Comparative study on excess distribution estimation in iid settings

Figures & data

Table 1. The polynomial convergence rates of the MSE of the estimators and the lengths.

Burr			$u^{\ast }=n^{1/8}$		$u^{\ast }=n^{1/4}$		$u^{\ast }=n^{1/2}$		$u^{\ast }=n^{3/4}$
c, ℓ	α	β	PE	NE	PE	NE	PE	NE	PE	NE
1/2,1/2	1/4	1/2	−1/8	−31/32	−1/4	−15/16	−1/2	–	−13/16	–
1,1/2	1/2	1	−1/4	−15/16	−1/2	−7/8	−3/4	–	−5/8	–
3,1/2	3/2	3	−3/4	−13/16	−5/8	–	−1/4	–	–	–
1/2,1	1/2	1/2	−1/8	−15/16	−1/4	−7/8	−1/2	–	−5/8	–
1,1	1	1	−1/4	−7/8	−1/2	–	−1/2	–	−1/4	–
3,1	3	3	−5/8	−5/8	−1/4	–	–	–	–	–
1/2,3	3/2	1/2	−1/8	−13/16	−1/4	–	−1/4	–	–	–
1,3	3	1	−1/4	−5/8	−1/4	–	–	–	–	–
3,3	9	3	–	–	–	–	–	–	–	–
Weibull			$u^{\ast }=n^{1/8}$		$u^{\ast }=n^{1/4}$		$u^{\ast }=n^{1/2}$		$u^{\ast }=n^{3/4}$
κ	γ	ρ	PE	NE	PE	NE	PE	NE	PE	NE
1/2	0	0	–	–	–	–	–	–	–	–
1	0	0	–	–	–	–	–	–	–	–
3	0	0	–	–	–	–	–	–	–	–
10	0	0	–	–	–	–	–	–	–	–
inv. Burr			$u^{\ast }=n^{1/8}$		$u^{\ast }=n^{1/4}$		$u^{\ast }=n^{1/2}$		$u^{\ast }=n^{3/4}$
c, ℓ	μ	σ	PE	NE	PE	NE	PE	NE	PE	NE
3, 2	−6	−2	−1/4	–	–	–	–	–	–	–
1, 2	−2	−2	−1/2	−3/4	−1/2	–	–	–	–	–
1/2, 2	−1	−2	−1/2	−7/8	−3/4	–	−1/2	–	−1/4	–
3, 1	−3	−1	−1/4	−5/8	−1/4	–	–	–	–	–
1, 1	−1	−1	−1/4	−7/8	−1/2	–	−1/2	–	−1/4	–
1/2, 1	−1/2	−1	−1/4	−15/16	−1/2	−7/8	−3/4	–	−5/8	–
3, 1/3	−1	−1/3	−1/12	−7/8	−1/6	–	−1/3	–	−1/4	–
1, 1/3	−1/3	−1/3	−1/12	−23/24	−1/6	−11/12	−1/3	–	−1/2	–
1/2, 1/3	−1/6	−1/3	−1/12	−47/48	−1/6	−23/24	−1/3	–	−1/2	–

Table 2. Scaled MISE values (×100) and sd values (×100) for the estimators.

