EXTENDED EXPONENTIATED WEIBULL DISTRIBUTION AND ITS ...

STATISTICA, anno LXX#, n. #, 201#

EXTENDED EXPONENTIATED WEIBULL DISTRIBUTIONAND ITS APPLICATIONS

Eisa Mahmoudi1

Department of Statistics, Faculty of Mathematical Sciences, Yazd University, Yazd 89195-

741, Iran.

RahmatSadat Meshkat


741, Iran.

Batool Kargar


741, Iran.

Debasis Kundu

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin

208016, India.

1. Introduction

The three-parameter exponentiated Weibull (EW) distribution introduced by Mud-holkar and Srivastava (1993) as an extension of the Weibull family, is a veryflexible class of probability distribution functions. The applications of the EWdistribution in reliability and survival studies were illustrated by Mudholkar et al.(1995). The EW distribution has the cumulative distribution function (CDF)

F (x;α, λ, γ) =(

1− e−λxγ)α

, x > 0,

and the associated probability density function (PDF) as

f(x;α, λ, γ) = αγλxγ−1e−λxγ

(1− e−λxγ

)α−1, x > 0,

where α > 0 and γ > 0 are shape parameters and λ > 0 is the scale parameter.Gupta and Kundu (1999) considered a special case of the EW distribution when

1 Corresponding Author. E-mail: [email protected]

2 Debasis Kundu

γ = 1, and called it as the generalized exponential (GE) distribution. The GEdistribution has received a considerable amount of attention in recent years. Thereaders are referred to a review article by Gupta and Kundu (2007) for a currentaccount on the generalized exponential distribution and a book length treatmentof different exponentiated distributions by Al-Hussaini and Ahsanullah (2015).

Mudholkar et al. (1996) presented a three-parameter generalized Weibull(GW) family that contains distributions with unimodal and bathtub shaped haz-ard rates. They showed that the distributions in this family are analyticallytractable and computationally manageable. The modeling and analysis of sur-vival data using this family of distributions have been discussed and illustrated interms of a lifetime data set and the results of a two-arm clinical trial. The CDFof the GW distribution is the following

F (y;α, λ, δ) = 1−(

1− δ(λy)1/α)1/δ

,

where 0 < y <∞ for δ ≤ 0 and 0 < y <1

λδαfor δ > 0.

Recently, Gupta and Kundu (2011) introduced a three-parameter extendedGE (EGE) distribution by adding a shape parameter to a GE distribution. TheEGE distribution contains many well known distributions such as exponential,GE, uniform, generalized Pareto and Pareto distributions as special cases. Inter-estingly, the EGE distribution has increasing, decreasing, unimodal and bathtubshaped hazard rate functions similar to the EWE distribution. The EGE distri-bution has the following CDF and PDF,

F (y;α, β, λ) =

{(1− (1− βλy)1/β)α, β 6= 0,(1− e−λy)α, β = 0,

and

f(y;α, β, λ) =

{αλ(1− βλy)1/β−1(1− (1− βλy)1/β)α−1, β 6= 0,αλe−λy(1− e−λy)α−1, β = 0,

respectively, when 0 < y <∞ for β ≤ 0 and 0 < y < 1βλ for β > 0.

It may be mentioned that EW, GW or EGE are very flexible class of distri-butions. Their hazard functions can take variety of shapes. For EW and EGEdistributions, the hazard functions can be increasing, decreasing, bathtub shapedor unimodal functions depending on the shape parameters. Although, the hazardfunctions can be very flexible, they cannot take decreasing-increasing-decreasing(DID) shapes. In many practical situations, it is observed that the hazard functioncan take DID shapes. Not too many lifetime distributions, at least not known tothe authors, the hazard functions can take five different types, namely increasing,decreasing, bathtub shaped, unimodal and DID shaped.

The main aim of this paper is to extend the EGE distribution to a four-parameter distribution by adding a new shape parameter. It contains EW and

EEW Distribution and its Applications 3

EGE as special cases. The new distribution is capable of modeling bathtub-shaped, upside-down bathtub (unimodal), increasing, decreasing and decreasing-increasing-decreasing (DID) hazard rate functions which are widely used in engi-neering for repairable systems. Hence, it can be used quite effectively for analyzinglifetime data of different types.

The rest of the paper is organized as follows. In Section 2, we introducethe EEW distribution and outline some sub-models of the distribution and then,discuss some of its properties. Some related issues are discussed in Section 3. InSection 4, we provide the estimation procedures and the asymptotic distributionsof the estimators. Simulation results and the analysis of a data set are providedin Section 5, and finally conclusions arrive in Section 6.

2. Definition and some properties

In this section we formally define the extended exponentiated Weibull family ofdistributions. We observe that several well known distributions can be obtainedas special cases of the proposed distribution. We also derive different propertiesand different measures of the proposed distribution in this section.

2.1. The Extended Exponentiated Weibull Distribution

The random variable Y is said to have an extended exponentiated Weibull (EEW)distribution, if the CDF of the random variable Y , denoted by F (y;α, β, γ, λ), isgiven by

F (y;α, β, γ, λ) =

{(1− (1− βλyγ)1/β)α, β 6= 0,

(1− e−λyγ )α, β = 0,(1)

where α, γ, λ > 0 and −∞ < β < ∞. Here, 0 < y < ∞ if β ≤ 0 and 0 < y <1

(βλ)1/γif β > 0. The PDF of Y can be expressed as

f(y;α, β, γ, λ) =

{αγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)α−1, β 6= 0,

αγλyγ−1e−λyγ

(1− e−λyγ )α−1, β = 0.

(2)A random variable X follows the EEW distribution with parameters α, β, γ andλ is denoted by X ∼ EEW (α, β, γ, λ).

Several well known distribution functions can be obtained as special cases ofthe EEW distribution depending on the values of α, β, γ. The details are presentedbelow.

(i) For γ = 1, the EEW distribution reduces to the EGE introduced and studiedby Gupta and Kundu (2011).

4 Debasis Kundu

(ii) For γ = 1, β = 0, α 6= 1, the EEW distribution reduces to the GE distribu-tion introduced by Gupta and Kundu (2007).

(iii) For γ = 1, β = 1 and α = 1, the EEW distribution reduces to the Uniformdistribution with CDF

F (y;λ) = λy.

(iv) The Pareto distribution with CDF

F (y;β, λ) = 1− (1− βλy)1/β ,

is a special case of the EEW distribution for γ = 1, α = 1 and β > 0.

(v) For γ = 1 and α = 1 b = 1, θ → 0+, the EEW distribution reduces to thegeneralized Pareto (GP) distribution with CDF

F (y;β, λ) =

{1− (1− βλy)1/β , β 6= 0,1− e−λy, β = 0.

(vi) For γ = 1, β 6= 0, α 6= 1, the EEW distribution reduces to the GW distri-bution introduced and analyzed by Mudholkar et al. (1996).

(vii) For γ 6= 1, α = 1 and β = 0, the EEW distribution reduces to the Weibulldistribution with CDF

F (y; γ, λ) = 1− e−λyγ

.

(viii) For γ 6= 1, α 6= 1 and β = 0, the EEW distribution reduces to the EWdistribution proposed by Nassar and Eissa (2003).

(ix) For γ = 2, α 6= 1 and β = 0, the EEW distribution reduces to the Burr Xdistribution with CDF

F (y;α, λ) = (1− e−λy2

)α.

2.1.1. Different shapes of EEW probability density and hazard functionIn this section we provide different shapes of the PDFs and hazard functions ofthe EEW distribution. Because of the complicated nature of the PDFs and hazardfunctions, it is difficult to obtain the shapes of the PDFs and hazard functionsanalytically. The following observations have been made graphically. For β < 0,the shape of the PDF of the EEW distribution can be decreasing or unimodaldepending on different values of parameters. For β > 0, the shape of the PDFsof the EEW distribution can be bath-tub shaped and increasing depending onthe different values of the shape parameters. The PDFs of the EEW for differentvalues of α, β and γ, where λ = 1 are plotted in Figure 1.


