Table 2. Open in new tab Simulation...

			\|$n=500$\|		\|$n=2000$\|
Estimation of \|$\Theta _{X}$\|
[0.05in]	6.931	Ideal	\|$\!\!\!\!-$\|0.034	0.245	0.247	0.950	\|$\!\!\!\!-$\|0.013	0.122	0.123	0.982
Case 1
0.1	6.931	EC	0.007	0.379	0.474	0.948	\|$\!\!\!\!-$\|0.007	0.168	0.197	0.978
0.3	6.931	EC	\|$\!\!\!\!-$\|0.041	0.498	0.637	0.942	\|$\!\!\!\!-$\|0.032	0.242	0.272	0.950
0.5	6.931	EC	\|$\!\!\!\!-$\|0.101	0.582	0.805	0.913	\|$\!\!\!\!-$\|0.104	0.283	0.345	0.948
0.7	6.931	EC	\|$\!\!\!\!-$\|0.211	0.632	0.894	0.922	\|$\!\!\!\!-$\|0.141	0.333	0.411	0.910
Case 2
0.1	6.931	EC	0.013	0.402	0.452	0.906	\|$\!\!\!\!-$\|0.007	0.176	0.195	0.964
0.3	6.931	EC	\|$\!\!\!\!-$\|0.033	0.440	0.603	0.940	\|$\!\!\!\!-$\|0.041	0.211	0.256	0.962
0.5	6.931	EC	\|$\!\!\!\!-$\|0.016	0.578	0.776	0.920	\|$\!\!\!\!-$\|0.064	0.286	0.324	0.936
0.7	6.931	EC	\|$\!\!\!\!-$\|0.059	0.700	0.966	0.890	\|$\!\!\!\!-$\|0.086	0.320	0.394	0.936
Case 3
0.1	6.931	EC	0.001	0.353	0.412	0.934	0.015	0.149	0.179	0.952
0.3	6.931	EC	\|$\!\!\!\!-$\|0.019	0.367	0.477	0.964	\|$\!\!\!\!-$\|0.012	0.186	0.211	0.958
0.5	6.931	EC	\|$\!\!\!\!-$\|0.026	0.425	0.549	0.970	\|$\!\!\!\!-$\|0.025	0.228	0.249	0.958
0.7	6.931	EC	\|$\!\!\!\!-$\|0.043	0.550	0.676	0.920	\|$\!\!\!\!-$\|0.013	0.265	0.293	0.938
Estimation of \|$\Theta _{\bar{Z}}$\|
Case 1
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.039	0.255	0.255	0.940	\|$\!\!\!\!-$\|0.009	0.114	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.032	0.283	0.263	0.968	\|$\!\!\!\!-$\|0.040	0.119	0.132	0.956
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.053	0.271	0.270	0.942	\|$\!\!\!\!-$\|0.013	0.142	0.134	0.964
0.7	6.017	Naive	\|$\!\!\!\!-$\|0.035	0.264	0.274	0.938	\|$\!\!\!\!-$\|0.011	0.135	0.136	0.974
Case 2
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.036	0.262	0.255	0.940	\|$\!\!\!\!-$\|0.011	0.125	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.026	0.257	0.265	0.952	\|$\!\!\!\!-$\|0.015	0.121	0.131	0.952
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.022	0.255	0.271	0.942	\|$\!\!\!\!-$\|0.008	0.126	0.134	0.954
0.7	6.017	Naive	\|$\!\!\!-$\|0.019	0.261	0.274	0.940	\|$\!\!\!\!-$\|0.011	0.121	0.136	0.962
Case 3
0.1	6.818	Naive	\|$\!\!\!\!-$\|0.023	0.251	0.253	0.958	\|$\!\!\!\!-$\|0.001	0.121	0.126	0.940
0.3	6.629	Naive	\|$\!\!\!\!-$\|0.023	0.243	0.258	0.958	0.000	0.135	0.129	0.942
0.5	6.476	Naive	\|$\!\!\!\!-$\|0.016	0.258	0.262	0.954	0.003	0.131	0.131	0.954
0.7	6.347	Naive	\|$\!\!\!\!-$\|0.018	0.274	0.267	0.938	\|$\!\!\!\!-$\|0.005	0.136	0.132	0.948

|$n=500$|

|$n=2000$|

|$\Sigma ^{1/2}$|

True

Method

MAD

RMAD

MAD

RMAD

Estimation of |$\Theta _{X}$|

[0.05in]