Burr	n = 2^8				n = 2^12
	PE	sd	NE	sd	PE	sd	NE	sd
c, ℓ	u = 2				$u=2^{3/2}$
1/2,1/2	0.085	0.161	0.066	0.074	0.417	0.565	0.004	0.005
1,1/2	0.056	0.073	0.079	0.080	0.043	0.709	0.006	0.006
3,1/2	0.139	0.182	0.152	0.153	0.013	0.017	0.018	0.019
1/2,1	0.077	0.096	0.112	0.113	0.023	0.381	0.008	0.008
1,1	0.129	0.155	0.167	0.164	0.010	0.012	0.014	0.014
3,1	0.515	0.727	0.341	0.397	0.072	0.092	0.080	0.084
1/2,3	0.718	0.975	0.760	0.800	0.049	0.062	0.070	0.069
1,3	2.078	2.786	1.268	1.397	0.184	0.408	0.197	0.207
3,3	8.467	5.770	4.048	3.513	–	–	–	–
c, ℓ	u = 2^2				u = 2^3
1/2,1/2	0.061	0.095	0.079	0.082	0.141	0.221	0.043	0.020
1,1/2	0.073	0.098	0.098	0.098	0.016	0.226	0.010	0.010
3,1/2	0.389	0.492	0.391	0.398	0.063	0.083	0.076	0.077
1/2,1	0.098	0.122	0.123	0.114	0.008	0.010	0.012	0.012
1,1	0.226	0.278	0.275	0.266	0.021	0.026	0.030	0.030
3,1	5.031	5.550	3.310	4.391	2.639	3.517	1.925	2.310
1/2,3	1.793	2.668	1.449	1.529	0.160	0.201	0.197	0.198
1,3	6.413	5.378	6.785	6.686	3.832	4.364	3.374	4.222
3,3	10.452	5.223	12.662	1.048	–	–	–	–
c, ℓ	u = 2^4				u = 2^6
1/2,1/2	0.068	0.087	0.201	0.157	0.135	0.156	0.326	0.055
1,1/2	0.137	0.175	0.524	0.245	0.019	0.051	1.125	0.167
3,1/2	4.634	5.841	4.511	6.116	2.534	3.493	2.448	2.695
1/2,1	0.172	0.243	0.683	0.279	0.021	0.048	1.281	0.173
1,1	0.907	1.152	1.161	0.946	0.021	0.048	1.281	0.173
3,1	8.952	3.599	32.555	20.262	10.875	2.193	39.079	18.143
1/2,3	7.158	7.335	9.564	11.507	3.685	4.930	3.861	5.144
1,3	9.905	3.787	39.100	18.296	10.896	2.682	41.768	16.961
3,3	15.334	2.374	47.800	0.000	–	–	–	–

Table 3. Scaled MISE values (×100) and sd values (×100) for the estimators.

inv. Burr	n = 2^8				n = 2^12
	PE	sd	NE	sd	PE	sd	NE	sd
c, ℓ	u = 2^2				u = 2^3
3, 2	26.100	0.000	2.464	0.321	24.700	0.000	1.891	0.087
1, 2	7.580	0.000	0.533	0.139	6.590	0.000	0.307	0.032
1/2, 2	0.159	0.179	1.405	0.275	0.013	0.018	0.453	0.061
3,1	10.400	0.000	0.719	0.091	9.390	0.000	0.477	0.055
1,1	0.744	0.075	0.510	0.165	0.381	0.004	0.152	0.021
1/2,1	0.412	0.481	3.696	0.887	0.071	0.100	1.695	0.184
3,1/3	0.777	1.205	0.504	0.332	0.037	0.054	0.203	0.077
1,1/3	0.965	0.635	2.396	1.169	0.186	0.254	1.237	0.291
1/2,1/3	15.160	2.634	8.225	0.749	8.872	6.600	3.487	1.728
c, ℓ	u = 2^4				u = 2^6
3, 2	23.500	0.000	0.717	0.002	21.500	0.000	0.952	0.007
1, 2	5.881	0.003	3.034	0.196	5.010	0.000	4.573	0.043
1/2, 2	0.448	0.363	11.084	0.991	0.069	0.102	15.179	0.631
3,1	8.573	0.005	2.251	0.068	7.280	0.000	3.688	0.041
1,1	0.290	0.104	6.982	0.377	0.063	0.009	9.445	0.103
1/2,1	4.097	4.593	15.750	0.782	2.168	2.461	19.648	3.885
3,1/3	0.893	1.252	7.575	0.925	0.099	0.144	11.678	0.753
1,1/3	2.708	3.437	14.700	1.138	2.793	3.459	18.732	3.054
1/2,1/3	17.040	8.314	25.870	0.389	10.516	7.941	26.731	0.876
c, ℓ	u = 2^6				u = 2^9
3, 2	19.900	0.000	7.125	0.016	17.900	0.000	9.620	0.000
1, 2	4.574	0.010	16.540	0.052	4.240	0.000	19.100	0.006
1/2, 2	4.367	4.818	32.050	1.002	1.525	2.540	34.679	0.212
3,1	6.370	0.012	14.760	0.052	5.250	0.001	17.805	0.022
1,1	0.368	0.651	26.410	0.074	0.053	0.128	29.613	0.050
1/2,1	17.667	9.352	37.460	0.052	28.320	6.218	39.951	0.272
3,1/3	10.731	10.665	29.110	0.152	2.337	4.374	33.221	0.165
1,1/3	20.466	6.344	36.810	0.120	28.531	6.543	39.767	0.199
1/2,1/3	16.780	19.468	42.390	0.032	30.408	12.21	44.101	0.018