2 4 6 8 10y

f Hy L

Α=1 , Γ=0.65

Α=0.45 , Γ=1.98

Α=0.45 , Γ=1

Α=0.45 , Γ=0.65

Β=-2.5 , Λ=1

2 4 6 8 10y

f Hy L

Α=1.85 , Γ=1.98

Α=1.85 , Γ=1

Α=1 , Γ=1.98

Β=-2.5 , Λ=1

0.00 0.02 0.04 0.06 0.08 0.10 0.12y

f Hy L

Α=1 , Γ=0.45

Α=0.35 , Γ=0.45

Β=2.5 , Λ=1

0.1 0.2 0.3 0.4y

f Hy L

Α=2.19 , Γ=1

Α=1 , Γ=1

Β=2.5 , Λ=1

Figure 1 – The PDFs of the EEW for different values of α, β and γ with λ = 1

The hazard rate and survival function of the EEW distribution are given by,respectively

h(y;α, β, γ, λ) =

αγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)α−1

1− (1− (1− βλyγ)1/β)α, β 6= 0,

αγλyγ−1e−λyγ

(1− e−λyγ )α−1

1− (1− e−λyγ )α, β = 0,

and

s(y;α, β, γ, λ) =

{1− (1− (1− βλyγ)1/β)α, β 6= 0,1− (1− e−λyγ )α, β = 0.

(3)

The hazard function of the EEW distribution can take different shapes, namelyincreasing, decreasing, DID, unimodal and bathtub shaped. Figure 2 provides thehazard functions of the EEW distribution for different values of α, β, γ, and λ.

Because of the complicated nature of the hazard function, we could not es-tablish the above results in its full generality, but the following results can be

6 Debasis Kundu

2 4 6 8 10y

H Hy L

Α=1 , Γ=0.56

Α=0.45 , Γ=1

Α=0.45 , Γ=0.56

Β=-3 , Λ=1

2 4 6 8 10

y

H Hy L

Α=1.94 , Γ=2.6

Α=1.94 , Γ=1

Α=1 , Γ=2.6

Β=-3 , Λ=1

0.02 0.04 0.06 0.08 0.10 0.12y

H Hy L

Α=1 , Γ=0.45

Α=0.34 , Γ=0.45

Β=2.5 , Λ=1

0.02 0.04 0.06 0.08 0.10 0.12y

H Hy L

Α=2.8 , Γ=0.45

Β=2.5 , Λ=1

0.5 1.0 1.5 2.0 2.5 3.0y

H Hy L

Α=2.8 , Γ=0.45

Β=-1.22 , Λ=0.36

Figure 2 – The hazard rate of the EEW for different values of α, β and γ with λ = 1

established. In all these cases without loss of generality, we have assumed that λ= 1.

Theorem 1. If γ = 1 and β < 0, then for


(a) α > 1, the hazard function of EEW is an unimodal shaped,

(b) 0 < α < 1 it is a decreasing function.

Proof. See Theorem 1 of Gupta and Kundu (2011).

Theorem 2. If γ = 1 and for

(a) β > 1, α > 1, the hazard function of EEW is an increasing function,

(b) 0 < α < 1, 0 < β < 1, it is a bathtub shaped.

Proof. See Theorem 2 of Gupta and Kundu (2011).

Theorem 3. If 0 < α < 1, β < 0 and 0 < γ < 1, then the hazard function ofEEW is a decreasing function.

Proof. See in the Appendix.

Theorem 4. If α > 1, β > 1 and γ > 1, then the hazard function of EEW isan increasing function.


Theorem 5. If α > 1, β < 0 and γ > 1, then the hazard function of EEW isan unimodal function.


The quantile of a distribution plays an important role for any lifetime distri-bution. In case of the EEW distribution, it is observed that the p-th quantilecan be obtained in explicit form. Hence, if we have maximum likelihood estima-tors (MLEs) of the unknown parameters of a EEW distribution, the MLE of thecorresponding p-th quantile estimator also can be easily obtained.

The p-th quantile of the EEW distribution is given by

Q(p;α, β, γ, λ) =

(

1

βλ(1− (1− p1/α)β))1/γ , β 6= 0,

(− 1

λlog(1− p1/α))1/γ , β = 0.

When p =1

2, the median of EEW is

M =

(

1

βλ(1− (1− 2−1/α)β))1/γ , β 6= 0,

(− 1

λlog(1− 21/α))1/γ , β = 0.

(4)

The mode of EEW distribution cannot be obtained in explicit form. It has to beobtained by solving non-linear equation, and it is not pursued here.

8 Debasis Kundu

2.2. Moments and Moment Generating Function

The moment and moment generating functions play important roles for analyzingany distributions functions. The moment generating function characterizes thedistribution function. Although, we could not obtain the moments in explicitforms, they can be obtained as infinite summation of beta functions. Differentmoments can be easily calculated using any standard mathematical softwares.The k-th moments and the k-th central moments of the EEW distribution canbe obtained by using the expansion (1 − z)α =

∑∞i=0

(αi

)(−1)izi, where α is not

integer. We have the following results:

µ′k = E(Y k) =

∞∑i=0

(α−1i

) (−1)iα

β1+k/γλk/γBe(1 + k/γ, (i+ 1)/β), β > 0,

∞∑i=0

(α−1i

) (−1)iα

(−β)1+k/γλk/γBe(1 + k/γ,−k/γ − (i+ 1)/β), β < 0,

∞∑i=0

(α−1i

) (−1)iα

λk/γ(i+ 1)−(1+k/γ)Γ(1 + k/γ), β = 0,

(5)

where Be(., .) is beta function. Using µk = E(Y − µ′k)k =∑ki=0

(kj

)µ′j(−µ′1)k−j ,

the k-th central moments can be obtained as

µk =

k∑j=0

∞∑i=0

(kj

)(α−1i

) (−1)k+i−jα

β1+j/γλi/γBe(1 + j/γ, (i+ 1)/β)µ′1

k−j, β > 0,

k∑j=0

∞∑i=0

(kj

)(α−1i

) (−1)k+i−jα

(−β)1+j/γλj/γBe(1 + j/γ,−j/γ − (i+ 1)/β)µ′1

k−j, β < 0,

k∑j=0

∞∑i=0

(kj

)(α−1i

) (−1)k+i−jα

λj/γ(i+ 1)−(1+j/γ)Γ(1 + j/γ)µ′1

k−j, β = 0.

Using the moments of different orders, moment generating function of EEW canbe easily obtained. If all the moments of a random variable exist, then the moment

generating function of Y can be written as MY (t) = E(etY ) =

∞∑k=0

tk

k!E(Y k). Thus,

we have

MY (t) =

∞∑k=0

∞∑i=0

tk

k!

(α−1i

) (−1)iα

β1+k/γλk/γBe(1 + k/γ, (i+ 1)/β), β > 0,

∞∑k=0

∞∑i=0

tk

k!

(α−1i

) (−1)iα

(−β)1+k/γλk/γBe(1 + k/γ,−k/γ − (i+ 1)/β), β < 0,

∞∑k=0

∞∑i=0

tk

k!

(α−1i

) (−1)iα

λk/γ(i+ 1)−(1+k/γ)Γ(1 + k/γ), β = 0.