6.931

Ideal

|$\!\!\!\!-$|0.034

0.245

0.247

0.950

|$\!\!\!\!-$|0.013

0.122

0.123

0.982

Case 1

0.1

6.931

0.007

0.379

0.474

0.948

|$\!\!\!\!-$|0.007

0.168

0.197

0.978

0.3

6.931

|$\!\!\!\!-$|0.041

0.498

0.637

0.942

|$\!\!\!\!-$|0.032

0.242

0.272

0.950

0.5

6.931

|$\!\!\!\!-$|0.101

0.582

0.805

0.913

|$\!\!\!\!-$|0.104

0.283

0.345

0.948

0.7

6.931

|$\!\!\!\!-$|0.211

0.632

0.894

0.922

|$\!\!\!\!-$|0.141

0.333

0.411

0.910

Case 2

0.1

6.931

0.013

0.402

0.452

0.906

|$\!\!\!\!-$|0.007

0.176

0.195

0.964

0.3

6.931

|$\!\!\!\!-$|0.033

0.440

0.603

0.940

|$\!\!\!\!-$|0.041

0.211

0.256

0.962

0.5

6.931

|$\!\!\!\!-$|0.016

0.578

0.776

0.920

|$\!\!\!\!-$|0.064

0.286

0.324

0.936

0.7

6.931

|$\!\!\!\!-$|0.059

0.700

0.966

0.890

|$\!\!\!\!-$|0.086

0.320

0.394

0.936

Case 3

0.1

6.931

0.001

0.353

0.412

0.934

0.015

0.149

0.179

0.952

0.3

6.931

|$\!\!\!\!-$|0.019

0.367

0.477

0.964

|$\!\!\!\!-$|0.012

0.186

0.211

0.958

0.5

6.931

|$\!\!\!\!-$|0.026

0.425

0.549

0.970

|$\!\!\!\!-$|0.025

0.228

0.249

0.958

0.7

6.931

|$\!\!\!\!-$|0.043

0.550

0.676

0.920

|$\!\!\!\!-$|0.013

0.265

0.293

0.938

Estimation of |$\Theta _{\bar{Z}}$|

Case 1

0.1

6.720

Naive

|$\!\!\!\!-$|0.039

0.255

0.940

|$\!\!\!\!-$|0.009

0.114

0.127

0.958

0.3

6.411

Naive

|$\!\!\!\!-$|0.032

0.283

0.263

0.968

|$\!\!\!\!-$|0.040

0.119

0.132

0.956

0.5

6.186

Naive

|$\!\!\!\!-$|0.053

0.271

0.270

0.942

|$\!\!\!\!-$|0.013

0.142

0.134

0.964

0.7

6.017

Naive

|$\!\!\!\!-$|0.035

0.264

0.274

0.938

|$\!\!\!\!-$|0.011

0.135

0.136

0.974

Case 2

0.1

6.720

Naive

|$\!\!\!\!-$|0.036

0.262

0.255

0.940

|$\!\!\!\!-$|0.011

0.125

0.127

0.958

0.3

6.411

Naive

|$\!\!\!\!-$|0.026

0.257

0.265

0.952

|$\!\!\!\!-$|0.015

0.121

0.131

0.952

0.5

6.186

Naive

|$\!\!\!\!-$|0.022

0.255

0.271

0.942

|$\!\!\!\!-$|0.008

0.126

0.134

0.954

0.7

6.017

Naive

|$\!\!\!-$|0.019

0.261

0.274

0.940

|$\!\!\!\!-$|0.011

0.121

0.136

0.962

Case 3

0.1

6.818

Naive

|$\!\!\!\!-$|0.023

0.251

0.253

0.958

|$\!\!\!\!-$|0.001

0.121

0.126

0.940

0.3

6.629

Naive

|$\!\!\!\!-$|0.023

0.243

0.258

0.958

0.000

0.135

0.129

0.942

0.5

6.476

Naive

|$\!\!\!\!-$|0.016

0.258

0.262

0.954

0.003

0.131

0.954

0.7

6.347

Naive

|$\!\!\!\!-$|0.018

0.274

0.267

0.938

|$\!\!\!\!-$|0.005

0.136

0.132

0.948

			\|$n=500$\|		\|$n=2000$\|
Estimation of \|$\Theta _{X}$\|
