Ordered Double Ranked Set Samples and Applications to Inference

Figures & data

Table 1 Means under ODRSS for different distributions μ^ODRSS_(r)(1 ⩽ r ⩽ m)

m	r	Exponential	Normal	Right Triangular	Rectangular
2	1	0.379762	−0.707860	−0.723609	−0.747806
2	2	1.620240	0.707860	0.723609	0.747806
3	1	0.232736	−1.011510	−1.091690	−1.057820
3	2	0.756055	0.000000	0.127562	0.000000
3	3	2.011210	1.011510	0.964133	1.057820
4	1	0.167926	−1.198450	−1.317940	−1.220770
4	2	0.501153	−0.328321	−0.257226	−0.417860
4	3	1.037920	0.328321	0.494394	0.417860
4	4	2.293000	1.198450	1.080770	1.220770
5	1	0.131399	−1.331560	−1.474290	−1.320590
5	2	0.376544	−0.537555	−0.524439	−0.675374
5	3	0.720279	0.000000	0.151823	0.000000
5	4	1.258820	0.537555	0.697460	0.675374
5	5	2.488430	1.331560	1.149440	1.320590

Table 2 Variances and covariances under ODRSS for different distributions, σ^ODRSS_{(r, s)}(1 ⩽ r < s ⩽ m)

m	r	s	Exponential	Normal	Right Triangular	Rectangular
2	1	1	0.105574	0.498934	0.684026	0.440786
2	1	2	0.044418	0.061241	0.070095	0.079214
2	2	2	1.125040	0.498934	0.268755	0.440786
3	1	1	0.035103	0.377156	0.487145	0.221987
3	1	2	0.013220	0.044542	0.067884	0.054986
3	1	3	0.002413	0.003699	0.004154	0.004785
3	2	2	0.142167	0.199377	0.262129	0.318043
3	2	3	0.060993	0.044542	0.034271	0.054986
3	3	3	1.151980	0.377156	0.113105	0.221987
4	1	1	0.017298	0.321365	0.375111	0.131262
4	1	2	0.006282	0.036885	0.058085	0.036179
4	1	3	0.001238	0.003653	0.005358	0.004763
4	1	4	0.000110	0.000180	0.000195	0.000224
4	2	2	0.051112	0.134566	0.217660	0.203849
4	2	3	0.019892	0.029967	0.039926	0.049983
4	2	4	0.004070	0.003653	0.003105	0.004763
4	3	3	0.154184	0.134566	0.130125	0.203849
4	3	4	0.068306	0.036885	0.019404	0.036179
4	4	4	1.162910	0.321365	0.061489	0.131262
5	1	1	0.010270	0.288308	0.304333	0.086270
5	1	2	0.003665	0.032404	0.049681	0.025063
5	1	3	0.000753	0.003447	0.005416	0.003895
5	1	4	8.44E-05	0.000232	0.000322	0.000299
5	1	5	0.003228	7.42E-06	7.75E-06	8.79E-06
5	2	2	0.026211	0.106894	0.182291	0.138278
5	2	3	0.009913	0.023737	0.037794	0.038399
5	2	4	0.002214	0.003466	0.004510	0.005642
5	2	5	0.009462	0.000232	0.000204	0.000299
5	3	3	0.057199	0.085561	0.118109	0.150739
5	3	4	0.023284	0.023737	0.025059	0.038399
5	3	5	0.022777	0.003447	0.002208	0.003895
5	4	4	0.160060	0.106894	0.077464	0.138278
5	4	5	0.103258	0.032404	0.012387	0.025063
5	5	5	1.063300	0.288308	0.038517	0.086270

Table 3 Values of the coefficients of the BLUEs-ODRSS for different distributions