3. Some Related Issues

3.1. Order Statistics

The CDF and PDF of the first order statistics are given by

F1:n(y) =

{1− (1− (1− (1− βλyγ)1/β)α)n, β 6= 0,

1− (1− (1− e−λyγ )α)n, β = 0,

and

f1:n(y) =

nαγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)α−1 β 6= 0,

×(1− (1− (1− βλyγ)1/β)α)n−1,

nαγλyγ−1e−λyγ

(1− e−λyγ )α−1(1− (1− e−λyγ )α)n−1, β = 0,

respectively. The CDF and PDF of the largest order statistics are given, respec-tively, by

Fn:n(y) =

{(1− (1− βλyγ)1/β)nα, β 6= 0,

(1− e−λyγ )nα, β = 0,

and

fn:n(y) =

{nαγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)nα−1, β 6= 0,

nαγλyγ−1e−λyγ

(1− e−λyγ )nα−1, β = 0.

The CDF and PDF of r-th order statistics, where 1 ≤ r ≤ n and β 6= 0, are givenby

Fr:n(y) =

∞∑i=0

n∑j=r

(n− ji

)(n

j

)(−1)i(1− (1− βλyγ)1/β)α(i+j),

fr:n(y) =n!αγλyγ−1

(r − 1)!(n− r)!

∞∑i=0

∞∑j=0

(n− ri

)(α(i+ r)− 1

j

)(−1)i+j(1− βλyγ)

j+1β −1,

respectively.

3.2. Mean Deviations

For a random variable Y with the PDF f(y), CDF F (y), mean µ = E(Y ) andM = Median(Y ), the mean deviation about the mean and the mean deviationabout the median are defined by

δ1 = E|Y − µ| = 2µF (µ)− 2I(µ),

andδ2 = E|Y −M | = µ− 2I(M),

respectively, where I(t) =

∫ t

0

yf(y)dy.

10 Debasis Kundu

Theorem 6. The mean deviation functions of the EEW distribution are

δ1 = 2µ(1− (1− βλµγ)1/β)α − 2

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1µγ+1(1− βλµγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλµγ),

and

δ2 = µ− 2

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1Mγ+1(1− βλMγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλMγ),

where β < 0, µ is the mean of the EEW distribution given in (5) , M is the medianof the EEW distribution given in (4), and the Hypergeometric function 2F1 can be

expressed as 2F1(a, b; c; z) =∑∞n=0

(a)n(b)n(c)n

zn

n! , where c does not equal to 0, -1, -2,

..., and

(q)n =

{1 if n = 0q(q + 1)...(q + n− 1) if n > 0.

Proof.F (y) = (1− (1− βλyγ)

1β )α,

and

f(y) = αγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)α−1

= αγλ(1− βλyγ)1β−1

∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

iβ

= αγλyγ−1∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1.

so,

I(µ) =

∫ µ

0

yf(y)dy

= αγλ

∞∑i=0

(α− 1

i

)(−1)i

∫ µ

0

yγ(1− βλyγ)i+1β −1dy

=

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1µγ+1(1− βλµγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλµγ)


Thus, we have

δ1 = 2µ(1− (1− βλyγ)1β )α − 2

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1µγ+1(1− βλµγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλµγ),

and

δ2 = µ− 2∑∞i=0

(α−1i

)(−1)i

αγλ

γ + 1Mγ+1(1− βλMγ)

i+1β

×2 F1(1, 1 +1

γ+ i+1

β ; 2 + 1γ ;βλMγ).

3.3. Probability Weighted Moment

The probability weighted moments (PWMs) makes use of the analytical relation-ship among the parameters and the so-called PWMs of probability distributionin calculating magnitudes for the parameters. Although, probability weightedmoments are useful to characterize a distribution. We use the PWMs to derive es-timates of the parameters and quantiles of the probability distribution. Estimatesbased on PWMs are often considered to be superior to standard moment-basedestimates. These concepts used when maximum likelihood estimates (MLE), areunavailable or difficult to compute; e.g. see Greenwood et al. (1979); Hosking(1986); Harvey et al. (2017) and Tarko (2018). The PWMs are defined by

τs,r = E(Y sF (Y )r) =

{ ∫∞0ysF (y)rf(y)dy, β < 0,∫ 1

(βλ)1/γ

0 ysF (y)rf(y)dy, β > 0,

where r and s are positive integers and F (y) and f(y) are the CDF and PDF ofdistribution. The following theorem gives the PWMs of the EEW distribution.

Theorem 7. The PWMs of the EEW distribution are

τs,r =

∑∞i=0

(rαi

)(−1)i

(−βλ)s−1γ Γ( s+1

γ )Γ(−β+sβ+iγβλ )

γΓ(− iβ )

, β < 0,

∑∞i=0

(rαi

)(−1)i

(−βλ)s−1γ Γ( i+ββ )Γ(− s+1

λ )

γΓ(− iβ + 1+s+r

γ ), β > 0,

Proof. Here, we only give the proof for β > 0, in which case 0 < y < 1(βλ)1/γ

.

Proof of the other case is similar. We have

τs,r =

∫ 1

(βλ)1/γ

0

αγλys+γ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)(r+1)α−1dy.

12 Debasis Kundu

By using the expansion (1− z)α =∑∞i=0

(αi

)(−1)izi, we have

τs,r =

∞∑i=0

((r + 1)α− 1

i

)(−1)i

∫ 1

(βλ)1/γ

0

αγλys+γ−1(1− βλyγ)i+1β −1

=

∞∑i=0

((r + 1)α− 1

i

)(−1)i

αλ(βλ)−s+γγ Γ( 1+i

β )Γ( s+γγ )

Γ( i+1β −

s+γγ )

,

where α, β, γ, and s are positive.

By use of PWMs, we can obtain the mean and variance of the distribution accord-ing to

E(Y ) = τ1,0, and V ar(Y ) = τ2,0 − τ21,0.

3.4. Bonferroni and Lorenz curve

The Bonferroni and Lorenz curves are fundamental tools for analyzing data aris-ing in economics and reliability. Also, these curves have many applications inother fields like demography, insurance and medicine; e.g. see Bonferroni (1930);Gail and Gastwirth (1978); Giorgi and Crescenzi (2001); Shanker et al. (2017)and Arnold and Sarabia (2018). The Lorenz curve is a function of the cumula-tive proportion of ordered individuals mapped onto the corresponding cumulativeproportion of their size. The Bonferroni curve is given by

BF (F (y)) =1

µF (y)

∫ y

0

tf(t)dt,

or equivalently given by BF (p) = 1µp

∫ p0F−1(t)dt, where p = F (y) and F−1(t) =

inf{y : F (y) ≥ t}. Also, the Lorenz curve is given by

LF (F (y)) = F (y).BF (F (y)) =1

µ

∫ y

0

tf(t)dt,

where µ = E(Y ).

Theorem 8. The Bonferroni and Lorenz curve of the EEW distribution aregiven by, respectively

BF (F (y)) =1

µF (y)

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1yγ+1(1− βλyγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλyγ),


and

LF (F (y)) =1

µ

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1yγ+1(1− βλyγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλyγ),

where β 6= 0, µ = E(Y ) is given in (5) and F (y) is given in (1).

Proof. Such as the proof of theorem 2, we have

F (y) = (1− (1− βλyγ)1β )α,

and

f(y) = αγλyγ−1(1− βλyγ)1/β−1(1− (1− βλyγ)1/β)α−1

= αγλ(1− βλyγ)1β−1

∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

iβ

= αγλyγ−1∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1.

Thus, we have

BF (F (y)) =1

µF (y)

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1yγ+1(1− βλyγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλyγ)

and

LF (F (y)) =1

µ

∞∑i=0

(α− 1

i

)(−1)i

αγλ

γ + 1yγ+1(1− βλyγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλyγ).