[0.05in]	6.931	Ideal	\|$\!\!\!\!-$\|0.034	0.245	0.247	0.950	\|$\!\!\!\!-$\|0.013	0.122	0.123	0.982
Case 1
0.1	6.931	EC	0.007	0.379	0.474	0.948	\|$\!\!\!\!-$\|0.007	0.168	0.197	0.978
0.3	6.931	EC	\|$\!\!\!\!-$\|0.041	0.498	0.637	0.942	\|$\!\!\!\!-$\|0.032	0.242	0.272	0.950
0.5	6.931	EC	\|$\!\!\!\!-$\|0.101	0.582	0.805	0.913	\|$\!\!\!\!-$\|0.104	0.283	0.345	0.948
0.7	6.931	EC	\|$\!\!\!\!-$\|0.211	0.632	0.894	0.922	\|$\!\!\!\!-$\|0.141	0.333	0.411	0.910
Case 2
0.1	6.931	EC	0.013	0.402	0.452	0.906	\|$\!\!\!\!-$\|0.007	0.176	0.195	0.964
0.3	6.931	EC	\|$\!\!\!\!-$\|0.033	0.440	0.603	0.940	\|$\!\!\!\!-$\|0.041	0.211	0.256	0.962
0.5	6.931	EC	\|$\!\!\!\!-$\|0.016	0.578	0.776	0.920	\|$\!\!\!\!-$\|0.064	0.286	0.324	0.936
0.7	6.931	EC	\|$\!\!\!\!-$\|0.059	0.700	0.966	0.890	\|$\!\!\!\!-$\|0.086	0.320	0.394	0.936
Case 3
0.1	6.931	EC	0.001	0.353	0.412	0.934	0.015	0.149	0.179	0.952
0.3	6.931	EC	\|$\!\!\!\!-$\|0.019	0.367	0.477	0.964	\|$\!\!\!\!-$\|0.012	0.186	0.211	0.958
0.5	6.931	EC	\|$\!\!\!\!-$\|0.026	0.425	0.549	0.970	\|$\!\!\!\!-$\|0.025	0.228	0.249	0.958
0.7	6.931	EC	\|$\!\!\!\!-$\|0.043	0.550	0.676	0.920	\|$\!\!\!\!-$\|0.013	0.265	0.293	0.938
Estimation of \|$\Theta _{\bar{Z}}$\|
Case 1
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.039	0.255	0.255	0.940	\|$\!\!\!\!-$\|0.009	0.114	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.032	0.283	0.263	0.968	\|$\!\!\!\!-$\|0.040	0.119	0.132	0.956
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.053	0.271	0.270	0.942	\|$\!\!\!\!-$\|0.013	0.142	0.134	0.964
0.7	6.017	Naive	\|$\!\!\!\!-$\|0.035	0.264	0.274	0.938	\|$\!\!\!\!-$\|0.011	0.135	0.136	0.974
Case 2
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.036	0.262	0.255	0.940	\|$\!\!\!\!-$\|0.011	0.125	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.026	0.257	0.265	0.952	\|$\!\!\!\!-$\|0.015	0.121	0.131	0.952
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.022	0.255	0.271	0.942	\|$\!\!\!\!-$\|0.008	0.126	0.134	0.954
0.7	6.017	Naive	\|$\!\!\!-$\|0.019	0.261	0.274	0.940	\|$\!\!\!\!-$\|0.011	0.121	0.136	0.962
Case 3
0.1	6.818	Naive	\|$\!\!\!\!-$\|0.023	0.251	0.253	0.958	\|$\!\!\!\!-$\|0.001	0.121	0.126	0.940
0.3	6.629	Naive	\|$\!\!\!\!-$\|0.023	0.243	0.258	0.958	0.000	0.135	0.129	0.942
0.5	6.476	Naive	\|$\!\!\!\!-$\|0.016	0.258	0.262	0.954	0.003	0.131	0.131	0.954
0.7	6.347	Naive	\|$\!\!\!\!-$\|0.018	0.274	0.267	0.938	\|$\!\!\!\!-$\|0.005	0.136	0.132	0.948

Table 2.

Open in new tab

Simulation results for estimation of |$\Theta _{X}$| and |$\Theta _{\bar{Z}}$|⁠.