	Exponential		Normal
m	γi	δi	γi	δi
2	1.306140	−0.806142	0.500000	−0.706354
2	−0.306140	0.806142	0.500000	0.706354
3	1.208800	−1.077820	0.257440	−0.494310
3	−0.110430	0.730484	0.485119	0.000000
3	−0.098370	0.347334	0.257440	0.494310
4	1.106790	−1.156320	0.153080	−0.360565
4	0.025046	0.466231	0.346920	−0.206754
4	−0.082770	0.495468	0.346920	0.206754
4	−0.049060	0.194622	0.153080	0.360565
5	1.014540	−1.170300	0.100001	−0.272908
5	0.095549	0.284434	0.251348	−0.254126
5	−0.020190	0.410634	0.297303	0.000000
5	−0.056070	0.351061	0.251348	0.254126
5	−0.033820	0.124167	0.100001	0.272908
	Right Triangular		Rectangular
m	γi	δi	γi	δi
2	0.500000	−0.690980	0.500000	−0.668623
2	0.500000	0.690981	0.500000	0.668623
3	0.355355	−0.419010	0.409164	−0.472668
3	0.279215	−0.165650	0.181673	0.000000
3	0.365429	0.584667	0.409164	0.472668
4	0.270311	−0.305730	0.350801	−0.396824
4	0.224410	−0.169500	0.149199	−0.037259
4	0.225303	−0.067950	0.149199	0.037259
4	0.279976	0.543187	0.350801	0.396824
5	0.215706	−0.238250	0.313222	−0.349765
5	0.184962	−0.158570	0.124953	−0.056420
5	0.192705	−0.104640	0.123650	0.000000
5	0.180623	−0.011240	0.124953	0.056420
5	0.226005	0.512701	0.313222	0.349765

Table 4 Variances–covariances of ORSS and ODRSS for different distributions

		Exponential		Normal		Right Triangular		Rectangular
m		ORSS	ODRSS	ORSS	ODRSS	ORSS	ODRSS	ORSS	ODRSS
2	V*1	0.330782	0.250030	0.340845	0.280087	0.340000	0.273243	0.333333	0.260000
	V*2	0.836735	0.741999	0.498674	0.436762	0.455300	0.387975	0.388889	0.323286
	V*3	0.119756	0.075905	0.169971	0.122331	0.130602	0.085427	0.129630	0.084054
3	V*1	0.089441	0.061395	0.171984	0.119656	0.167133	0.118599	0.151013	0.102776
	V*2	0.341057	0.263943	0.220854	0.182503	0.172496	0.132135	0.131572	0.097052
	V*3	0.016851	0.008325	0.037983	0.021838	0.021306	0.010953	0.019869	0.009975
4	V*1	0.039434	0.025649	0.103622	0.063286	0.099690	0.064400	0.083705	0.052234
	V*2	0.189775	0.130998	0.132210	0.102365	0.091974	0.065153	0.063714	0.043554
	V*3	0.004560	0.001883	0.013700	0.006478	0.006143	0.002681	0.005333	0.002275
5	V*1	0.021672	0.013354	0.069231	0.038059	0.066423	0.039678	0.052496	0.030675
	V*2	0.121515	0.079244	0.089951	0.065228	0.057458	0.038226	0.036767	0.023905
	V*3	0.001695	0.000632	0.006227	0.002483	0.002371	0.000912	0.001930	0.000733

Table 5 REs of the BLUEs-ODRSS with respect to the BLUEs-ORSS

m	RE	Exponential	Normal	Right Triangular	Rectangular
2	RE1	1.32299	1.2169	1.2443	1.2821
	RE2	1.1277	1.1418	1.1735	1.2029
	RE3	1.5777	1.3894	1.5288	1.5422
3	RE1	1.4568	1.4373	1.4092	1.4693
	RE2	1.2922	1.2101	1.3055	1.3557
	RE3	2.0242	1.7394	1.9452	1.9919
4	RE1	1.5375	1.6374	1.5480	1.6025
	RE2	1.4487	1.2916	1.4117	1.4629
	RE3	2.4220	2.1148	2.2914	2.3443
5	RE1	1.6229	1.8190	1.6740	1.7113
	RE2	1.5334	1.3790	1.5031	1.5380
	RE3	2.6821	2.5085	2.6007	2.6321