3.5. Renyi and Shannon entropy

In information theory, entropy is a measure of the uncertainty associated with arandom variable. The Shannon entropy is a measure of the average informationcontent one is missing when one doesn’t know the value of the random variable. Auseful generalization of Shannon entropy is the Renyi entropy. The Renyi entropyis important in ecology and statistics. It is also important in quantum information,

14 Debasis Kundu

where it can be used as a measure of entanglement; e.g. see Shannon (1948); Renyi(1961); Seo and Kang (1970); Kayal and Kumar (2013); Kayal et al. (2015) andKang et al. (2012).The Renyi and Shannon entropy of the EEW distribution are given, respectively,by

IR(r) =1

1− rlog

∫y

fr(y)dy

=

1

1−r log∑∞i=0

(r(α−1)

i

)(−1)i

(−βλ)r−rγ−1

γ (αγλ)γΓ(− γ+1β + γ−1

γ )

γΓ(− r−rβ+iβ ), β < 0,

11−r log

∑∞i=0

(r(α−1)

i

)(−1)i 1

1+r(γ−1) (αγλ)γ(βλ)−1+r(γ−1)

γ

×2F1(− i+r−rββ , 1+r(γ−1)γ ; 1+r(γ−1)+γ

γ ;βλ((βλ)−1/γ)γ), β > 0,

and

E(− log f(y)) = log(αγλ)− α− 1

α+ (1− γ)

∞∑i=0

i∑j=0

(ij

)(−1)j

iµ′1j

+ (1− 1

β)

∞∑i=0

(βλ)i

iµ′iγ ,

where µ′k = E(Y k) is given in (5).

3.6. Residual life function

Given that a component survives up to time t ≥ 0, the residual life function is theperiod beyond t, and defined by the conditional variable Y − t|Y > t. We obtainthe r-th order moment of the residual life of the EEW distribution via the generalformula

mr(t) = E[(Y − t)r|Y > t],

in two cases; β < 0 and β > 0. Let β < 0, thus the r-th residual life is given by

mr(t) =1

s(t)

∫ ∞t

(y − t)rf(y)dy =1

s(t)

∞∑i=0

r∑j=0

(α− 1

i

)(r

j

)(−1)r+i−j

× tγ+

(i+1)γβ αγ(−βγ)

i+1β

jβ + γ + iγ2F1(

β + i− 1

β,− jβ + γ + iγ

βγ; 1 − i+ 1

β− j

β;t− γ

βλ),

and for β > 0, the r-th residual life is given by

mr(t) =1

s(t)

∫ 1

(βλ)1/γ

t

(y − t)rf(y)dy =1

s(t)

∞∑i=0

r∑j=0

(α− 1

i

)(r

j

)(−1)r+i−j

αγtγ−j

jβ + γ + iγ

×[−e−iπjβ+γ+iγ

βγ (−βλ)−j 2F1(β − i− 1


βγ; 1 − i+ 1

β− j

β; eiπ)

+tjβ+γ+iγ−βγ

β (−βλ)i+1β 2F1(

β − i− 1


βγ;− jβ + γ + iγ − βγ

βγ;t−γ

βλ)],


where s(t) (the survival function of Y ) is given in (3).The mean residual life (MRL) function is a helpful tool in model building, andit has been used for both parametric and nonparametric building. It is very im-portant in any lifetime modeling as it characterizes the distribution. Lifetime canexhibit increasing MRL (IMRL) or decreasing MRL (DMRL) shaped. MRL func-tions that first increasing (decreasing) and then decreasing (increasing) are usuallycalled upside-down bathtub (bathtub) shaped, UMRL (BMRL). the relationshipbetween the behavior of the two functions of a distribution was studied by manyauthors such as Ghitany (1998), Mi (1995), Park (1985), Shanbhag (1970)and Tang et al. (1999). The MRL function for the EEW distribution obtainsby setting r = 1 in the above equations and it is given in the following theorem.Although, the MRL function of a EEW cannot be obtained in explicit form, it canbe obtained in terms of infinite series.

Theorem 9. The MRL function of the EEW distribution is

m1(t) =1

s(t)[

∞∑i

(α− 1

i

)(−1)i

αγtβ+γ+iγ

β (−βλ)i+1β

β + γ + iγ

×2F1(β − i− 1

β,−β + γ + iγ

βγ;γ − 1

γ− i+ 1

β;t−γ

βλ)

−∞∑i=0

(α− 1

i

)(−1)i

tα(1− βλtγ)i+1β

i+ 1],

where β < 0, and

m1(t) =1

s(t)[

∞∑i

(α− 1

i

)(−1)i

αγλ

γ + 1tγ+1(1− βλtγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλtγ)

−∞∑i=0

(α− 1

i

)(−1)i

tα

i+ 1(1− βλtγ)

i+1β ].

16 Debasis Kundu

Proof. when β < 0, we have

m1(t) =1

s(t)

∫ ∞t

(y − t)f(y)dy =1

s(t)[

∫ ∞t

yf(y)dy −∫ ∞t

tf(y)dy]

=1

s(t)[

∫ ∞t

αγλyγ∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1dy

−t∫ ∞

0

αγλyγ−1∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1dy]

=1

s(t)[

∞∑i

(α− 1

i

)(−1)i

αγtβ+γ+iγ

β (−βλ)i+1β

β + γ + iγ

×2F1(β − i− 1

β,−β + γ + iγ

βγ;γ − 1

γ− i+ 1

β;t−γ

βλ)

−∞∑i=0

(α− 1

i

)(−1)i

tα(1− βλtγ)i+1β

i+ 1],

and when β > 0, we have

m1(t)=1

s(t)

∫ 1/(βλ)1γ

t

(y − t)f(y)dy =1

s(t)[

∫ 1/(βλ)1γ

t

yf(y)dy −∫ 1/(βλ)

1γ

t

tf(y)dy]

=1

s(t)[

∫ 1/(βλ)1γ

t

αγλyγ∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1dy

−t∫ 1/(βλ)

1γ

0

αγλyγ−1∞∑i=0

(α− 1

i

)(−1)i(1− βλyγ)

i+1β −1dy]

=1

s(t)[

∞∑i

(α− 1

i

)(−1)i

αγλ

γ + 1tγ+1(1− βλtγ)

i+1β

×2F1(1, 1 +1

γ+i+ 1

β; 2 +

1

γ;βλtγ)−

∞∑i=0

(α− 1

i

)(−1)i

tα

i+ 1(1− βλtγ)

i+1β ].

On the other hand, we analogously discuss the reversed residual life and someof its properties. The reversed residual life can be defined as the conditionalrandom variable t − Y |Y ≤ t which denotes the time elapsed from the failure ofa component given that its life is less than or equal to t. Also in reliability, themean reversed residual life and the ratio of two consecutive moments of reversedresidual life characterize the distribution uniquely.

The r-th order moment of the reversed residual life for the EEW distributioncan be obtained via the general formula

µr(t) = E[(t− y)r|Y ≤ t],


in two cases; β < 0 and β > 0. Let β < 0, thus the r-th reversed residual life isgiven by

µr(t) =1

F (t)

∫ t

0

(t− y)rf(y)dy =1

F (t)

∞∑i=0

r∑j=0

(α− 1

i

)(r

j

)(−1)r+i−jαe−

iπ(γ+r−j)γ

×[tj(βλ)

j−rγ Γ( j−rγ + i+1

β )Γ(γ+r−jγ )

βΓ(β−i−1β )

+γ(βλ)

i+1β

rβ − jβ + γ + iγeiπ(rβ−jβ+γ+iγ)

βγ

×tr+(i+1)γβ

2F1(β − i− 1

β,j − rγ− j + 1

β;γ + j − r

γ− i+ 1

β;t−γ

βλ)], (6)

and for β > 0 the r-th reversed residual life is given by

µr(t) =1

F (t)

∫ 1

(βλ)1/γ

0

(t− y)rf(y)dy

=1

F (t)

∞∑i=0

r∑j=0

(α− 1

i

)(r

j

)(−1)r+i−j

αλtj(βλ)−γ+r−jγ Γ( i+1

β )Γ(γ+r−jγ )

Γ( i+1β + γ+r−j

γ ),

where t > 1(βλ)1/γ

, and for t < 1(βλ)1/γ

, the r-th reversed residual life is same as

the equation (6).The mean and second moment of the reversed residual life of the EEW distributioncan be obtained by setting r = 1, 2 in above equations. Also, by using µ1(t)and µ2(t), we obtained the variance of the reversed residual life of the EEWdistribution.