Table 4. Scaled MISE values (×100) and sd values (×100) for the estimators.

Weibull	n = 2^8				n = 2^12
	PE	sd	NE	sd	PE	sd	NE	sd
κ, C	u = 2^2				u = 2^3
1/2, 1	0.229	0.658	0.246	0.257	0.020	0.021	0.021	0.021
1, 1	0.425	0.603	0.301	0.348	–	–	–	–
3, 1	–	–	–	–	–	–	–	–
10, 1	–	–	–	–	–	–	–	–
κ, C	u = 2^4				u = 2^6
1/2, 1	0.402	0.538	0.414	0.453	0.051	0.066	0.063	0.065
1, 1	4.078	4.156	2.208	2.883	–	–	–	–
3, 1	–	–	–	–	–	–	–	–
10, 1	–	–	–	–	–	–	–	–
κ, C	u = 2^5				u = 2^9
1/2, 1	4.272	4.354	3.590	4.205	6.682	5.021	11.531	9.561
1, 1	4.078	4.156	36.876	17.292	–	–	–	–
3, 1	–	–	–	–	–	–	–	–
10, 1	–	–	–	–	–	–	–	–

Table 5. Scaled MISE values (×100) and sd values (×100) for the estimators.

Weibull	n = 2^8				n = 2^12
	PE	sd	NE	sd	PE	sd	NE	sd
κ, C	$u=(\ln n{)}^{1/\kappa }$
1/2, 1	8.013	5.481	13.924	10.731	7.882	5.686	20.619	16.486
1, 1	7.673	5.664	7.162	6.460	7.255	5.602	10.879	8.304
3, 1	7.799	6.144	1.710	1.207	7.626	5.940	1.308	1.785
10, 1	7.949	6.228	22.972	0.617	7.705	6.095	9.079	0.482
κ, C	$u=(\ln n{)}^{1/\kappa }$
1/2, 1/2	5.690	6.383	4.910	6.881	3.603	4.349	2.729	3.361
1, 1/2	6.902	5.369	5.265	3.227	5.112	2.103	4.897	1.692
3, 1/2	25.110	1.504	27.714	1.080	25.471	0.470	27.283	0.441
10, 1/2	41.092	0.084	43.492	0.077	42.322	0.060	44.371	0.045
κ, C	$u=(\ln n{)}^{1/\kappa }$
1/2, 1/5	7.846	4.927	6.414	3.756	6.702	1.960	6.399	1.771
1, 1/5	20.343	2.531	20.126	2.388	20.116	0.834	20.085	0.923
3, 1/5	40.452	0.242	43.002	0.310	41.155	0.099	43.483	0.112
10, 1/5	43.112	0.032	43.800	0.000	44.100	0.005	44.600	0.000

Figure 1. Abisko rainfall amount (in mm) from 1/1/1913 to 1/1/2015.

Figure 2. The estimated ED functions of Abisko rainfall amount (in mm) from 1/1/1913 to 1/1/2015 data by the non parametric approach (solid line) and by the fitting to the GPD (dashed line).

Figure 3. The estimated ED functions in [7, 14] of the Abisko rainfall amount (in mm) by the nonparametric approach (solid line) and by the fitting to the GPD (dashed line).