Trimmed L-moments (1,0) for the generalized Pareto distribution

Figures & data

Fig. 1 Fitting the GPA distribution to annual maximum flows at the streamgauge Slim at Slim River, Perak, Malaysia.

Table 1  Bias of x(F) estimator fitting the GPA distribution to generated GPA samples for all sample sizes

k	Method	n = 15			n = 25			n = 50
		F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99
−0.4	L-moments	1.041	1.254	1.374	0.700	0.846	0.925	0.472	0.574	0.632
	TL-moments (1,0)	1.185	1.496	1.701	0.801	1.022	1.168	0.536	0.686	0.785
−0.3	L-moments	0.537	0.567	0.543	0.355	0.370	0.348	0.152	0.139	0.107
	TL-moments (1,0)	0.660	0.755	0.784	0.432	0.490	0.502	0.200	0.214	0.204
−0.2	L-moments	0.291	0.253	0.187	0.133	0.095	0.041	0.070	0.047	0.014
	TL-moments (1,0)	0.387	0.391	0.356	0.191	0.177	0.141	0.100	0.090	0.068
−0.1	L-moments	0.116	0.053	−0.022	0.049	0.004	0.016	0.034	0.012	−0.013
	TL-moments (1,0)	0.188	0.148	0.089	0.087	0.057	−0.046	0.053	0.038	0.017
0.1	L-moments	0.006	−0.045	−0.094	−0.004	−0.035	−0.064	−0.002	−0.016	−0.031
	TL-moments (1,0)	0.042	−0.001	−0.047	0.013	−0.015	−0.042	0.007	−0.006	−0.018
0.2	L-moments	−0.012	−0.053	−0.091	−0.015	−0.038	−0.059	−0.011	−0.023	−0.034
	TL-moments (1,0)	0.011	−0.026	−0.062	−0.001	−0.021	−0.041	−0.005	−0.014	−0.024
0.3	L-moments	−0.032	−0.067	−0.098	−0.016	−0.034	−0.051	−0.009	−0.019	−0.027
	TL-moments (1,0)	−0.015	−0.049	−0.078	−0.006	−0.023	−0.037	−0.004	−0.012	−0.019
0.4	L-moments	−0.029	−0.055	−0.077	−0.016	−0.031	−0.043	−0.007	−0.014	−0.019
	TL-moments (1,0)	−0.012	−0.036	−0.057	−0.006	−0.018	−0.028	−0.002	−0.008	−0.013

Table 2  RMSE of x(F) estimator fitting the GPA distribution to generated GPA samples for all sample sizes

k	Method	n = 15			n = 25			n = 50
		F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99
−0.4	L-moments	5.075	6.934	8.651	4.200	5.801	7.28	3.228	4.514	5.701
	TL-moments (1,0)	5.105	6.908	8.569	4.210	5.775	7.22	3.210	4.475	5.645
−0.3	L-moments	3.788	5.088	6.272	3.068	4.142	5.115	2.332	3.174	3.933
	TL-moments (1,0)	3.796	5.062	6.216	3.053	4.103	5.057	2.314	3.148	3.903
−0.2	L-moments	2.686	3.527	4.282	2.192	2.883	3.492	1.597	2.100	2.537
	TL-moments (1,0)	2.664	3.481	4.216	2.172	2.852	3.456	1.589	2.091	2.532
−0.1	L-moments	1.961	2.517	3.005	1.576	2.015	2.391	1.127	1.434	1.694
	TL-moments (1,0)	1.932	2.471	2.948	1.563	1.997	2.372	1.121	1.429	1.691
0.1	L-moments	1.016	1.241	1.429	0.788	0.956	1.093	0.550	0.663	0.752
	TL-moments (1,0)	0.997	1.217	1.405	0.776	0.941	1.077	0.541	0.650	0.738
0.2	L-moments	0.756	0.907	1.030	0.566	0.674	0.760	0.399	0.470	0.526
	TL-moments (1,0)	0.735	0.879	1.000	0.549	0.652	0.735	0.384	0.451	0.504
0.3	L-moments	0.552	0.655	0.738	0.416	0.487	0.543	0.286	0.332	0.368
	TL-moments (1,0)	0.532	0.628	0.709	0.395	0.460	0.512	0.271	0.313	0.345
0.4	L-moments	0.417	0.489	0.547	0.309	0.357	0.394	0.212	0.243	0.267
	TL-moments (1,0)	0.393	0.458	0.512	0.287	0.328	0.361	0.196	0.222	0.243

