International Mathematical Forum, Vol. 11, 2016, no. 19, 943 - 959

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/imf.2016.6793

The Odd Generalized Exponential Modified

Weibull Distribution

Yassmen Y. Abdelall

Department of Mathematical Statistics

Institute of Statistical Studies and Research

Cairo University, Egypt

Copyright © 2016 Yassmen Y. Abdelall. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

In this paper we propose a new distribution, called the odd generalized

exponential modified Weibull distribution. Some mathematical properties of the

new distribution are studied. The method of maximum likelihood is used for

estimating the model parameters and the observed Fisher's information matrix is

derived. We illustrate the usefulness of the proposed model by application to real

data.

Keywords: Modified Weibull distribution, moments, maximum likelihood

estimation, order statistics

1. Introduction

The modified Weibull (MW) distribution is one of the most important

distributions in lifetime modeling, and some well-known distributions such as the

exponential, Rayleigh, linear failure rate and Weibull distributions are special

cases of it. This distribution was introduced by Lai, Xie, and Murthy (2003) to

which we refer the reader for a detailed discussion as well as applications of the

MW distribution (in particular, the use of the real data set representing failure

times to illustrate the modeling and estimation procedure). Also Sarhan and

Zaindin (2009) introduced the modified Weibull distribution. It can be used to

describe several reliability models. It has three parameters, two scale and one

shape parameters. Recently, Carrasco et al. (2008) extended the MW distribution

by adding another shape parameter and introducing a four parameter generalized

MW (GMW) and log-GMW (LGMW).

944 Yassmen Y. Abdelall

Recently Tahir et al. (2015) proposed a new class of univariate distributions called

the odd generalized exponential (OGE) family and studied each of the OGE-

Weibull (OGE-W) distribution, the OGE-Fréchet (OGE-Fr) distribution and the

OGE-Normal (OGE-N) distribution. This method is flexible because the hazard

rate shapes could be increasing, decreasing, bathtub and upside down bathtub.

In this article we present a new distribution from the odd generalized exponential

distribution and modified Weibull distribution called the Odd Generalized

Exponential-Modified Weibull (OGE-MW) distribution using new family of

univariate distributions proposed by Tahir et al. (2015).

A random variable X is said to have generalized exponential (GE) distribution

with parameters , if the cumulative distribution function (cdf) is given by

.0,0,0,1,; xexF x (1)

The Odd Generalized Exponential family by Tahir et al. (2015) is defined as

follows. Let );( xG is the cdf of any distribution depends on parameter and thus

the survival function is );(1);( xGxG , then the cdf of OGE-family is

defined by replacing x in CDF of GE in Equation (1) by );(

);(

xG

xGto get

.0,0,0,0,1,,; );(

);(

xexF xG

xG

Where , are two additional parameters. This paper is outlined as follows. In

Section 2, we define the cumulative distribution function, density

function,reliability function and hazard function of the Odd Generalized

Exponential-Modified Weibull (OGE-MW) distribution. In Section 3, we

introduce the statistical properties include, the quantile function, the median

andthe moments. Section 4 discusses the distribution of the order statistics for

(OGE-MW) distribution. Moreover, maximum likelihood estimation of the

parameters is determined in Section 5. Finally, an application of (OGE-MW)

using a real data set is presented in Section 6.

2. The OGE-MW Distribution

2.1 OGE-MW specifications

In this section we define new five parameters distribution called Odd Generalized

Exponential-Modified Weibull distribution with parameters ,,,, and

written as OGE-MW(Θ), where the vector Θ is defined by Θ = ( ,,,, ) .

Odd generalized exponential modified Weibull distribution 945

A random variable X is said to have OGE-MW with parameters ,,,, and if

its cumulative distribution function (cdf) given as follows

.0,,,,,0,1;1

xexFxxe

(2)

where ,, are scale parameters and , are shape parameters. Hence, the

corresponding probability density function (pdf) is

.0,,,,,0,1;

111

1

xeeexxfxxxx ee

xx

(3)

2.2 Survival and hazard functions

If a random variable X has cdf in (2), then the corresponding survival function is

given by

.111

xxe

exS

The hazard function of OGE-MW( ) is defined as follow

.

