$\rho$

$\rho ,\beta$

$\epsilon ~N\left(0,{0.5}^2\right)$

$\epsilon ~U\left[-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}\right]$

$\epsilon ~\frac{1}{8}{\chi }^2\left(8\right)-1$

Mean

SD

Mean

SD

Mean

SD

Rook

−0.9

$\rho$

−0.8998

0.0066

−0.8991

0.0083

−0.8992

0.0076

$\beta$

3.0013

0.0283

3.0022

0.0478

3.0017

0.0281

−0.5

$\rho$

−0.5007

0.0154

−0.4992

0.0215

−0.4996

0.0162

$\beta$

2.9997

0.0320

2.9988

0.0463

2.9971

0.0345

−0.1

$\rho$

−0.1003

0.0159

−0.0991

0.0273

−0.1005

0.0153

$\beta$

3.0022

0.0400

2.9981

0.0457

3.0016

0.0387

0

$\rho$

0.0004

0.0160

0.0009

0.0284

0.0001

0.0141

$\beta$

3.0002

0.0427

3.0005

0.0450

3.0004

0.0396

0.1

$\rho$

0.0998

0.0137

0.1000

0.0274

0.1002

0.0148

$\beta$

3.0024

0.0430

2.9968

0.0453

2.9995

0.0430

0.5

$\rho$

0.4998

0.0091

0.4988

0.0219

0.4999

0.0086

$\beta$

2.9992

0.0461

2.9998

0.0441

2.9991

0.0446

0.9

$\rho$

0.9002

0.0018

0.8996

0.0082

0.9002

0.0019

$\beta$

2.9968

0.0450

3.0009

0.0489

2.9970

0.0473

Queen

−0.9

$\rho$

−0.8988

0.0215

−0.8999

0.0219

−0.8992

0.0217

$\beta$

3.0003

0.0343

3.0011

0.0352

3.0014

0.0333

−0.5

$\rho$

−0.5016

0.0209

−0.5005

0.0200

−0.5003

0.0205

$\beta$

3.0008

0.0375

2.9995

0.0382

2.9979

0.0386

−0.1

$\rho$

−0.1009

0.0173

−0.0998

0.0167

−0.1006

0.0167

$\beta$

3.0034

0.0431

2.9992

0.0398

3.0017

0.0411

0

$\rho$

−0.0002

0.0169

−5.9283e−004

0.0162

0.0002

0.0154

$\beta$

3.0011

0.0445

3.0014

0.0426

3.0001

0.0421

0.1

$\rho$

0.0998

0.0142

0.1018

0.0158

0.1000

0.0154

$\beta$

3.0023

0.0437

2.9954

0.0457

2.9997

0.0441

0.5

$\rho$

0.4998

0.0089

0.5007

0.0088

0.4999

0.0084

$\beta$

2.9993

0.0453

2.9982

0.0457

2.9992

0.0435

0.9

$\rho$

0.9001

0.0018

0.9001

0.0018

0.9002

0.0471

$\beta$

2.9978

0.0446

2.9967

0.0476

2.9978

0.0474