Table 3  Bias of x(F) estimator fitting the GPA distribution to generated Wakeby samples for all sample sizes

Dist.	Method	n = 15			n = 25			n = 50
		F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99
WA1	L-moments	0.396	0.551	0.661	0.396	0.598	0.756	0.347	0.575	0.762
	TL-moments (1,0)	0.272	0.294	0.286	0.207	0.248	0.265	0.108	0.145	0.168
WA2	L-moments	0.395	0.582	0.725	0.437	0.659	0.833	0.444	0.688	0.882
	TL-moments (1,0)	0.159	0.212	0.243	0.148	0.226	0.284	0.112	0.202	0.275
WA3	L-moments	0.065	0.113	0.150	0.055	0.133	0.199	0.074	0.177	0.266
	TL-moments (1,0)	0.057	0.080	0.095	0.022	0.072	0.114	0.017	0.086	0.148
WA4	L-moments	0.268	0.365	0.436	0.284	0.401	0.489	0.309	0.442	0.542
	TL-moments (1,0)	0.076	0.085	0.086	0.056	0.078	0.092	0.053	0.084	0.108
WA5	L-moments	0.046	0.083	0.114	0.047	0.101	0.146	0.056	0.123	0.178
	TL-moments (1,0)	0.029	0.053	0.071	0.013	0.050	0.081	0.013	0.062	0.104
WA6	L-moments	0.153	0.224	0.277	0.166	0.242	0.298	0.179	0.258	0.317
	TL-moments (1,0)	0.082	0.137	0.180	0.092	0.155	0.203	0.101	0.168	0.220

Table 4  RMSE of x(F) estimator fitting the GPA distribution to generated Wakeby samples for all sample sizes

Dist.	Method	n = 15			n = 25			n = 50
		F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99	F = 0.98	F = 0.987	F = 0.99
WA1	L-moments	2.381	3.082	3.686	1.916	2.473	2.942	1.433	1.863	2.222
	TL-moments (1,0)	2.219	2.905	3.516	1.780	2.331	2.815	1.318	1.736	2.100
WA2	L-moments	1.397	1.733	2.005	1.124	1.402	1.623	0.864	1.112	1.311
	TL-moments (1,0)	1.327	1.691	2.002	1.055	1.339	1.575	0.771	0.981	1.154
WA3	L-moments	1.425	1.773	2.063	1.107	1.362	1.569	0.791	0.974	1.122
	TL-moments (1,0)	1.412	1.779	2.092	1.108	1.381	1.607	0.798	0.992	1.151
WA4	L-moments	0.707	0.859	0.979	0.578	0.710	0.811	0.469	0.598	0.697
	TL-moments (1,0)	0.663	0.831	0.973	0.520	0.646	0.750	0.374	0.465	0.539
WA5	L-moments	0.748	0.890	1.002	0.573	0.680	0.764	0.401	0.478	0.541
	TL-moments (1,0)	0.750	0.903	1.027	0.576	0.691	0.783	0.405	0.483	0.546
WA6	L-moments	0.312	0.371	0.417	0.258	0.321	0.369	0.224	0.294	0.348
	TL-moments (1,0)	0.322	0.381	0.427	0.248	0.299	0.340	0.189	0.243	0.287

Fig. 2 Map showing northern Peninsular Malaysia.

Fig. 3 L-moments diagram for annual maximum stream flows of northern Peninsular Malaysia.

Fig. 4 TL-moments (1,0) diagram for annual maximum stream flows of northern Peninsular Malaysia.