4. Parametric inference

In this section, we consider the parametric inference of the unknown parametersα, β, γ and λ of the EEW distribution. The parametric inferences will be discussedbased on likelihood method in two situations; censored and full data.

4.1. Likelihood method based on censored data

In most of survival analysis and reliability studies, the censored data are oftenencountered. Let Yi be the random variable from EEW distribution with theparameters vector Θ = (α, β, γ, λ). A simple random censoring procedure is onein which each element i is assumed to have a lifetime Yi and a censoring timeCi, where Yi and Ci are independent random variables. Suppose that the dataincluding n independent observations yi = min(Yi, Ci) for i = 1, · · · , n. Thedistribution of Ci does not depend on any of the unknown parameters of Yi. Thecensored log-likelihood `c(Θ) is given by

18 Debasis Kundu

`c(Θ) = r logα+ r log γ + r log λ+ (γ − 1)∑i∈F

log yi + (1

β− 1)

∑i∈F

log(1− βλyγi )

+(α− 1)∑i∈F

log(1− (1− βλyγi )1/β) +∑i∈C

log[1− (1− (1− βλyγi )1/β)α],

where r is the number of failures and F and C denote the uncensored and censoredsets of observations, respectively.

By differentiating the log-likelihood function with respect to α, β, γ and λ,

respectively, components of score vector U(Θ) = (∂`c(Θ)∂α , ∂`c(Θ)

∂β , ∂`c(Θ)∂γ , ∂`c(Θ)

∂λ )

are derived as

∂`c(Θ)

∂α=r

α+∑i∈F

log(

1 − (1 − βλyγi )1/β)

−∑i∈C

(1 − (1 − βλyγi )1/β

)αlog(

1 − (1 − βλyγi )1/β)

1 −(

1 − (1 − βλyγi )1/β)α

∂`c(Θ)

∂β=− 1

β2

∑i∈F

log (1 − βλyγi ) −(

1

β− 1

)∑i∈F

λyγi1 − βλyγi

−(α− 1)∑i∈F

(1 − βλyγi )1/β(− log(1−βλyγi )

β2 − λyγi

β(1−βλyγi )

)1 − (1 − βλyγi )1/β

+∑i∈C

α (1 − βλyγi )1/β(

1 − (1 − βλyγi )1/β)α−1

(− log(1−βλyγi )

β2 − λyγi

β(1−βλyγi )

)1 −

(1 − (1 − βλyγi )1/β

)α∂`c(Θ)

∂γ=r

γ−(

1

β− 1

)∑i∈F

βλyγi log(yi)

1 − βλyγi+ (α− 1)

∑i∈F

λyγi log(yi) (1 − βλyγi )1β−1

1 − (1 − βλyγi )1/β

+∑i∈F

log(yi) −∑i∈C

αλyγi log(yi) (1 − βλyγi )1β−1(

1 − (1 − βλyγi )1/β)α−1

1 −(

1 − (1 − βλyγi )1/β)α

∂`c(Θ)

∂λ=r

λ+

(1

β− 1

)∑i∈F

− βyγi1 − βλyγi

+ (α− 1)∑i∈F

y(i)γ (1 − βλyγi )1β−1

1 − (1 − βλyγi )1/β

−∑i∈C

αyγi (1 − βλyγi )1β−1(

1 − (1 − βλyγi )1/β)α−1

1 −(

1 − (1 − βλyγi )1/β)α .

4.2. Likelihood method based on complete data


Let y1, ..., yn be the random sample of size n from EEW distribution. The log-likelihood function is given by

`(α, β, γ, λ) =n logα+ n log γ + n log λ+ (γ − 1)

n∑i=1

log yi

+(1

β− 1)

n∑i=1

log(1− βλyγi ) + (α− 1)

n∑i=1

log(1− (1− βλyγi )1/β).

The first derivatives of the log-likelihood function with respect to α, β, γ and λ aregiven in Appendix, and the maximum likelihood estimators (MLEs) of parameterscan be obtained by maximizing this function. For given β, γ and λ, the MLE of αcan be obtained as

α(β, γ, λ) = − n∑ni=1 log(1− (1− βλyγi )1/β)

.

By maximizing the profile log-likelihood function `(α(β, γ, λ), β, γ, λ), with respectto β, γ, λ, the MLEs of β, γ and λ can be obtained. Now we will discuss theasymptotic properties of the MLEs in two situations; β < 0 and β > 0.The Regular Case: When β < 0, the situation is exactly the same as the gen-eralized Weibull case discussed by Mudholkar et al. (1996). In this case EEWsatisfies all the regularity properties of the parametric family. Then asymptoti-cally, as n→∞, √

n(Θ−Θ) −→ N4(0, I−1(Θ)),

where Θ = (α, β, γ, λ), Θ = (α, β, γ, λ), N4 denotes the four-variate normal dis-tribution and I(Θ) is the Fisher information matrix. The observed informationmatrix is

In (Θ) = −

Iαα Iαβ Iαγ IαλIαβ Iββ Iβγ IβλIαγ Iβγ Iγγ IγλIαλ Iβλ Iγλ Iλλ

,where

Iθiθj =∂2`

∂θi∂θj, i, j = 1, 2, 3, 4

and they are provided in the Appendix.The Non-Regular Case: When β > 0, we propose a reparametrization of

α, β, γ, λ as (α, β, γ, φ), where φ =1

(βλ)1γ

, then λ =φ−γ

β. The PDF (2) and CDF

(1) can be written as

f(y;α, β, γ, φ) = αγφ−γ

βyγ−1(1− (

y

φ)γ)

1β−1(1− (1− (

y

φ)γ)

1β )α−1, (7)

20 Debasis Kundu

andF (y;α, β, γ, φ) = (1− (1− (

y

φ)γ)

1β )α,

respectively, for 0 < y < φ and 0 otherwise. The corresponding quantile functionbecomes

Q(y;α, β, γ, φ) = (φγ(1− (1− u 1α )β))

1γ .

Based on a random sample y1, ..., yn from (7), the MLEs can be obtained bymaximizing the log-likelihood function:

`(α, β, γ, φ) =n logα+ n log γ − nγ log φ− n log β + (γ − 1)

n∑i=1

log y(i)

+(1

β− 1)

n∑i=1

log(1− (y(i)

φ)γ) + (α− 1)

n∑i=1

log(1− (1− (y(i)

φ)γ)

1β ).

(8)

It is immediate from (8) that for fixed 0 < α < 1, 0 < β < 1 and γ > 0, asφ ↓ y(n), `(α, β, γ, φ)→∞. Thus, in this case the MLEs do not exist.

To estimate the unknown parameters, first estimate the parameter φ by itsconsistent estimator φ = y(n). The modified log-likelihood function based on theremaining (n−1) observations after ignoring the largest observation and replacingφ by φ = y(n) is

`(α, β, γ, φ) = (n− 1) logα+ (n− 1) log γ − (n− 1)γ log y(n) − (n− 1) log β

+(γ − 1)

n−1∑i=1

log y(i) + (1

β− 1)

n−1∑i=1

log(1− (y(i)

y(n))γ)

+(α− 1)

n−1∑i=1

log(1− (1− (y(i)

y(n))γ)

1β ).

The modified MLE of α and λ, for fixed β and γ, can be obtained as

α(β, γ) = − n− 1

1− (1− (y(i)y(n)

)γ)1β

and

λ =y−γ(n)

β,

respectively. Therefore, in this case the modified MLE of β and λ can be obtainedby solving the optimization problem from the modified log-likelihood function ofβ and γ.For the propose of statistical inference, an understanding of the distribution of φis necessary. This is given in the following theorem.


