Journal of Biostatistics & Biometrics

Article / Research Article

"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

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.

1.       Tadikamalla PR (1980) A look at the Burr and related distributions. International Statistical Review/Revue Internationale de Statistique 48: 337-344.

2.       Saran J, Pushkarna N (1999) Moments of order statistics from doubly truncated Lomax distribution. Journal of Statistical Research 33: 57-66.

3.       Begum AA, Parvin S (2002) Moments of order statistics from Doubly truncated Burr XII distribution. J. of Statist. Res 36: 179-190.

4.       Usta I (2013) Different estimation methods for the parameters of the extended Burr XII distribution. Journal of Applied Statistics 40: 397-414.

5.       Shao Q, Wong H, Xia J, Ip WC (2004) Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis. Hydrological Sciences Journal 49: 685- 701.

6.       Olapade AK (2008) On a six-parameter generalized Burr XII distribution, Electronic Journal of Statistics Mathematical Statistics 1-5.

7.       Paranaíba PF, Ortega EM, Cordeiro GM, Pescim RR (2011) The beta Burr XII distribution with application to lifetime data. Computational Statistics & Data Analysis 55: 1118-1136.

8.       Paranaíba PF, Ortega EM, Cordeiro GM,Pascoa MAD (2013) The Kumaraswamy Burr XII distribution: theory and practice. Journal of Statistical Computation and Simulation 83: 2117-2143.

9.       Akhtar MT, Khan AA (2014) Bayesian analysis of generalized log-Burr family with R. Springer Plus 3: 1.

10.    Korkmaz MC, Erişoglu M (2014) The Burr XII-Geometric Distribution. Journal of Selcuk University Natural and Applied Science 3: 75-87.

11.    Gomes AE, da-Silva CQ, Cordeiro GM (2015) Two extended Burr models: Theory and practice. Communications in Statistics-Theory and Methods44: 1706-1734.

12.    Okasha MK, Matter MY (2015) On the Three-Parameter Burr Type XII Distribution and its Application to Heavy Tailed Lifetime Data. Journal: Journal of Advances in Mathematics. 10: 3429- 3442.

13.    Silva RB, Cordeiro GM (2015) The Burr XII power series distributions: A new compounding family. Brazilian Journal of Probability and Statistics 29: 565-589.

14.    Thupeng WM (2016) Use of the Three-parameter Burr XII Distribution for Modelling Ambient Daily Maximum Nitrogen Dioxide Concentrations in the Gaborone Fire Brigade. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)26: 18-32.

15.    Abouelmagd THM, Al-mualim S, Afify AZ, Ahmad M, Al-Mofleh H (2018) The Odd Lindley Burr XII Distribution with Applications. Pak. J. Statist34: 15-32.

16.    Muhammad M (2016) A generalization of the Burr XII-Poisson distribution and its applications. Journal of Statistics Applications & Probability5: 29-41.

17.    Cadena M (2017) Extensions of the Burr Type XII distribution and Applications. arXiv preprint 1705: 10374.

18.    Ghosh I, Bourguignon M (2017) A New Extended Burr XII Distribution. Austrian Journal of Statistics46: 33-39.

19.    Guerra RR, Pena-Ramirez FA, Cordeiro GM (2017) The gamma Burr XII distribution: Theory and application. Journal of Data Science15.

20.    Kumar D (2017) The Burr type XII distribution with some statistical properties. Journal of Data Science15.

21.    Mdlongwa P, Oluyede B, Amey A, Huang S (2017) The Burr XII modified Weibull distribution: model, properties and applications. Electronic Journal of Applied Statistical Analysis10: 118-145.

22.    Mead ME, Afify AZ (2017) On five-parameter Burr XII distribution: properties and applications. South African Statist J51: 67-80.

23.    Yari G, Tondpour Z (2017) The new Burr distribution and its application. Mathematical Sciences11: 47-54.

24.    Korkmaz MC, Yousof HM, Rasekhi M, Hamedani GG (2018) The Odd Lindley Burr XII Model: Bayesian Analysis, Classical Inference and Characterizations. Journal of Data Science 16: 327-353.

25.    Bhatti FA, Hamedani GG, Ahmad M (2018) On Modified Log Burr XII Distribution. JIRSS-Journal of the Iranian Statistical Society17: 57-89.

26.    Gurvich MR, DiBenedetto AT, Ranade SV (1997) A new statistical distribution for characterizing the random strength of brittle materials. Journal of Material Science 32: 2559-2564.

27.    Alzaatreh A, Lee C, Famoye F (2013) A new method for generating families of continuous distributions. Metron71: 63-79.

28.    Bourguignon M, Silva RB, Cordeiro GM (2014) The Weibull-G Family of Probability Distributions. Journal of Data Science 12: 53-68.

29.    Bhattacharyya GK, Johnson RA (1974) Estimation of reliability in a multicomponent stress-strength model. Journal of the American Statistical Association69: 966-970.

30.    Glänzel WA (1990) Some consequences of a characterization theorem based on truncated moments, Statistics 21: 613 - 618.

31.    Nagarajah H (1988) Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values Sankhya Series A 50: 70-73.

32.    Arnold BC, Balakrishnan N, Nagarajah H (1998) Records, John Wiley New York.

33.    Khan AH,Alzaid AA (2004) Characterization of distributions through linear regression of non-adjacent generalized order statistics, Journal of Applied Statistical Science 13: 123-136.

34.    Athar H, Noor Z (2014) Characterization of probability distributions by conditional expectations of functions of record statistics, Journal of the Egyptian Mathematical Society 22: 275-279.


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