Table 6 ODRSS- and ORSS-based 90% (bold 95%) confidence intervals for the pth quantile with ECL

m	Design	p→	0.3	0.4	0.5	0.6	0.7
3	ORSS	[r, s]			[1,3]
		ECL			89.06
	ODRSS	[r, s]		[1,3]	[1,3]	[1,3]
		ECL		90.68	94.83	90.68
4	ORSS	[r, s]		[1,4]	[1,4]	[1,4]
		ECL		94.86	97.48	94.86
	ODRSS	[r, s]	[1,3]	[1,3]		[2,4]	[2,4]
		ECL	89.86	92.97		92.97	89.86
		[r, s]	[1,4]	[1,4]	[1,4]	[1,4]	[1,4]
		ECL	90.53	98.00	99.55	98.00	90.53
5	ORSS	[r, s]	[1,4]	[1,4]	[1,4]
		ECL	92.60	96.46	91.09
		[r, s]	[1,5]	[1,5]	[1,5]	[1,5]	[1,5]
		ECL	92.81	98.37	99.54	98.37	92.81
		[r, s]			[2,5]	[2,5]	[2,5]
		ECL			91.09	96.46	92.60
	ODRSS	[r, s]	[1,3]		[2,4]		[3,5]
		ECL	92.99		91.89		92.99
		[r, s]	[1,4]	[1,4]	[1,4]
		ECL	96.63	99.35	95.93
		[r, s]	[1,5]	[1,5]	[1,5]	[1,5]	[1,5]
		ECL	96.64	99.70	99.98	99.70	96.64
		[r, s]			[2,5]	[2,5]	[2,5]
		ECL			95.93	99.35	96.63

Table 7 ODRSS- and ORSS-based 90% and 95% upper confidence intervals for the pth quantile with ECL

			0.1		0.2		0.3		0.4		0.5		0.6		0.7
m	Design	p→	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%
2	ORSS	[${\it{t_u}}$]	2	2	2	2	2	2	2
		ECL	99.81	99.81	98.56	98.56	95.41	95.41	89.76
	ODRSS	[${\it{t_u}}$]	2	2	2	2	2	2	2
		ECL	99.96	99.96	99.44	99.44	97.46	97.46	92.86
3	ORSS	[${\it{t_u}}$]	2	2	2	2	3	3	3	3	3	3
		ECL	99.21	99.21	94.53	94.53	99.62	99.62	98.23	98.23	94.53	94.53
	ODRSS	[${\it{t_u}}$]	2	2	2	2	3	3	3	3	3	3	3
		ECL	99.77	99.77	96.99	96.99	99.95	99.95	99.52	99.52	97.41	97.41	91.16
4	ORSS	[${\it{t_u}}$]	2	2	3	3	3	3	3	4	4	4	4	4
		ECL	98.06	98.06	99.69	99.69	97.55	97.55	90.63	99.79	98.74	98.74	95.07	95.07
	ODRSS	[${\it{t_u}}$]	2	2	2	3	3	3	3	3	4	4	4	4	4
		ECL	99.26	99.26	91.53	99.97	99.32	99.32	94.96	94.96	99.77	99.77	98.01	98.01	90.53
5	ORSS	[${\it{t_u}}$]	2	2	3	3	3	4	4	4	4	5	5	5	5
		ECL	96.28	96.28	98.84	98.84	92.22	99.78	98.08	98.08	91.33	99.77	98.38	98.38	92.82
	ODRSS	[${\it{t_u}}$]	2	2	3	3	3	3	4	4	4	4	5	5	5	5
		ECL	98.26	98.26	99.81	99.81	96.35	96.35	99.65	99.65	95.94	95.94	99.7	99.7	96.64	96.64

