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Table 2 Skewness and kurtosis of KGD for some values of α , β and q

From: The Kumaraswamy-geometric distribution

  

q=0.4

q=0.6

q=0.8

α

β

Skewness

Kurtosis

Skewness

Kurtosis

Skewness

Kurtosis

0.4

0.4

2.4529

8.4883

2.3789

8.0612

2.3360

7.8215

 

0.6

2.8022

10.920

2.6534

9.9332

2.5656

9.3787

 

0.8

3.1918

14.035

2.9430

12.172

2.7954

11.133

 

1.5

4.9662

32.736

4.0792

23.245

3.5894

18.553

 

2.0

6.8722

60.013

5.0708

35.519

4.1724

25.214

 

4.0

30.524

990.63

12.671

196.82

7.1320

72.567

0.6

0.4

2.2480

7.2663

2.1956

7.0004

2.1684

6.8666

 

0.6

2.4465

8.4649

2.3387

7.8659

2.2823

7.5649

 

0.8

2.6694

9.9363

2.4883

8.8459

2.3932

8.3020

 

1.5

3.6723

17.839

3.0582

13.119

2.7529

10.999

 

2.0

4.6998

27.856

3.5337

17.209

2.9979

13.043

 

4.0

14.964

238.38

6.6621

54.450

4.0916

23.691

0.8

0.4

2.1208

6.5743

2.0847

6.4108

2.0685

6.3383

 

0.6

2.2269

7.1361

2.1498

6.7675

2.1151

6.6063

 

0.8

2.3522

7.8283

2.2198

7.1642

2.1603

6.8772

 

1.5

2.9553

11.535

2.5010

8.8497

2.3055

7.8006

 

2.0

3.5861

16.072

2.7466

10.413

2.4048

8.4499

 

4.0

9.2202

90.141

4.3332

22.881

2.8543

11.487

1.5

0.4

1.9024

5.5182

1.9015

5.5273

1.9038

5.5384

 

0.6

1.8485

5.2051

1.8400

5.2063

1.8433

5.2269

 

0.8

1.8105

4.9322

1.7862

4.9087

1.7894

4.9374

 

1.5

1.8191

4.4671

1.6645

4.1552

1.6529

4.1894

 

2.0

1.9494

4.6197

1.6281

3.8307

1.5890

3.8375

 

4.0

3.3608

11.0492

1.7629

3.5961

1.4574

3.0767

2.0

0.4

1.8322

5.2176

1.8440

5.2733

1.8503

5.2983

 

0.6

1.7260

4.6855

1.7442

4.7835

1.7563

4.8321

 

0.8

1.6352

4.2020

1.6551

4.3386

1.6733

4.4121

 

1.5

1.4620

3.0364

1.4309

3.2239

1.4664

3.3814

 

2.0

1.4564

2.6068

1.3301

2.7152

1.3700

2.9254

 

4.0

2.0958

3.6392

1.1838

1.6496

1.1546

1.9922

4.0

0.4

1.7344

4.8228

1.7559

4.9001

1.7627

4.9251

 

0.6

1.5605

4.0589

1.6019

4.1951

1.6151

4.2403

 

0.8

1.4043

3.3984

1.4675

3.5897

1.4874

3.6530

 

1.5

1.0049

1.9031

1.1406

2.2381

1.1833

2.3432

 

2.0

0.8155

1.2870

0.9925

1.7112

1.0500

1.8310

 

4.0

0.4669

-0.1633

0.6666

0.8077

0.7728

0.9349

6.0

0.4

1.7069

4.7081

1.7246

4.7731

1.7310

4.7964

 

0.6

1.5197

3.8941

1.5522

4.0017

1.5641

4.0428

 

0.8

1.3558

3.2107

1.4034

3.3521

1.4207

3.4078

 

1.5

0.9572

1.7788

1.0513

1.9565

1.0843

2.0406

 

2.0

0.7743

1.2870

0.8984

1.4383

0.9394

1.5299

 

4.0

0.3102

0.6548

0.5863

0.5921

0.6444

0.6837