Journal of Biostatistics & Biometrics

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On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications

Fiaz Ahmad Bhatti1*, Gholamhossein Hamedani2, Haitham M. Yousof3, Azeem Ali1, Munir Ahmad1

1National College of Business Administration and Economics, Lahore, Pakistan

2Marquette University, Milwaukee, WI, USA

3Department of Statistics Mathematics and Insurance, Benha University, Al Qalyubia Governorate, Egypt

*Corresponding author: Fiaz Ahmad Bhatti, National College of Business Administration and Economics, Lahore, Pakistan. Email: fiazahmad72@gmail.com

Received Date: 06 September, 2018; Accepted Date: 25 September, 2018; Published Date: 05 October, 2018

Citation: Bhatti FA, Hamedani G, Yousof HM, Ali A, Ahmad M (2018) On Modified Burr XII-Inverse Exponential Distribution: Properties, Characterizations and Applications. J Biostat Biom: JBSB-106. DOI: 10.29011/JBSB-106.100006

Abstract

 

In this paper, a flexible lifetime distribution with increasing, increasing and decreasing and modified bathtub hazard rate called Modified Burr XII-Inverse Exponential (MBXII-IE) is introduced. The density function of MBXII-IE has exponential, left-skewed, right-skewed and symmetrical shapes. Descriptive measures such as moments, moments of order statistics, incomplete moments, inequality measures, residual life function and reliability measures are theoretically established. The MBXII-IE distribution is characterized via different techniques. Parameters of MBXII-IE distribution are estimated using maximum likelihood method. The simulation study is performed to illustrate the performance of the Maximum Likelihood Estimates (MLEs) of the parameters of the MBXII-IE distribution. The potentiality of MBXII-IE distribution is demonstrated by its application to real data sets: fracture toughness, taxes revenue’s data and coal mining disaster data.

Keywords: Characterizations; Moments; Maximum Likelihood Estimation; Reliability



Figure 1: Plots of pdf of the MBXII-IE distribution for selected parameter values.



Figure 2: Plots of hrf of the MBXII-IE distribution for selected parameter values.



Figure 3: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for fracture toughness.



Figure 4: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for Tax Revenue.



Figure 5: Fitted pdf, cdf, survival and pp plots of the MBXII-IE distribution for coal mining disasters data.

1

α

β

γ

λ

MBXII-IE distribution

2

α

β

1

λ

BXII-IE distribution

3

α

1

1

λ

Lomax-IE distribution

4

1

β

1

λ

Log-logistic-IE distribution

5

α

β

γ→0

λ

Weibull-IE distribution

Table 1: Sub-models of the MBXII-IE Distribution.


Sample

Statistics

 

 

n=50

Means 

0.508

0.5275

0.4009

1.5208

0.8235

0.9994

1.1443

1.3154

 Bias

0.008

0.0275

-0.0991

  0.6208

0.0235

0.0994

0.1443

0.8154

MSE

0.0691

0.6688

14.5017

3.0359

1.0307

8.3995

318.6126

3.4638

  n=100

Means

0.5047

0.5165

 0.4233

1.1072

0.7974

0.9218

0.6404

0.8864

Bias

0.0047

0.0165

-0.0767

0.2072

-0.0026

  0.0218

-0.3596

0.3864

MSE

0.0335

0.0542

0.1008

0.5377

0.2057

0.8449

2.629

1.2674

  n=200

Means

0.5009

0.5092

0.4617

0.9831

0.7882

0.926

0.7403

0.6479

Bias

9e-04

0.0092

-0.0383

0.0831

-0.0118

0.026

-0.2597

0.1479

MSE

0.0177

0.0229

0.0401

0.1467

0.1524

0.1817

0.2849

0.2877

n=300

Means

0.5016

0.5071

0.4804

0.9509

0.7864

 0.9253

0.8063

0.5879

Bias

0.0016

0.0071

-0.0196

0.0509

-0.0136

0.0253

-0.1937

0.0879

MSE

0.0117

0.0132

0.0266

0.0875

0.1211

0.1281

0.1785

0.1394

n=500

Means

0.5001

0.5029

0.4854

  0.932

0.7921

0.9131 

0.8687

0.5514

 

Bias

1e-04

0.0029

-0.0146

  0.032

-0.0079

0.0131

-0.1313

0.0514

 

MSE

0.007

0.0071

0.0153

0.0481

0.0841

0.077

0.096

 0.064

Table 2: Means, Bias and MSEs of the MBXII-IE distribution (0.5, 0.5, 0.5, 0.9) and (0.8, 0.9, 1, 0.5).