11

1

)(

)(

1

111

1

xx

xxxx

e

eexx

e

eeex

xS

xfxh

Figure 1, 2 and 3 illustrates some of the possible shapes of the pdf, cdf and hazard

function of OGE-MW distribution for some values of the parameters ,,,,

and

946 Yassmen Y. Abdelall

0 2 4 6 8 100

0.2

0.4

0.6

f x 0.25 0.7 0.3 0.2 2( )

f x 0.2 0.2 0.4 0.6 2( )

f x 0.25 0.6 0.2 0.1 2.5( )

f x 0.4 0.3 0.6 0.8 1.5( )

f x 0.3 0.4 0.9 0.5 3( )

x

Figure 1. The pdf’s of various OGE-MW distributions.

0 2 4 6 8 100

0.5

1

F x 0.5 0.4 0.8 0.5 0.9( )

F x 0.4 0.2 0.4 0.2 1( )

F x 0.2 0.6 0.2 0.6 1( )

F x 0.3 0.3 0.6 0.8 1.4( )

F x 0.2 0.7 0.5 0.4 1( )

x

Figure 2.The cdf of various OGE-MW distributions.

1 1.5 2 2.5 3 3.5 40

8

16

24

32

40

h t 1 0.7 0.5 0.4 1.5( )

h t 1.5 0.6 0.4 0.3 1.5( )

h t 1.5 0.8 0.6 0.7 1.5( )

h t 1.5 0.3 0.7 1 1.5( )

h t 2 0.5 0.3 0.8 2( )

t

Figure 3. The hazard function of various OGE-MW distributions.

Odd generalized exponential modified Weibull distribution 947

Note that theOGE-MW distribution is very flexible model that approaches to

different distributions when its parameters are changed. The OGE-MW

distribution contains as special-models with the following well known

distributions. In particular, for 0 we have the odd generalized exponential-

Weibull (OGE-W) distribution as discussed in Tahir et al. (2015). The odd

generalized exponential-exponential(OGE-E) distribution is clearly a special case

for 1,0 and 1 as discussed in Maiti and Pramanik (2015).for 2/

and 2 we have the odd generalized exponential-linear failure rate (OGE-

LFR) distribution as discussed in El-Damcese et al. (2015). When 0 and

2 then the resulting distribution is the odd generalized exponential-Rayleigh

(OGE-R) distribution.

3. Statistical Properties

This section is devoted for studying somestatistical properties for the odd

generalized exponential-modified

Weibull (OGE-MW), specifically quantile, median and the moments.

3.1 Quantile and Median of OGE-MW

The quantile function qx of OGE-MW(Θ) distribution is given by using

qxF q )( (4)

Substituting from (2) into (4), qx is the real solution of the following equation

,10,0

1ln

1ln

1

q

q

xx qq

(5)

The above equation has no closed form solution in qx , so we have to use a

numerical technique such as a Newton- Raphson method to get the quantile. By

putting 5.0q in Equation (5) we can get the median of odd generalized

exponential modified Weibull distribution.

3.2 Moments

Moments are necessary and important in any statistical analysis, especially in

applications. It can be used tostudy the most important features and characteristics

of a distribution (e.g., tendency, dispersion, skewness and kurtosis). In this

subsection, we will derive the rth moments of the OGE-MW(Θ) distribution as

infinite series expansion.

948 Yassmen Y. Abdelall

Theorem1.

The rth moment of a random variable X ~ OGE-MW(Θ), where Θ = ( ,,,,

) is given

LrLrLrLr

i j

j

k L

LjLjkji

r

kj

Lr

kj

Lr

Lj

kji

k

j

i

11

1

!!

111

1

1

0 0 0 0

1'

Proof: The rth moment of a random variable X with pdff(x) is defined by

0

' )( dxxfx r

r (6)

Substituting from (3) into (6), we obtain

.10

111

1'

dxeeexxxxxx ee

xxr

r

(7)

Since 1101

xxe

e for 0x , we obtain

0

11

1

11

1i

eii

e xxxx

ei

e

. (8)

Substituting from (8) into (7), we get

.11

0 0

111'

i

eixxri

r dxeexxi

xx

Using series expansion of

11

xxei

e , we obtain

.1!

11

1

0 0 0

11

'

i j

jxxxxr

jjji

r dxeexxj

i

i

Using binomial expansion of jxxe 1 , we obtain

Odd generalized exponential modified Weibull distribution 949

0 0 0

1'

!

11

1

i j

j

k

jjkji

rj

i

k

j

i

.0

111

dxeexx xkjxkjr

Using series expansion of xkje 1 , we obtain

0 0 0 0

1'

!!

111

1

i j

j

k L

LjLjkji

rLj

kji

k

j

i

.0

11

0

1

dxexdxex xkjLrxkjLr

By using the definition of gamma function in the form, see Zwillinger (2014),

.0,,0

1

xzdttexz zxtz

Finally, we obtain the rth moment of OGE-MW as follows

.

