A simple and efficient method for finding the closest generalized lambda distribution to a specific model

Figures & data

Table 1. Hellinger distances for the pdQ method with different weighting methods for step 1 and either distributional least squares $(K=5)$ or quartile matching for step 2

	Distributional least squares $(K=5)$				Quartile matching
Distribution	No weighting	Left	Right	Both	No weighting	Left	Right	Both
Uniform (2,5)	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Normal (0,1)	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
Lognormal (0,1)	0.0499	0.0191	0.0733	0.0327	0.0313	0.0066	0.0577	0.0091
Exponential (1)	0.0133	0.0058	0.0000	0.0122	0.0069	0.0052	0.0000	0.0049
${\chi }_5^2$	0.0019	0.0026	0.0025	0.0025	0.0016	0.0022	0.0021	0.0019
Pareto (7)	0.0222	0.0098	0.0000	0.0191	0.0030	0.0053	0.0000	0.0024
Gamma (2)	0.0026	0.0031	0.0036	0.0033	0.0019	0.0024	0.0025	0.0022
Beta (2,3)	0.0004	0.0006	0.0006	0.0005	0.0004	0.0006	0.0006	0.0005
Weibull (2)	0.0001	0.0000	0.0001	0.0001	0.0001	0.0000	0.0001	0.0001
Cauchy (0,1)	0.0003	0.0003	0.0003	0.0004	0.0003	0.0003	0.0003	0.0003
Frechet (0,1,1)	0.1255	0.0697	0.1622	0.1057	0.0263	0.0116	0.1078	0.0215
Dagum (1,2,3)	0.0072	0.0059	0.0102	0.0077	0.0028	0.0033	0.0034	0.0032
Singh-Maddala (1,2,3)	0.0002	0.0003	0.0002	0.0002	0.0001	0.0002	0.0002	0.0002

Table 2. Hellinger distances comparing the performance of the moments, percentile matching and pdQ with quartile matching methods

Distribution	Moments	Percentile	pdQ
Uniform (2,5)	0.0000	0.0000	0.0000^a
Normal (0,1)	0.0000	0.0001	0.0001^a
Lognormal (0,1)	0.0166	0.0189	0.0066^b
Exponential (1)	0.0137	0.0074	0.0000^c
${\chi }_5^2$	0.0098	0.0046	0.0016^a
Pareto (7)	0.0067	0.0077	0.0000^c
Gamma (2)	0.0157	0.0062	0.0019^a
Beta (2,3)	0.0004	0.0007	0.0004^a
Weibull (2)	0.0000	0.0001	0.0000^b
Cauchy (0,1)	NA	0.0002	0.0003^a
Frechet (0,1,1)	NA	0.0215	0.0116^b
Dagum (1,2,3)	NA	0.0099	0.0028^a
Singh-Maddala (1,2,3)	0.0018	0.0007	0.0001^a

^aNo weighting ^bleft weighting ^cright weighting.

Figure 1. Plots of the true and approximating GLD probability densities for several distributions using the moment matching (green), percentile matching (blue) and pdQ (red) methods. The true probability density is the black curve.

Figure 2. Plots of the pdQ for several distributions using the percentile matching (blue) and pdQ (red) methods. The true pdQ is the black curve.

Table 3. Parameters chosen for the generalized lambda distribution using the pdQ method and quartile matching

Distribution	l1	l2	l3	l4	Weight
Uniform (2,5)	3.5	0.6667	1	1	None
Normal (0,1)	0	1.4420	0.1469	0.1469	None
Lognormal (0,1)	0.8038	1.8141	0.7589	−0.7082	Left
Exponential (1)	0	1	30209	0	Right
${\chi }_5^2$	4.0559	0.4977	0.5167	−0.1470	None
Pareto (7)	1	7	19695	−0.1429	Right
Gamma (2)	1.5291	1.1391	0.5840	−0.1867	None
Beta (2,3)	0.3770	5.3836	0.4958	0.2637	None
Weibull (2)	0.8056	2.7931	0.4393	0.0862	Left
Cauchy (0,1)	0	2.3025	-0.8418	-0.8418	None
Frechet (0,1,1)	1.0644	1.7144	0.7167	−1.4507	Left
Dagum (1,2,3)	1.7568	1.5779	0.4593	−0.7716	None
Singh-Maddala (1,2,3)	0.4782	4.4903	0.4647	−0.1661	None

Table 4. Parameters chosen for the generalized beta distribution with Hellinger distance using pdQ method with quartile matching to choice the best parameters

Distribution	Location	Scale	Shape 1	Shape 2	Hellinger	Weight
Uniform (2,5)	2	5	1	1	0.0000	None
Normal (0,1)	−81.1991	81.2063	3296.9272	3297.2195	0.0000	None
Lognormal (0,1)	0.0358	35,968,259	1.0784	28,694,641	0.0139	Left
Exponential (1)	0	79,194,935	1	79,194,934	0.0000	None
${\chi }_5^2$	0	4,141,322	2.5	2,070,656	0.0000	None
Pareto (7)	0.9973	115,128.1	0.9354	680,085.2	0.0122	Left
Gamma (2)	0	6,421,984	2	6,421,980	0.0000	None
Beta (2,3)	0	1	2	3	0.0000	None
Weibull (2)	−0.0518	3.3441	2.7716	7.2798	0.0007	None
Cauchy (0,1)	−837.5752	829.6024	158,813.3	157,301.6	0.0749	None
Frechet (0,1,1)	−0.1662	6,466,193	0.8929	2,370,934	0.1090	Left
Dagum (1,2,3)	0.4140	32,616,990	1.5629	26,243,197	0.0184	Both
Singh-Maddala (1,2,3)	−0.0160	339,986.2	2.7880	1,592,214	0.0011	Left
Triangular (0,1,0)	0	1	2	1	0.0000	None