Model

W

A

K-S

p-value

MBXII-IE

0.0181133061 (0.022788744)

4.1601365928

(0.413218923)

0.0000000001

(0.002022918)

1.5161119726

(0.352629351)

0.1182952 

0.7247703

0.0796 (0.4387)

BXII-IE

10.836337 (8.8614837)

  2.939956 (0.5268154)

1

  5.420295 (1.3394846)

0.2103398

1.295879

0.1186 (0.07024)

L-IE

88.15512 (30.158622)

1

1

20.65397              (1.627536)

0.3337133

2.026292

0.1147             (0.08714)

LL-IE

1

(4.867383)

0.38248098

1

2.975707 (0.06835492)

0.4064326 

2.522655

0.332    (8.12e-12)

 

Table 3: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for fracture toughness.


Model

 AIC

CAIC

BIC

HQIC

-l

MBXII-IE

346.419

346.7698

357.5355

350.933

169.2095

BXII-IE

351.0785

351.2872

359.4159

354.4641

172.5393

L-IE

356.4294

356.5328

361.9876

358.6864

176.2147

LL-IE

370.1331

370.2365

375.6913

372.3901

183.0665

Table 4: Goodness-of-fit statistics for fracture toughness.


Model

W

A

K-S            (p-value)

MBXII-IE

0.01574783 (0.017879592)

2.66428346 (0.356379272)

0.00150912 (0.002280467)

0.54190197

(0.168851792)

0.08121186

0.4526733

0.1009 (0.2603)

BXII-IE

3.405735

(1.5676440)

1.865183

(0.3395644)

1

2.831844

(0.6706193)

0.2254017

1.182635

0.2072

(0.0003741)

Lomax-IE

9.196698

(2.1114083)

1

1

6.214104

(0.5991982)

0.3149302

1.693847

  0.1703 (0.006058)

LL-IE

1

2.770423

(0.23340747)

1

1.676333

(0.07349948)

0.3281241

 1.739495

0.3634 (6.722e-12)

Table 5: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for Tax Revenue.


Model

AIC

CAIC

BIC

HQIC

-l

MBXII-IE

289.7468

290.1678

300.1675

293.9642

140.8734

BXII-IE

300.7174

300.9674

308.5329

303.8805

147.3587

L-IE

304.8322

304.956

310.0426

306.941

150.4161

LL-IE

307.8828

308.0065

313.0932

309.9915

151.9414

Table 6: Goodness-of-fit statistics for Tax Revenue.


Model

W

A

K-S

p-value

MBXII-IE

0.014003462 (0.010293280)

0.974040069 (0.114284919)

0.002542779

(0.001955081)

2.319100102 (1.400766759)

0.06743529

0.4651009

0.1142      (0.1167)

BXII-IE

0.5257374 (0.1056558)

0.9798724 (0.1313540)

20.8102228 (4.3210242)

1

0.804424

4.659193

0.5874     (<2.2e-16)

L-IE

0.5129513 (0.0608668)

1

1

20.4062733 (3.3094989)

0.8043515

 4.653696

 0.592     (<2.2e-16)

LL-IE

1

0.6939298 (0.05621336)

1

36.3055045 (4.43815735)

0.9376481

5.457812

0.521      (<2.2e-16)

Table 7: MLEs, their standard errors (in parentheses) and Goodness-of-fit statistics for coal mining disasters.


Model

AIC

 CAIC

 BIC

 HQIC

-l

MBXII-IE

1409.089

1409.474

1419.854

1413.455

700.5445

BXII-IE

1493.264

1493.493

1501.338

1496.539

743.6322

L-IE

1491.287

1491.4

1496.67

1493.47

743.6435

LL-IE

1503.023

1503.136

1508.406 1

505.206

749.5115

Table 8: Goodness-of-fit statistics for coal mining disasters.

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