11

1

!!

111

1

1

0 0 0 0

1

'

LrLrLrLr

i j

j

k L

LjLjkji

r

kj

Lr

kj

Lr

Lj

kji

k

j

i

This completes the proof. □

4. Order Statistics

Let nXXX ,...,, 21 be a simple random sample of size n from OGE-MW(Θ)with

cumulative distribution function );( xF and probability density function );( xf

given by (2) and (3) respectively. Let nnnn XXX ::2:1 ..., denote the order

statistics obtained from this sample. The probability density function of nrX : is

given by

,);(1);();()1,(

1;

1

:

rnr

nr xFxFxfrnrB

xf

(9)

950 Yassmen Y. Abdelall

where );( xf and );( xF are the pdf and cdf of OGE-MW(Θ) distribution given

by (2) and (3) respectively and B(., .) is the beta function, also we define first

order statistics nn XXXX ;...;;min 21:1 ,and the last order statistics as

nnn XXXX ;...;;max 21: . Since 1);(0 xF for ,0x we can use the

binomial expansion of rnxF

);(1 given as follows

.);()1();(10

iirn

i

rnxF

i

rnxF

(10)

Substituting from (10) into (9), we obtain

.);()1();()1,(

1;

0

1

:

rn

i

rii

nr xFi

rnxf

rnrBxf (11)

Substituting from (2) and (3) into (11), we obtain

.))(,,,,;()(!)(!)1(!

!)1(,,,,;

0

:

rn

i

i

nr irxfirirnri

nxf

(12)

Relation (12) shows that ,,,,;: xf nr is the weighted average of the odd

generalized exponential-modified Weibull with different shape parameters.

5. Estimation and Inference

Now, we discuss the estimation of the OGE-MW ),,,,( parameters by

using the method of maximum likelihood based on a complete sample.

5.1 Maximum likelihood estimators

Let nXXX ,...,, 21 be a random sample of size n from OGE-MW(Θ), where Θ = (

,,,, ) , then the likelihood function l of this sample is defined as

.),,,,;(1

i

n

i

xfl

(13)

Substituting from (3) into (13), we get

.11

111

1

n

i

eexx

i

ixixixix

ii eeexl

The log-likelihood function, L, becomes:

Odd generalized exponential modified Weibull distribution 951

.1ln1

1lnlnln

1

1

111

1

n

i

e

n

i

xxn

i

ii

n

i

i

ixix

ii

e

exxxnnL

(14)

The maximum likelihood estimates of the parameters are obtained by

Differentiating the log-likelihood function L, with respect to the parameters

,,,, and setting the result to zero.

,0,,,,

1,,,ln11,,,

11

n

i i

in

i

ix

xx

nL

(15)

,0

,,,,

,,,1,,,

1

1111

1

n

i i

iin

i

ii

n

i i

n

i

ix

xxxx

xx

L

(16)

,0

,,,,

,,,1,,,

1111

1

1

n

i i

iin

i

ii

n

i i

in

i

ix

xxxx

x

xx

L

(17)

,0

,,,,

,,,ln1

,,,lnln

ln

1

111

11

1

n

i i

iii

n

i

iii

n

i i

iiin

i

ii

x

xxx

xxxx

xxxxx

L

(18)

and

,01ln1

1

n

i

e ixix

enL

(19)

Where the nonlinear functions ,,,ix and ,,,,ix are given by

,,,,

ii xx

i ex

1,,,,1

ixix

e

i ex .

952 Yassmen Y. Abdelall

From equation (19), we obtain the maximum likelihood estimate of in a closed

form as follow

.

1ln1

1

n

i

e ixix

e

n

(20)

Substituting from (20) into (15), (16), (17) and (18), we get the MLEs of

,,,, by solving the following system of non-linear equations

,0

,,,,

1,,,

ln11,,,11

n

ii

in

i

i

x

x

xn

,0

,,,,

,,,

1,,,1

111 11

n

ii

iin

i

ii

n

ii

n

i

i

x

xx

xx

x

x

,0

,,,,

,,,

1,,,111 1

1

1

n

ii

iin

i

ii

n

ii

in

i

i

x

xx

xx

x

xx

,0

,,,,

,,,ln

1

,,,lnln

ln

1

11 1

11

1

n

ii

iii

n

i

iii

n

ii

iiin

i

ii

x

xxx

xxx

x

xxxxx

where ,,,,

ii xx

i ex and

1,,,,

1

ixixe

i ex . These

equations cannot be solved analytically and statistical software can be used to

solve the equations numerically. We can use iterative techniques such as Newton

Raphson type algorithm to obtain the estimate

.