Theorem 10. i) The marginal distribution of φ = y(n) is given by

p(φ ≤ t) = (1− (1− (t

φ)γ)

1β )nα.

ii) Asymptotically as n→∞,

nβ((y(n)

φ)γ − 1)→ −Xβ ,

where X ∼ Exp(α), with mean 1α .

Proof. i)

p(φ ≤ t) = p(Y(n) ≤ t) = (p(Y ≤ t))n = (1− (1− (t

φ)γ)

1β )nα.

ii)

Y(n)d=Q(U(n)) = φ(1− (1− U

1α

(n))β)

1γ ,

soY(n)

φ= (1− (1− U

1α

(n))β)

1γ ,

then we have

nβ((Y(n)

φ)γ − 1) = −(n(1− U

1α

(n)))β .

Hence,

p(n(1− U1α

(n)) ≤ t) = p(1− U1α

(n) ≤t

n) = p(−U

1α

(n) ≤t

n− 1) = p(U

1α

(n) ≥ 1− t

n)

= p(U(n) ≥ (1− t

n)α) = 1− ((1− t

n)α)n = 1− (1− t

n)nα,

as n −→∞, we have

p(n(1− U1α

(n)) ≤ t) −→ 1− e−αt.

Thusnβ((

y(n)

φ)γ − 1)→ −Xβ , where X ∼ Exp(α).

5. Simulation Experiments and Data Analysis

5.1. Simulation Experiments

In this section, we perform some simulation studies, just to verify how the MLEswork for different sample sizes and different parameter values for the proposed

22 Debasis Kundu

EEW model. The results are obtained from 1000 Monte Carlo replications fromsimulations carried out using the software R.We have used the following parameter sets:Model 1: α = 1, β = 2, γ = 1.5, λ = 1,Model 2: α = 2.35, β = 2, γ = 1.21, λ = 1,Model 3: α = 0.8, β = −0.50, γ = 0.8, λ = 1,Model 4: α = 1.57, β = 0.81, γ = 2.5, λ = 1.5,Model 5: α = 0.48, β = −0.78, γ = 0.35, λ = 1Model 6: α = 1.35, β = 1.57, γ = 1.49, λ = 1.87and different sample sizes, namely: n = 50, 100, 150, 200, 300 and 400.We report the average estimates and the associated square root of mean squarederrors (RMSEs). The results are presented in Table 1. From the results presentedin Table 1 the following points are quite clear. (i) It is quite clear that the MLEsare working quite well. As the sample size increases the standard deviation andthe square root of mean squared errors decrease. (ii) This verifies the consistencyproperties of the MLEs. For all practical purposes, MLEs can be used quiteeffectively for estimating the unknown parameters of the proposed EEW model.

EEW Distribution and its Applications 23TABLE

1TheMLEs,

Std

andRMSE

n(α,β,γ,λ

)M

LE

Std

RM

SE

αβ

γλ

αβ

γλ

αβ

γλ

50

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

76

1.7

69

1.0

69

1.9

09

0.1

53

0.2

77

0.1

35

0.4

88

0.1

71

0.3

61

0.4

52

1.0

32

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.0

73

1.9

18

1.8

01

4.7

09

0.9

21

0.3

03

1.0

84

7.3

11

0.9

62

0.3

14

1.2

35

8.1

98

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.7

98

-0.9

27

1.3

54

1.0

72

2.2

15

1.5

99

1.2

48

0.8

52

2.4

30

1.6

55

1.3

66

0.8

55

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

70

0.8

17

1.2

74

5.8

96

0.4

63

0.1

39

0.3

42

3.2

84

0.5

04

0.1

39

1.2

73

5.4

87

(0.4

8,

-0.7

8,

0.3

5,

1.0

0)

1.7

96

-0.8

21

1.0

75

0.7

76

4.4

09

1.1

96

0.8

67

0.8

28

4.6

01

1.1

97

1.1

31

0.8

58

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.3

02

1.4

80

1.1

88

4.4

62

0.3

44

0.2

41

0.2

72

2.6

72

0.3

47

0.2

58

0.4

06

3.7

23

100

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

60

1.8

32

1.0

22

1.7

01

0.0

91

0.1

94

0.0

51

0.2

13

0.1

09

0.2

56

0.4

80

0.7

33

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.1

24

1.9

71

1.5

10

2.9

13

0.6

57

0.1

89

0.8

89

3.7

83

0.6

95

0.1

91

0.9

38

4.2

39

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.5

09

-0.4

71

0.9

93

1.0

50

1.2

88

0.6

65

0.6

60

0.6

48

1.4

70

0.6

65

0.6

88

0.6

50

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

47

0.8

15

1.2

17

5.1

62

0.3

30

0.1

01

0.2

63

1.9

63

0.3

97

0.1

01

1.3

10

4.1

55

(0.4

8,

-0.7

8,

0.3

5,

.1.0

0)

0.8

82

-0.8

98

1.0

27

0.7

57

0.8

40

1.2

07

0.7

10

0.5

59

0.9

31

1.2

12

0.8

84

0.6

10

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.2

78

1.5

28

1.1

33

3.8

73

0.2

28

0.1

99

0.2

31

2.2

69

0.2

39

0.2

04

0.4

24

3.0

27

150

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

34

1.9

00

1.0

35

1.6

64

0.0

73

0.1

70

0.0

75

0.2

48

0.0

81

0.1

97

0.4

71

0.7

09

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.1

03

1.9

68

1.4

42

2.5

23

0.5

86

0.1

86

0.7

86

2.9

50

0.6

36

0.1

88

0.8

20

3.3

20

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.2

37

-0.5

37

1.0

40

1.9

50

0.9

87

0.5

63

0.5

99

0.5

53

1.0

79

0.5

65

0.6

45

0.5

55

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

33

0.8

25

1.2

13

5.0

51

0.3

00

0.0

84

0.2

65

1.8

63

0.3

83

0.0

86

1.3

14

4.0

10

(0.4

8,

-0.7

8,

0.3

5,

1.0

0)

0.9

98

-0.8

17

0.9

91

0.7

79

1.3

38

0.8

51

0.6

25

0.6

14

1.4

35

0.8

52

0.7

95

0.6

53

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.2

69

1.5

24

1.1

19

3.6

32

0.2

14

0.1

34

0.1

97

1.4

03

0.2

29

0.1

41

0.4

20

2.2

53

200

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

31

1.9

11

1.0

18

1.6

12

0.0

59

0.1

45

0.0

41

0.1

53

0.0

66

0.1

70

0.4

84

0.6

31

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.1

78

1.9

98

1.2

93

1.8

16

0.5

04

0.1

78

0.5

63

1.2

92

0.5

33

0.1

78

0.5

69

1.5

29

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.1

48

-0.4

67

0.9

64

0.9

31

0.7

39

0.4

42

0.4

27

0.4

68

0.8

17

0.4

43

0.4

58

0.4

73

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

39

0.8

18

1.2

17

5.0

49

0.3

07

0.0

76

0.2

56

1.7

21

0.3

85

0.0

77

1.3

08

3.9

44

(0.4

8,

-0.7

8,

0.3

5,

1.0

0)

0.6

54

-0.7

55

0.9

83

0.6

44

0.3

72

0.6

51

0.4

74

0.3

76

0.4

11

0.6

52

0.6

77

0.5

17

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.2

71

1.5

44

1.1

27

3.6

33

0.2

01

0.1

32

0.2

11

1.3

79

0.2

16

0.1

35

0.4

19

2.2

39

300

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

23

1.9

32

1.0

22

1.5

99

0.0

44

0.1

19

0.0

43

0.1

38

0.0

50

0.1

37

0.4

80

0.6

15

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.2

02

1.9

94

1.2

79

1.7

88

0.4

85

0.1

37

0.5

36

1.3

42

0.5

07

0.1

37

0.5

40

1.5

56

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.0

79

-0.4

63

0.9

71

0.9

16

0.6

06

0.3

56

0.3

85

0.4

26

0.6

68

0.3

58

0.4

21

0.4

32

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

48

0.8

15

1.2

08

4.9

90

0.3

00

0.0

58

0.2

59

1.6

75

0.3

73

0.0

58

1.3

17

3.8

71

(0.4

8,

-0.7

8,

0.3

5,

1.0

0)