Table 8 ODRSS- and ORSS-based 90% and 95% lower confidence intervals for the pth quantile with ECL

			0.3		0.4		0.5		0.6		0.7		0.8		0.9
m	Design	p→	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%	90%	95%
2	ORSS	[${\it{t_l}}$]							1		1	1	1	1	1	1
		ECL							89.76		95.41	95.41	98.56	98.56	99.81	99.81
	ODRSS	[${\it{t_l}}$]							1		1	1	1	1	1	1
		ECL							92.86		97.46	97.46	99.44	99.44	99.96	99.96
3	ORSS	[${\it{t_l}}$]					1	1	1	1	1	1	2	2	2	2
		ECL					94.53	94.53	98.23	98.23	99.62	99.62	94.53	94.53	99.21	99.21
	ODRSS	[${\it{t_l}}$]			1		1	1	1	1	1	1	2	2	2	2
		ECL			91.16		97.41	97.41	99.52	99.52	99.95	99.95	96.99	96.99	99.77	99.77
4	ORSS	[${\it{t_l}}$]			1	1	1	1	2	1	2	2	2	2	3	3
		ECL			95.07	95.07	98.74	98.74	90.63	99.79	97.55	97.55	99.69	99.69	98.06	98.06
	ODRSS	[${\it{t_l}}$]	1		1	1	1	1	2	2	2	2	3	2	3	3
		ECL	90.53		98.01	98.01	99.77	99.77	94.96	94.96	99.32	99.32	91.53	99.97	99.26	99.26
5	ORSS	[${\it{t_l}}$]	1		1	1	2	1	2	2	3	2	3	3	4	4
		ECL	92.82		98.38	98.38	91.33	99.77	98.08	98.08	92.22	99.78	98.84	98.84	96.28	96.28
	ODRSS	[${\it{t_l}}$]	1	1	1	1	2	2	2	2	3	3	3	3	4	4
		ECL	96.64	96.64	99.7	99.7	95.94	95.94	99.65	99.65	96.35	96.35	99.81	99.81	98.26	98.26

Table 9 Two-sided tolerance intervals 90% (bold 95%) that cover γ proportion of the population with ETL

m	Design	γ→	0.1	0.2	0.3	0.4	0.5	0.6
3	ORSS	[r, s]	[1,3]	[1,3]
		ETL	98.82	95.15
	ODRSS	[r, s]	[1,3]	[1,3]	[1,3]
		ETL	99.46	97.51	93.22
4	ORSS	[r, s]	[1,3]	[1,3]	[1,4]	[1,4]
		ETL	97.53	90.07	97.37	92.83
		[r, s]		[1,4]
		ETL		99.31
		[r, s]	[2,4]	[2,4]
		ETL	97.53	90.07
	ODRSS	[r, s]	[1,3]	[1,3]	[1,4]	[1,4]	[1,4]
		ETL	98.77	94.23	99.17	96.79	90.36
		[r, s]		[1,4]
		ETL		99.85
		[r, s]	[2,4]	[2,4]
		ETL	98.77	94.23
5	ORSS	[r, s]	[1,3]	[1,4]	[1,4]	[1,5]	[1,5]
		ETL	95.88	98.15	93.19	97.95	93.49
		[r, s]	[2,4]		[1,5]
		ETL	95.50		99.50
		[r, s]	[3,5]	[2,5]	[2,5]
		ETL	95.88	98.15	93.19
	ODRSS	[r, s]	[1,3]	[1,3]	[1,4]	[1,4]	[1,5]	[1,5]
		ETL	97.86	89.58	97.13	89.37	97.41	89.94
		[r, s]		[1,4]
		ETL		99.49
		[r, s]	[3,5]	[3,5]	[2,5]	[2,5]
		ETL	97.86	89.58	97.13	89.37
		[r, s]	[2,4]	[2,5]
		ETL	97.56	99.49