	0.1430	0.1040	0.1630	0.0748	0.398
200	0.0774	0.0800	0.0931	0.0501	0.184
700	0.0483	0.0626	0.0600	0.0361	0.037
		$\mu =1$	$\sigma =0.5$
50	0.172	0.1930	0.1400	0.2260	0.4560
200	0.236	0.1530	0.0899	0.1460	0.121
700	0.285	0.0989	0.0506	0.0794	0.119
		$\mu =1$	$\sigma =0.1$
50	3.67	0.2620	1.380	0.3980	0.431
200	4.29	0.1250	0.986	0.1720	0.353
700	4.58	0.0838	0.652	0.0878	0.137
		$\mu =5$	$\sigma =1$
50	1.14	0.1450	0.2390	0.2430	0.4580
200	1.23	0.0815	0.1210	0.0899	0.2530
700	1.29	0.0686	0.0745	0.0540	0.0203
		$\mu =5$	$\sigma =0.5$
50	2.40	0.1860	0.600	0.2510	0.4340
200	2.68	0.1180	0.444	0.1440	0.2020
700	2.80	0.0743	0.296	0.0804	0.0597
		$\mu =5$	$\sigma =0.1$
50	15.7	0.884	5.95	0.3980	0.4810
200	17.1	1.021	4.51	0.1570	0.2630
700	17.7	1.104	3.34	0.0656	0.0178

Figure 1. Boxplots of the relative values RE for the bandwidth selectors for the estimation of densities $\mu =0,1,5$ and $\sigma =1,0.5,0.1$. The sample size varies from 100 to 2000.

Table 4. $RE\left(\hat{h}\right)$ for lognormal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$
		$\mu =0$	$\sigma =1$
50	0.901	0.933	0.905	0.920	0.923
200	0.949	0.972	0.949	0.935	0.956
700	0.969	0.987	0.978	0.988	0.990
		$\mu =0$	$\sigma =0.5$
50	0.880	0.883	0.856	0.885	0.890
200	0.938	0.844	0.886	0.887	0.893
700	0.962	0.817	0.942	0.943	0.950
		$\mu =0$	$\sigma =0.1$
50	0.627	0.218	0.798	0.699	0.802
200	0.561	0.096	0.830	0.708	0.825
700	0.517	0.046	0.947	0.789	0.889
		$\mu =1$	$\sigma =1$
50	0.824	0.937	0.911	0.941	0.950
200	0.973	0.971	0.967	0.977	0.980
700	0.978	0.986	0.971	0.983	0.989
		$\mu =1$	$\sigma =0.5$
50	0.907	0.836	0.838	0.840	0.876
200	0.906	0.784	0.882	0.887	0.902
700	0.831	0.687	0.919	0.887	0.908
		$\mu =1$	$\sigma =0.1$
50	0.248	0.199	0.796	0.799	0.800
200	0.097	0.082	0.885	0.803	0.810
700	0.048	0.038	0.950	0.932	0.940
		$\mu =5$	$\sigma =1$
50	0.395	0.351	0.824	0.813	0.835
200	0.239	0.244	0.942	0.860	0.899
700	0.221	0.179	0.968	0.907	0.934
		$\mu =5$	$\sigma =0.5$
50	0.234	0.214	0.872	0.863	0.881
200	0.128	0.113	0.924	0.896	0.905
700	0.082	0.071	0.952	0.911	0.972
		$\mu =5$	$\sigma =0.1$
50	0.139	0.137	0.785	0.753	0.823
200	0.051	0.050	0.811	0.748	0.873
700	0.022	0.021	0.850	0.750	0.902

Table 5. $E\left(\hat{h}\right)$ for lognormal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$	$h_{MISE}$
			$\mu =0$	$\sigma =1$
50	0.451	0.452	0.512	0.518	0.520	0.507
200	0.356	0.354	0.387	0.396	0.383	0.364
700	0.282	0.280	0.299	0.308	0.293	0.281
			$\mu =0$	$\sigma =0.5$
50	0.329	0.316	0.389	0.416	0.343	0.345
200	0.253	0.236	0.271	0.314	0.248	0.234
700	0.196	0.182	0.197	0.244	0.188	0.201
			$\mu =0$	$\sigma =0.1$
50	0.133	0.092	0.105	0.337	0.075	0.072
200	0.086	0.057	0.060	0.282	0.053	0.061
700	0.067	0.041	0.043	0.219	0.040	0.058
			$\mu =1$	$\sigma =1$
50	0.504	0.498	0.398	0.576	0.588	0.439
200	0.401	0.398	0.426	0.442	0.431	0.417
700	0.319	0.320	0.340	0.346	0.333	0.309
			$\mu =1$	$\sigma =0.5$
50	0.408	0.362	0.438	0.487	0.373	0.354
200	0.322	0.269	0.296	0.369	0.265	0.236
700	0.255	0.204	0.213	0.287	0.199	0.186
			$\mu =1$	$\sigma =0.1$
50	0.339	0.171	0.103	0.453	0.075	0.089
200	0.278	0.104	0.059	0.341	0.053	0.072
700	0.222	0.065	0.042	0.269	0.040	0.051
			$\mu =5$	$\sigma =1$
50	1.300	0.753	0.743	1.410	0.610	0.616
200	0.992	0.500	0.477	1.070	0.441	0.509
700	0.770	0.362	0.352	0.831	0.336	0.350
			$\mu =5$	$\sigma =0.5$
50	1.270	0.596	0.458	1.370	0.369	0.325
200	0.962	0.375	0.290	1.040	0.262	0.211
700	0.749	0.256	0.210	0.808	0.197	0.163
			$\mu =5$	$\sigma =0.1$
50	1.260	0.521	0.102	1.360	0.075	0.070
200	0.953	0.262	0.060	1.030	0.053	0.059
700	0.743	0.172	0.042	0.802	0.040	0.038

