Estimates of Inequality Indices Based on Simple Random, Ranked Set, and Systematic Sampling

Table 15

Relative efficiency of absolute Lorenz index for exponential (𝜆=3) distribution.

𝑟	𝑚	MSE (RSS)	MSE (SRS)	MSE (SYS)	𝐸(1)	𝐸(2)	𝐸(3)
2	3	0.0001772438	0.0002139371	9.98993𝑒−05	1.207022	2.141528	1.774225
	4	0.0001160937	0.001597166	4.811047𝑒−05	1.375756	3.319789	2.413065
	5	8.333168𝑒−05	0.0001291083	2.622109𝑒−05	1.549330	4.923834	3.178040
	6	6.31974𝑒−05	0.0001062572	1.622325𝑒−05	1.681354	6.549686	3.895483
5	3	7.01547𝑒−05	8.522752𝑒−05	3.934262𝑒−05	1.214851	2.166290	1.783173
	4	4.750068𝑒−05	6.440414𝑒−05	1.899583𝑒−05	1.355857	3.390436	2.500585
	5	3.310539𝑒−05	5.144982𝑒−05	1.046542𝑒−05	1.554122	4.916173	3.163312
	6	2.532269𝑒−05	4.341858𝑒−05	6.32867𝑒−06	1.714612	6.860617	4.001266
20	3	1.813884𝑒−05	2.141139𝑒−05	1.017992𝑒−05	1.180417	2.103296	1.781825
	4	1.161498𝑒−05	1.671353𝑒−05	4.712122𝑒−06	1.438963	3.546922	2.464915
	5	8.348938𝑒−06	1.282813𝑒−05	2.611685𝑒−06	1.536498	4.911821	3.196763
	6	6.300591𝑒−06	1.076669𝑒−05	1.601593𝑒−06	1.708839	6.722488	3.933953
100	3	3.576201𝑒−06	4.282342𝑒−06	1.949306𝑒−06	1.197455	2.196855	1.834602
	4	2.350533𝑒−06	3.236420𝑒−06	9.673106𝑒−07	1.376887	3.345792	2.429967
	5	1.667317𝑒−06	2.622885𝑒−06	5.363255𝑒−07	1.573117	4.890472	3.108778
	6	1.220823𝑒−06	2.203291𝑒−06	3.159623𝑒−07	1.804759	6.973272	3.863825
200	3	1.776957𝑒−06	2.109820𝑒−06	1.023774𝑒−06	1.187322	2.060826	1.735693
	4	1.175403𝑒−06	1.581642𝑒−06	4.78013𝑒−07	1.345617	3.308784	2.458935
	5	8.277293𝑒−07	1.306533𝑒−06	2.650969𝑒−07	1.578454	4.928511	3.122365
	6	6.383724𝑒−07	1.065240𝑒−06	1.576827𝑒−07	1.668682	6.755592	4.048462
500	3	7.048948𝑒−07	8.251703𝑒−07	4.060795𝑒−07	1.170629	2.032041	1.735854
	4	4.695757𝑒−07	6.461275𝑒−07	1.943934𝑒−07	1.375981	3.323814	2.415595
	5	3.350298𝑒−07	5.157602𝑒−07	1.047704𝑒−07	1.539446	4.922766	3.197752
	6	2.468869𝑒−07	4.345264𝑒−07	6.523285𝑒−08	1.760022	6.661159	3.784702