			\|$n=500$\|				\|$n=2000$\|
\|$\Sigma ^{1/2}$\|	True	Method	B	MAD	RMAD	CP	B	MAD	RMAD	CP
Estimation of \|$\Theta _{X}$\|
[0.05in]	6.931	Ideal	\|$\!\!\!\!-$\|0.034	0.245	0.247	0.950	\|$\!\!\!\!-$\|0.013	0.122	0.123	0.982
Case 1
0.1	6.931	EC	0.007	0.379	0.474	0.948	\|$\!\!\!\!-$\|0.007	0.168	0.197	0.978
0.3	6.931	EC	\|$\!\!\!\!-$\|0.041	0.498	0.637	0.942	\|$\!\!\!\!-$\|0.032	0.242	0.272	0.950
0.5	6.931	EC	\|$\!\!\!\!-$\|0.101	0.582	0.805	0.913	\|$\!\!\!\!-$\|0.104	0.283	0.345	0.948
0.7	6.931	EC	\|$\!\!\!\!-$\|0.211	0.632	0.894	0.922	\|$\!\!\!\!-$\|0.141	0.333	0.411	0.910
Case 2
0.1	6.931	EC	0.013	0.402	0.452	0.906	\|$\!\!\!\!-$\|0.007	0.176	0.195	0.964
0.3	6.931	EC	\|$\!\!\!\!-$\|0.033	0.440	0.603	0.940	\|$\!\!\!\!-$\|0.041	0.211	0.256	0.962
0.5	6.931	EC	\|$\!\!\!\!-$\|0.016	0.578	0.776	0.920	\|$\!\!\!\!-$\|0.064	0.286	0.324	0.936
0.7	6.931	EC	\|$\!\!\!\!-$\|0.059	0.700	0.966	0.890	\|$\!\!\!\!-$\|0.086	0.320	0.394	0.936
Case 3
0.1	6.931	EC	0.001	0.353	0.412	0.934	0.015	0.149	0.179	0.952
0.3	6.931	EC	\|$\!\!\!\!-$\|0.019	0.367	0.477	0.964	\|$\!\!\!\!-$\|0.012	0.186	0.211	0.958
0.5	6.931	EC	\|$\!\!\!\!-$\|0.026	0.425	0.549	0.970	\|$\!\!\!\!-$\|0.025	0.228	0.249	0.958
0.7	6.931	EC	\|$\!\!\!\!-$\|0.043	0.550	0.676	0.920	\|$\!\!\!\!-$\|0.013	0.265	0.293	0.938
Estimation of \|$\Theta _{\bar{Z}}$\|
Case 1
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.039	0.255	0.255	0.940	\|$\!\!\!\!-$\|0.009	0.114	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.032	0.283	0.263	0.968	\|$\!\!\!\!-$\|0.040	0.119	0.132	0.956
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.053	0.271	0.270	0.942	\|$\!\!\!\!-$\|0.013	0.142	0.134	0.964
0.7	6.017	Naive	\|$\!\!\!\!-$\|0.035	0.264	0.274	0.938	\|$\!\!\!\!-$\|0.011	0.135	0.136	0.974
Case 2
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.036	0.262	0.255	0.940	\|$\!\!\!\!-$\|0.011	0.125	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.026	0.257	0.265	0.952	\|$\!\!\!\!-$\|0.015	0.121	0.131	0.952
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.022	0.255	0.271	0.942	\|$\!\!\!\!-$\|0.008	0.126	0.134	0.954
0.7	6.017	Naive	\|$\!\!\!-$\|0.019	0.261	0.274	0.940	\|$\!\!\!\!-$\|0.011	0.121	0.136	0.962
Case 3
0.1	6.818	Naive	\|$\!\!\!\!-$\|0.023	0.251	0.253	0.958	\|$\!\!\!\!-$\|0.001	0.121	0.126	0.940
0.3	6.629	Naive	\|$\!\!\!\!-$\|0.023	0.243	0.258	0.958	0.000	0.135	0.129	0.942
0.5	6.476	Naive	\|$\!\!\!\!-$\|0.016	0.258	0.262	0.954	0.003	0.131	0.131	0.954
0.7	6.347	Naive	\|$\!\!\!\!-$\|0.018	0.274	0.267	0.938	\|$\!\!\!\!-$\|0.005	0.136	0.132	0.948