Table 6. $E|\hat{h}/h_{MISE}-1|$ for lognormal mixture $f\left(x\right)=0.5\varphi \left(x\right)+0.5{\varphi }_{\sigma }(x-\mu )$.

n	${\hat{h}}_{NR}$	${\hat{h}}_{LSCV}$	${\hat{h}}_{{\mathcal{D}}_{1.1}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.5}CV}$	${\hat{h}}_{{\mathcal{D}}_{1.9}CV}$
		$\mu =0$	$\sigma =1$
50	0.143	0.165	0.090	0.081	0.078
200	0.079	0.102	0.056	0.051	0.049
700	0041.	0.065	0.045	0.050	0.046
		$\mu =0$	$\sigma =0.5$
50	0.133	0.167	0.175	0.221	0.207
200	0.067	0.095	0.122	0.268	0.202
700	0.052	0.062	0.071	0.295	0.195
		$\mu =0$	$\sigma =0.1$
50	0.777	0.307	409	4.010	0.532
200	0.629	0.109	0.153	4.320	0.321
700	678	0.058	0.087	4.511	0.029
		$\mu =1$	$\sigma =1$
50	0.156	0.178	0.097	0.088	0.076
200	0.078	0.109	0.050	0.041	0.039
700	0.043	0.054	0.041	0.036	0.
		$\mu =1$	$\sigma =0.5$
50	0.165	0.149	0.207	0.306	0.295
200	0.217	0.091	0.155	0.394	0.281
700	0.279	0.052	0.086	0.442	0.248
		$\mu =1$	$\sigma =0.1$
50	3.510	1.280	0.394	5.020	0.943
200	4.24	0.957	0.136	5.440	0.675
700	4.570	0.646	0.080	5.760	0.430
		$\mu =5$	$\sigma =1$
50	1.140	0.239	0.243	1.310	0.430
200	1.250	0.132	0.096	1.430	0.270
700	1.291	0.076	0.056	1.474	0.094
		$\mu =5$	$\sigma =0.5$
50	2.430	0.616	258	2.712	0.756
200	2.683	0.433	0.129	2.975	0.415
700	2.802	0.298	0.079	3.102	0.234
		$\mu =5$	$\sigma =0.1$
50	15.8	5.930	0.380	17.10	0.342
200	17.00	4.520	0.146	18.40	0.132
700	17.75	3.332	0.081	19.20	0.063

Figure 2. Estimated density of $CO2$ per capita in 2008 using the different bandwidths. ${\hat{h}}_{D_{1.1}CV}$ (solid line); ${\hat{h}}_{D_{1.9}CV}$ (dashed line); ${\hat{h}}_{D_{1.5}CV}$ (dotted line); ${\hat{h}}_{LSCV}$, (dotdash line) and ${\hat{h}}_{N{R}_2}$ (longdash line).

Figure 3. Estimated density of repair times (hours) for an airborne communication transceiver: ${\hat{h}}_{D_{1.1}CV}$ (solid line); ${\hat{h}}_{D_{1.9}CV}$ (dashed line); ${\hat{h}}_{D_{1.5}CV}$ (dotted line); ${\hat{h}}_{LSCV}$, (dotdash line) and ${\hat{h}}_{N{R}_2}$, normal reference (longdash line).

Figure 4. Estimated density of repair times (hours) for an airborne communication transceiver using the different bandwidths: ${\hat{h}}_{D_{1.1}CV}$ (solid line); ${\hat{h}}_{D_{1.9}CV}$ (dashed line); ${\hat{h}}_{D_{1.5}CV}$ (dotted line); ${\hat{h}}_{LSCV}$, (dotdash line) and ${\hat{h}}_{N{R}_2}$, normal reference (longdash line).