Odd generalized exponential modified Weibull distribution 953

5.2 Asymptotic confidence bounds

In this subsection, we derive the asymptotic confidence intervals of the unknown

parameters ,,,, and . As the sample size n , then

),,,,(

approaches a multivariate normal vector with

zero means and covariance matrix ,1

0

I where 1

0

I is the inverse of the

observedinformation matrix which defined as follows

1

2

2

2

2

2

2

2

2

2

2

2

2

22

2

222

222

2

22

2222

2

2

1

0

L

L

L

L

L

L

L

L

L

L

LLLLL

LLLLL

LLLLL

I

Var

Var

Var

Var

Var

,cov

,cov,cov

,cov

,cov

,cov

,cov

,cov

,cov,cov,cov,cov

,cov,cov,cov,cov

,cov,cov,cov,cov

(21)

The second partial derivatives included in 1

0

I are given as follows

,22

2

nL

,

,,,,

,,,

1

2

n

ii

ii

x

xxL

954 Yassmen Y. Abdelall

,

,,,,

,,,ln

1

2

n

ii

iii

x

xxxL

,

,,,,

,,,

1

2

n

ii

ii

x

xxL

,

,,,,

1,,,

1

2

n

ii

i

x

xL

,

,,,,

1,,,

11

2

1,,,2

22

2

n

i

i

x

i

x

exnL

i

1,,,1

2

n

i

ii xxL

,

,,,,

1,,,

1

,,,,

,,,

1

1,,,

n

ii

x

i

i

ii

x

ex

x

xxi

1,,,1

2

n

i

ii xxL

,

,,,,

1,,,

1

,,,,

,,,

1

1,,,

n

i

i

x

i

i

ii

x

ex

x

xxi

Odd generalized exponential modified Weibull distribution 955

1,,,ln1

2

n

i

iii xxxL

,

,,,,

1,,,

1

,,,,

,,,ln

1

1,,,

n

i

i

x

i

i

iii

x

ex

x

xxxi

1,,,1

2

1

2

1

1

2

2

n

i

ii

n

i i

i xxx

xL

,

,,,,

,,,

1

,,,,

,,,

1

1,,,2

n

i

i

x

i

i

ii

x

ex

x

xxi

1,,,1

1

121

12

n

i

ii

n

ii

i xxx

xL

,

,,,,

,,,

1

,,,,

,,,

1

1,,,1

n

i

i

x

i

i

ii

x

ex

x

xxi

1,,,ln1

lnln1

ln1

1

111

1

1

12

n

i

iiii

n

i

ii

n

i i

ii

i

i

i

xxxx

xxx

xxx

x

xL

,

,,,,

,,,

1

,,,,

,,,ln

1

1,,,

n

i

i

x

ii

i

i

iii

x

exx

x

x

xxxi

956 Yassmen Y. Abdelall

1,,,1

1

2

1212

2

n

i

ii

n

ii

xxx

L

,

,,,,

,,,

,,,,

,,,

1

1,,,

n

i

i

x

ii

i

i

ii

x

exx

x

x

xxi

,,,lnln11

1

121

12

i

n

i

ii

n

i

i

i

i xxxxx

xL

,

,,,,

,,,

1

,,,,

,,,ln

11

1,,,1

n

i

i

x

i

i

iii

x

ex

x

xxxi

1,,,ln1

lnln1

lnln2

1

2

1

2

11

2

2

1

1

2

2

n

i

iiii

n

i

ii

n

i i

i

ii

i

i

xxxx

xxx

xxx

x

xL

,

,,,,

,,,

1

,,,,

,,,ln

1

1,,,2

n

i

i

x

ii

i

i

iii

x

exx

x

x

xxxi

where ,,,,

ii xx

i ex and

1,,,,

1

ixixe

i ex .

The asymptotic 1100 % confidence intervals of ,,,, and are

,2

Varz ,2

Varz ,2

Varz ,2

Varz and

Odd generalized exponential modified Weibull distribution 957

Varz2

respectively, where 2

z is the upper

2

th percentile of the

standard normal distribution.

6. Data Analysis

In this section, we perform an application to real data to illustrate that the OGE-

MW can be a good lifetime model, comparing with many known distributions

such as the Exponential (E), Generalized Exponential (GE), Linear Failure Rate

(LFR), and Weibull (W). Consider the data have been obtained from Aarset [1],

and widely reported in many literatures. It represents the lifetimes of 50 devices,

and also, possess a bathtub-shaped failure rate property, Table 1.