			\|$n=500$\|				\|$n=2000$\|
\|$\Sigma ^{1/2}$\|	True	Method	B	MAD	RMAD	CP	B	MAD	RMAD	CP
Estimation of \|$\Theta _{X}$\|
[0.05in]	6.931	Ideal	\|$\!\!\!\!-$\|0.034	0.245	0.247	0.950	\|$\!\!\!\!-$\|0.013	0.122	0.123	0.982
Case 1
0.1	6.931	EC	0.007	0.379	0.474	0.948	\|$\!\!\!\!-$\|0.007	0.168	0.197	0.978
0.3	6.931	EC	\|$\!\!\!\!-$\|0.041	0.498	0.637	0.942	\|$\!\!\!\!-$\|0.032	0.242	0.272	0.950
0.5	6.931	EC	\|$\!\!\!\!-$\|0.101	0.582	0.805	0.913	\|$\!\!\!\!-$\|0.104	0.283	0.345	0.948
0.7	6.931	EC	\|$\!\!\!\!-$\|0.211	0.632	0.894	0.922	\|$\!\!\!\!-$\|0.141	0.333	0.411	0.910
Case 2
0.1	6.931	EC	0.013	0.402	0.452	0.906	\|$\!\!\!\!-$\|0.007	0.176	0.195	0.964
0.3	6.931	EC	\|$\!\!\!\!-$\|0.033	0.440	0.603	0.940	\|$\!\!\!\!-$\|0.041	0.211	0.256	0.962
0.5	6.931	EC	\|$\!\!\!\!-$\|0.016	0.578	0.776	0.920	\|$\!\!\!\!-$\|0.064	0.286	0.324	0.936
0.7	6.931	EC	\|$\!\!\!\!-$\|0.059	0.700	0.966	0.890	\|$\!\!\!\!-$\|0.086	0.320	0.394	0.936
Case 3
0.1	6.931	EC	0.001	0.353	0.412	0.934	0.015	0.149	0.179	0.952
0.3	6.931	EC	\|$\!\!\!\!-$\|0.019	0.367	0.477	0.964	\|$\!\!\!\!-$\|0.012	0.186	0.211	0.958
0.5	6.931	EC	\|$\!\!\!\!-$\|0.026	0.425	0.549	0.970	\|$\!\!\!\!-$\|0.025	0.228	0.249	0.958
0.7	6.931	EC	\|$\!\!\!\!-$\|0.043	0.550	0.676	0.920	\|$\!\!\!\!-$\|0.013	0.265	0.293	0.938
Estimation of \|$\Theta _{\bar{Z}}$\|
Case 1
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.039	0.255	0.255	0.940	\|$\!\!\!\!-$\|0.009	0.114	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.032	0.283	0.263	0.968	\|$\!\!\!\!-$\|0.040	0.119	0.132	0.956
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.053	0.271	0.270	0.942	\|$\!\!\!\!-$\|0.013	0.142	0.134	0.964
0.7	6.017	Naive	\|$\!\!\!\!-$\|0.035	0.264	0.274	0.938	\|$\!\!\!\!-$\|0.011	0.135	0.136	0.974
Case 2
0.1	6.720	Naive	\|$\!\!\!\!-$\|0.036	0.262	0.255	0.940	\|$\!\!\!\!-$\|0.011	0.125	0.127	0.958
0.3	6.411	Naive	\|$\!\!\!\!-$\|0.026	0.257	0.265	0.952	\|$\!\!\!\!-$\|0.015	0.121	0.131	0.952
0.5	6.186	Naive	\|$\!\!\!\!-$\|0.022	0.255	0.271	0.942	\|$\!\!\!\!-$\|0.008	0.126	0.134	0.954
0.7	6.017	Naive	\|$\!\!\!-$\|0.019	0.261	0.274	0.940	\|$\!\!\!\!-$\|0.011	0.121	0.136	0.962
Case 3
0.1	6.818	Naive	\|$\!\!\!\!-$\|0.023	0.251	0.253	0.958	\|$\!\!\!\!-$\|0.001	0.121	0.126	0.940
0.3	6.629	Naive	\|$\!\!\!\!-$\|0.023	0.243	0.258	0.958	0.000	0.135	0.129	0.942
0.5	6.476	Naive	\|$\!\!\!\!-$\|0.016	0.258	0.262	0.954	0.003	0.131	0.131	0.954
0.7	6.347	Naive	\|$\!\!\!\!-$\|0.018	0.274	0.267	0.938	\|$\!\!\!\!-$\|0.005	0.136	0.132	0.948

EC, error corrected; B, empirical bias based on the median; MAD, empirical median absolute deviation; RMAD, resampling median absolute deviation; CP, empirical coverage probability of 95% confidence interval.

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