0.6

36

-0.7

76

0.9

77

0.6

48

0.3

49

0.6

07

0.4

14

0.3

59

0.3

82

0.6

07

0.6

32

0.5

03

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.2

41

1.5

67

1.1

34

3.5

90

0.1

85

0.1

11

0.2

04

1.2

65

0.2

14

0.1

11

0.4

10

2.1

35

400

(1.0

0,

2.0

0,

1.5

0,

1.0

0)

1.0

22

1.9

41

1.0

13

1.5

75

0.0

37

0.1

00

0.0

39

0.1

20

0.0

43

0.1

16

0.4

88

0.5

87

(2.3

5,

2.0

0,

1.2

1,

1.0

0)

2.2

16

1.9

84

1.2

15

1.6

41

0.4

09

0.1

20

0.4

62

1.1

03

0.4

31

0.1

21

0.4

62

1.2

76

(0.8

0,

-0.5

0,

0.8

0,

1.0

0)

1.0

47

-0.4

56

0.9

70

0.9

10

0.5

51

0.3

30

0.3

49

0.3

62

0.6

04

0.3

33

0.3

88

0.3

73

(1.5

7,

0.8

1,

2.5

0,

1.5

0)

1.3

53

0.8

20

1.1

86

4.7

72

0.2

93

0.0

55

0.2

36

1.4

17

0.3

64

0.0

56

1.3

35

3.5

65

(0.4

8,

-0.7

8,

0.3

5,

1.0

0)

0.6

14

-0.6

97

0.9

42

0.6

22

0.2

77

0.3

96

0.3

17

0.3

04

0.3

08

0.4

04

0.5

44

0.4

85

(1.3

5,

1.5

7,

1.4

9,

1.8

7)

1.2

73

1.5

66

1.1

12

3.4

64

0.1

63

0.7

80

0.1

94

1.2

24

0.1

80

0.0

79

0.4

25

2.0

10

24 Debasis Kundu

5.2. Data Analysis

In this part, we fit the EEW distribution to the real data set and also comparethe fitted EEW with some sub-models such as: the EW, W, EGE, GE and Ex-ponential distributions, to show the superiority of the EEW distribution. In fact,it is observed that empirical hazard function of the data indicates that the dataare coming from a lifetime distribution which has a DID shaped hazard functionand the proposed distribution provides the best fit than many existing lifetimedistributions. Therefore, the proposed distribution provides another option to apractitioner to use it for data analysis purposes. The results are obtained by usingthe function optim form package stats4 in R.

As a data set, we consider the 101 data points represent the stress-rupture lifeof kevlar 49/epoxy strands which were subjected to constant sustained pressureat the 70% stress level until all had failed, so that we have complete data withexact times of failure, which are shown by Andrews and Herzberg (1985). Coorayand Ananda (2008) used this data in fitting generalization of the half-normaldistribution.

The TTT plot of this data set in Figure 3 display a decreasing-increasing-decreasing (DID) hazard rate function. The MLEs of the parameters, -2log-likelihood, AIC (Akaike Information Criterion), the Kolmogorov-Smirnov teststatistic (K-S), the Anderson-Darling test statistic (AD), the Cramer-von Misestest statistic (CM) and Durbin-Watson test statistic (DW) are displayed in Table2. The CM and DW test statistics are described in details in Chen and Balakrish-nan (1995) and Watson (1961), respectively. In general, the smaller the valuesof KS, AD, CM and WA, the better the fit to the data. From the values of thesestatistics, we conclude that the EEW distribution provides a better fit to this datathan the EW, W, EGE, GE and Exponential distributions.

Figure 3 – TTT Plots of Andrews and Herzberg data.


TABLE 2MLEs, KS, p-value, -2 Log L, AD, CM, DW and AIC statistics for Andrews and

Herzberg data.

Dist MLE K-S AIC p-value -2 Log L AD CM DW

EEW α = 0.1004, β = −3.4749 0.0588 205.7 0.8760 197.7 0.4550 0.1575 0.1486

γ = 6.9363, λ = 0.0081

EGE α = 0.8766, β = −0.0178 0.0909 211.6 0.3745 205.6 1.0560 0.2722 0.2634

λ = 0.9113

GE α = 0.8663, λ = 0.8883 0.0887 209.6 0.4041 205.6 1.0209 0.2634 0.2566EW α = 0.7930, γ = 1.0604 0.0844 211.6 0.4674 205.6 0.9553 0.2473 0.2434

λ = 0.8113

E λ = 0.9758 0.0888 209.0 0.4035 207.0 1.2481 0.2469 0.2457

W γ = 0.9259, λ = 1.0094, 0.0965 210.0 0.3033 206.0 1.1221 0.2789 0.2715

Plots of the estimated PDF and CDF of the EEW, EW, W, EGE, GE andExponential models fitted to the data set corresponding to Table 2, are given inFigures 4 and 5. Also, Figure 6 is displayed the QQ-plot of the EEW model to thedata set. These plots suggest that the EEW distribution is superior to the otherdistributions in terms of model fitting.

Figure 4 – Plots of fitted PDF and CDF of EEW, EGE, GE, EW, Exponential andWeibull models for Andrews and Herzberg data.

6. Conclusions and Some Future Work

We propose the Extended Exponentiated Weibull (EEW) distribution to general-ized the Extended Generalized Exponential (EGE) distribution by adding a new

26 Debasis Kundu

Figure 5 – Estimated distribution function versus the empirical distribution from thefitted EEW, EGE, GE, EW, E and Weibull models for Andrews and Herzberg data.


0 2 4 6

02

46

8

qq plot for EEW

Theoretical Quantile

Sam

ple

Quantile

Figure 6 – QQ-plot of EEW model for Andrews and Herzberg data.

shape parameter. The PDF of the EEW distribution can take various shapesdepending on it’s parameter values. The hazard rate function of the EEW dis-tribution can take i) increasing, ii) decreasing, iii) unimodal, iv) bathtub and v)decreasing-increasing-decreasing (DID) shaped, depending on it’s parameter val-ues. Therefore, it is quite flexible and can be used effectively in modeling survivaldata and reliability problems. Application of the EEW distribution to the realdata set is given to show that the new distribution provides consistently betterfit than the some of its sub-models. Now a natural question is how to choose thecorrect model, i.e. whether we should choose the full four-parameter model orone of the sub models. As we have mentioned before, we may use some testingof hypothesis namely whether the shape parameters take some specific values asindicated in Section 2, or we can use some of the information theoretic criteria tochoose the correct model.

It should be mentioned that in this paper we have mainly discussed about theclassical inference of the unknown parameters. It will be interesting to developthe Bayesian inference of the unknown parameters. One may think of taking in-dependent gamma priors of α, γ and δ and normal priors on β. It is expectedthat the Bayes estimates or the associated highest posterior density credible inter-vals cannot be obtained in closed form. One may try to use importance samplingmethod or Markov Chain Monte Carlo methods to compute Bayes estimates andto construct highest posterior density credible intervals. Another important de-velopment can be to provide inference both classical and Bayesian when we havecensored data. More work is needed along these directions.

28 Debasis Kundu

Acknowledgements:

The authors would like to thank two unknown reviewers for their constructivecomments which had helped to improve the paper significantly.