Table 1: The data from Aarset [1].

0.1 0.2 1 1 1 1 1 2 3 6 7 11

12 18 18 18 18 18 21 32 36 40 45 46

47 50 55 60 63 63 67 67 67 67 72 75

79 82 82 83 84 84 84 85 85 85 85 85

86 86

The maximum likelihood estimates (MLEs) of the unknown parameters for the

five models is given in Table 2.

Table 2.MLEs of the parameters

MLE of the parameter(s) The Model

022.0

)(E

901.0,021.0

),( GE

4104.2,014.0

),( LFR

949.0,022.0

),( W

946.0,02.0,012.0,448.0,793.0

),,,,( OGEMW

The values of log-likelihood functions (-L), Akaike Information Criteria (AIC),

Bayesian Information Criteria (BIC), and the Consistent Akaike Information

Criteria (CAIC) are given in Table 3 for the five models in order to verify which

distribution fits better to these data.

958 Yassmen Y. Abdelall

Table 3.The -L, AIC, BIC, and CAIC for devices data.

The Model

Measures

L AIC BIC CAIC

E 241.090 484.179 486.091 484.263

GE 240.386 484.771 488.595 485.026

LFR 238.064 480.128 483.952 480.383

W 241.002 486.004 489.828 486.259

OGE-MW 233.393 476.786 486.346 478.150

Based on Table 2and 3, it is shown that OGE-MW ),,,,( model provide

better fit to the data rather than other distributions which we compared with

because it has the smallest value of AIC, BIC, CAIC.

Substituting the MLEs of the unknown parameters ),,,,( into (21), we get

estimation of the variance covariance matrix as the following:

6

5

5

4

8

6

5

4

5

4

86856

54543

5463

1

0

10976.4

10968.5

10968.5

10044.1

10766.3

10914.5

1014.2

10568.2

10851.7

10047.9

10766.310914.510446.210713.210623.7

1014.210568.210713.21052.210239.3

10851.710047.910623.710239.3014.0

I

The approximate 95% two sided confidence intervals of the unknown parameters

,,,, and are [0.714,1.178], [0,0.043], [0,0.02], [0.428,0.468], and

[0.789,0.797], respectively.

7. Conclusions

In this paper, we have introduced a new five-parameter model called odd

generalized exponential modified Weibull (OGE-MW) distribution and studied its

different properties. Some statistical properties of this distribution have been

derived and discussed. We provide the pdf, the cdf, and the hazard rate function

for the new model also we provide an explicit expression for the moments. The

distributions of the order statistics are discussed. Both point and asymptotic

confidence interval estimates of the parameters are derived using maximum

likelihood method and we obtained the observed fisher information matrix. We

use application on set of real data to compare the OGE-MW with other known

distributions such as Exponential (E), Generalized Exponential (GE), Linear

Failure Rate (LFR), and Weibull (W). Applications on set of real data showed that

the OGE-MW is the best distributionfor fitting these data sets compared with

other distributions considered in this article.

Odd generalized exponential modified Weibull distribution 959

References

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Reliability, 36 (1987), no. 1, 106-108.

http://dx.doi.org/10.1109/tr.1987.5222310

[2] J. M. Carrasco, E. M. Ortega and G. M. Corderio, A generalization modified

weibull distribution for lifetime modeling, Computational Statistics and Data

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[3] M. A. El-Damcese, A. Mustafa, B. S. El-Desouky and M. E. Mustafa, The odd

generalized exponential linear failure rate distribution, Journal of Statistics

Applications & Probability, 5 (2016), 299-309.

http://dx.doi.org/10.18576/jsap/050211

[4] C. D. Lai, M. Xie and D.N.P. Murthy, A modified Weibull distribution, IEEE

Transactions on Reliability, 52 (2003), 33-37.

http://dx.doi.org/10.1109/tr.2002.805788

[5] S. S Maiti and S. Pramanik, Odds generalized exponential-exponential

distribution, Journal of Data Science, 13 (2015), 733-754.

[6] A. M. Sarhan, and M. Zaindin, Modified Weibull distribution, Applied

Sciences, 11 (2009), 123-136.

[7] M. H. Tahir, G. M. Cordeiro, M. Alizadeh, M. Mansoor, M. Zubair and G. G.

Hamedani, The odd generalized exponential family of distributions with

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Received: July 22, 2016; Published: October 3, 2016