Appendix

A. Hazard Function

To prove Theorems 3, 4 and 5 we need the following lemma.

Lemma 1: Let U be a non-negative absolutely continuous random variable withthe PDF, CDF and hazard function as fU , FU and hU , respectively. For θ > 0, letus define V = Uθ. Then the shape of the hazard function of V will be the sameas the shape of the function g(u) = hU (u)u1−θ, for u > 0.

Proof: If we denote the PDF, CDF and the hazard function of V as fV (·), FV (·)and hV (·), respectively, then after some calculations, it can be that for v > 0

hV (v) =1

θhU (v1/θ)v(1/θ)(1−θ).

Since v1/θ is an increasing function and it increases from zero to infinity as vincreases from zero to infinity, for θ > 0, the result follows. 2

Proof of Theorem 3: First observe that if V ∼ EEW(α, β, γ, λ), then U =V γ ∼ EGE(α, β, λ). Now let us use Lemma 1 with θ = 1/γ. Using part (b) ofTheorem 1 and Lemma 1, the result immediately follows. 2

Proof of Theorem 4: Using part (a) of Theorem 4 and Lemma 1, the resultimmediately follows. 2

Proof of Theorem 5: From Theorem 3, it follows that if U ∼ EGE(α, β, λ),then hU (u) is unimodal. If for u > 0, g(u) = hU (u)u1−1/γ , then

g′(u) = h′U (u)u1−1/γ +(1− 1/γ)hU (u)

u1/γ.

Therefore, the sign of g′(u) will be the same as the sign of

p(u) = h′U (u)u+ (1− 1/γ)hU (u).

Since hU (u)→ 0 as u→∞, and γ > 1, then p(u) changes sign only once. Hence,hV (v) is also unimodal. 2


B. The first derivatives of the log-likelihood function

The first derivatives of the log-likelihood function with respect to α, β, γ and λare given by

∂`

∂α=

n

α+

n∑i=1

log(1− (1− βλyγi )1β ),

∂`

∂β= −

∑ni=1 log(1− βλyγi )

β2− (

1

β− 1)

n∑i=1

λyγi1− βλyγi

+(α− 1)

n∑i=1

(1− βλyγi )1β (

log(1−βλyγi )

β2 +λyγi

β(1−βλyγi ))

1− (1− βλyγi )1β

,

∂`

∂γ=

n

γ+

n∑i=1

log(yi)− (1

β− 1)

n∑i=1

βλ log(yi)yγi

1− βλyγi

+(α− 1)

n∑i=1

λ log(yi)yγi (1− βλyγi )

1β−1

1− (1− βλyγi )1β

,

∂`

∂λ=

n

λ− (

1

β− 1)

n∑i=1

βyγi1− βλyγi

+ (α− 1)

n∑i=1

yγi (1− βλyγi )1β−1

1− (1− βλyγi )1β

.

The elements of the observed information matrix are

Iαα =∂2`

∂α2=−nα2

,

30 Debasis Kundu

Iββ =∂2`

∂β2=

2∑ni=1 log(1− βλyγi )

β2+ (

1

β− 1)

n∑i=1

− λ2y2γi

(1− βλyγi )2

−2∑ni=1−

λyγi1−βλyγi

β2+ (α− 1)

n∑i=1

(−(1− βλyγi )

2β (− log(1−βλyγi )

β2 − λyγiβ(1−βλyγi )

)2

(1− (1− βλyγi )1β )2

−(1− βλyγi )

1β (

2 log(1−βλyγi )

β3 − λ2y2γiβ(1−βλyγi )2

+2λyγi

β2(1−βλyγi ))

1− (1− βλyγi )1β

−(1− βλyγi )


β2 − λyγiβ(1−βλyγi )2

)2

1− (1− βλyγi )1β

),

Iγγ =− n

γ2+∂2`

∂γ2= (

1

β− 1)

n∑i=1

(−β2λ2 log(yi)

2y2γi

(1− βλyγi )2− βλ log(yi)

2yγi1− βλyγi

)

+(α− 1)

n∑i=1

(−λ2 log(yi)

2y2γi (1− βλyγi )

2β−2

(1− (1− βλyγi )1β )2

−( 1β − 1)βλ2 log(yi)

2y2γi (1− βλyγi )

1β−2

1− (1− βλyγi )1β

+λ log(yi)

2yγi (1− βλyγi )1β−1

1− (1− βλyγi )1β

),

Iλλ=∂2`

∂λ2= − n

λ2+ (

1

β− 1)

n∑i=1

− β2y2γi

(1− βλyγi )2

+(α− 1)

n∑i=1

(− y2γi (1− βλyγi )

2β−2

(1− (1− βλyγi )1β )2−

( 1β − 1)βy2γ

i (1− βλyγi )1β−2

1− (1− βλyγi )1β

),

Iαβ =∂2`

∂α∂β=

n∑i=1

−(1− βλyγi )



)

1− (1− βλyγi )1β

,

Iαγ =∂2`

∂α∂γ=

n∑i=1

λ log(yi)yγi (1− βλyγi )

1β−1

1− (1− βλyγi )1β

,

Iαλ=∂2`

∂α∂λ=

n∑i=1

yγi (1− βλyγi )1β−1

1− (1− βλyγi )1β

,

Iβγ =∂2`

∂β∂γ= −

∑ni=1−

βλ log(yi)yγi

1−βλyγiβ2

+ (1

β− 1)

n∑i=1

(−βλ2 log(yi)y

2γi

(1− βλyγi )2− λ log(yi)y

γi

1− βλyγi)

+(α− 1)

n∑i=1

(λ log(yi)y

γi (1− βλyγi )

2β−1(− log(1−βλyγi )


)

(1− (1− βλyγi )1β )2

+λ2 log(yi)y

2γi (1− βλyγi )

1β−2

1− (1− βλyγi )1β

+λ log(yi)y




)

1− (1− βλyγi )1β

),

Iβλ=∂2`

∂β∂λ=

∑ni=1

βyγi1−βλyγiβ2

+ (1

β− 1)

n∑i=1

(− βλy2γi

(1− βλyγi )2− yγi

1− βλyγi)

+(α− 1)

n∑i=1

(yγi (1− βλyγi )



)

(1− (1− βλyγi )1β )2

+λy2γ

i (1− βλyγi )1β−2

1− (1− βλyγi )1β

+yγi (1− βλyγi )



)

1− (1− βλyγi )1β

),


Iγλ=∂2`

∂γ∂λ= (

1

β− 1)

n∑i=1

(−β2λ log(yi)y

2γi

(1− βλyγi )2− β log(yi)y

γi

1− βλyγi)

+(α− 1)

n∑i=1

(−λ log(yi)y2γi (1− βλyγi )

2β−2

(1− (1− βλyγi )1β )2

−( 1β − 1)βλ log(yi)y

2γi (1− βλyγi )

1β−2

1− (1− βλyγi )1β

+log(yi)y


1β−1

1− (1− βλyγi )1β

).

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Summary

In this paper, we introduce a univariate four-parameter distribution. Several known dis-tributions like exponentiated Weibull or extended generalized exponential distributioncan be obtained as special case of this distribution. The new distribution is quite flexible

34 Debasis Kundu

and can be used quite effectively in analysing survival or reliability data. It can havea decreasing, increasing, decreasing-increasing-decreasing (DID), upside-down bathtub(unimodal) and bathtub-shaped failure rate function depending on its parameters. Weprovide a comprehensive account of the mathematical properties of the new distribution.In particular, we derive expressions for the moments, mean deviations, Renyi and Shan-non entropy. We discuss maximum likelihood estimation of the unknown parameters ofthe new model for complete sample using the profile and modified likelihood functions.One empirical application of the new model to real data are presented for illustrativepurposes.

Keywords: Probability weighted moments; Renyi and Shannon entropy; Extended gen-eralized exponential distribution; Regular family of distributions

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