kumaraswamy weibull-generated family of … kumaraswamy weibull-generated family of distributions...

Advances and Applications in Statistics © 2016 Pushpa Publishing House, Allahabad, India Published Online: May 2016 http://dx.doi.org/10.17654/AS048030205 Volume 48, Number 3, 2016, Pages 205-239 ISSN: 0972-3617

Received: February 14, 2016; Revised: March 1, 2016; Accepted: March 20, 2016 2010 Mathematics Subject Classification: 62-XX. Keywords and phrases: Kumaraswamy Weibull distribution, Weibull-G family, moments, order statistics, maximum likelihood estimation.

Communicated by K. K. Azad

KUMARASWAMY WEIBULL-GENERATED FAMILY OF DISTRIBUTIONS WITH APPLICATIONS

Amal S. Hassan1 and M. Elgarhy1,2 1Institute of Statistical Studies and Research (ISSR) Department of Mathematical Statistics Cairo University Egypt e-mail: [email protected]

2Department of Mathematical Statistics Burraydah Colleges Qassim, Saudi Arabia e-mail: [email protected]

Abstract

In this paper, we introduce and study a new family of continuous distributions called Kumaraswamy Weibull-generated ( )GKwW -

family of distributions which is an extension of the Weibull-G family of distributions proposed by Bourguignon in [3]. The new family includes several known models. We obtain general explicit expressions for the quantile function, moments, probability weighted moments, generating function, mean deviation and order statistics. Making use of maximum likelihood method, we discuss the model

Amal S. Hassan and M. Elgarhy 206

parameters. Furthermore, besides showing the usefulness of our proposed family of distributions by considering a three real data set, evidence of this family to outperform other classes of lifetime models has been noticed.

1. Introduction

Over the past decades, several statistical distributions have been extensively used and applied in several areas such as engineering, actuarial, medical sciences, demography, etc. However, in many situations, the classical distributions are not suitable for describing and predicting real world phenomena. Recently, there has been an increased interest in defining new generators or generalized classes of univariate continuous distributions by introducing additional shape parameter(s) to baseline model. The new generating families extended the well-known distributions and at the same time provide great flexibility in modeling data in practice.

Eugene et al. [8] introduced a new class of generated distributions based on beta distribution, called the beta-generated family of distributions. Zografos and Balakrishnan [14] proposed gamma generated family. Cordeiro and de Castro [4], introduced Kumaraswamy generalized distributions. From an arbitrary parent cumulative distribution function (cdf) ( ),xG Ristic and

Balakrishnan [13] proposed an alternative gamma generator for any parent distribution ( ).xG Pescim et al. [11] proposed the Kummer beta-generated

family of distributions for any parent cdf. Cordeiro et al. [5], proposed the exponentiated generalized class of distributions. Alzaatreh et al. [1], extended the beta-generated family of distributions by using any non-negative continuous random variable T as the generator, in place of the beta random variable. They defined this class as follows:

( ) ( )( )( )

∫=xGW

dttrxF0

, (1)

where ( )tr is the probability density function (pdf) of a non-negative

Kumaraswamy Weibull-generated Family of Distributions … 207

continuous random variable T. In this new class, the distribution of the random variable T is the generator and the upper limit is the transformation

( )( ) ( )[ ].1log xGxGW −−= This generated family of distributions is called

“T-X distribution”. It is clear that one can define a different upper limit for generating different types of T-X distributions. The transformation ( )( )xGW

satisfies the following two conditions: ( )( ) [ )∞∈ ,0xGW and it is a

monotonic non decreasing function.

Recently, Bourguignon et al. [3], proposed the Weibull- ( )GWG - family

of distributions. They considered a continuous distribution ( )⋅G with density

function ( )⋅g and the Weibull cdf ( )βα−−= tetR 1 ( )0for >t with positive

parameters α and .β Based on this density, by replacing ( )( )xGW in (1)

with ( ) ( ) [ ( ) ( )],;1;,;; ξ−=ξξξ xGxGxGxG they defined cdf of their

family by

( )( )( )

( )( )∫ ξ−ξ β

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−α−−β −=αβ=ξβα

β;1;

0;1

;1 ,1,,; xG

xGxG

xGt edtetxF

,0,, >βα⊆∈ RDx

where ( )ξ;xG is a baseline cdf, which depends on a parameter vector .ξ

In this paper, a new extension of the GW - family of distributions called

Kumaraswamy Weibull-G family of distributions is proposed. The rest of this article is organized as follows: In Section 2, the Kumaraswamy Weibull-G family of distributions is defined. Some general mathematical properties of the family are discussed in Section 3. Some members of the proposed family are discussed in Section 4. In Section 5, the estimation of the model parameters is performed by the method of maximum likelihood. An illustrative application based on real data is examined in Section 6. Finally, concluding remarks are presented in Section 7.


2. The Kumaraswamy Weibull-generated Family

In this section, the new class of distributions, called GKwW - family of

distributions is introduced.

Cordeiro et al. [6] presented the KwW distribution with the following cdf and pdf:

( ) [ [ ] ] ,0,,,;0;111 >βα>−−−=βα− batetR bat

and

( ) [ ] [ [ ] ] ;111 111 −α−−α−α−−β βββ−−−αβ= batatt eeetabtr

.0,,,;0 >βα> bat (2)

We obtain the new family by replacing the generator ( )tr defined in (1)

by the pdf generator defined in (2) as follows:

( ) [ ] [ [ ] ]( )( )∫ ξ−ξ

−α−−α−α−−β βββ−−−αβ= ;1

;

0111 .111xG

xGbatatt dteeetabxF

Then, the distribution function of GKwW - family takes the following form:

( ) [ (( )( ) ) ] ,0,,,;0;111 ;1;

>βα>−−−=

β

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−baxexF baxG

xG

(3)

where 0,, >βba are the three shape parameters and 0>α is the scale

parameter. The cdf (3) provides a wider family of continuous distributions. The pdf corresponding to (3) is given by

( ) ( )( ) ( )( )( )

( )( ) [

( )( ) ] 1;1;

;1;

1

11

;1;; −⎥⎦

⎤⎢⎣⎡

ξ−ξ

α−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−

+β

−βββ

−ξ−

ξξαβ= axGxG

xGxG

eexG

xgxGabxf

[ (( )( ) ) ] .0,,,;0;11 1;1;

>βα>−−× −⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−β

baxe baxGxG

(4)

Hereafter, a random variable X with pdf (4) is denoted by .-~ GKwWX


Note that.

(i) For ,1=b the GKwW - distribution reduces to a new family, called

exponentiated Weibull-generated ( )GEW - family of distributions.

(ii) For 1== ba the GKwW - distribution reduces to GW - family of

distributions which is obtained by Bourguignon et al. [3].

The survival, hazard, reversed hazard and cumulative hazard rate functions are obtained, respectively, as follows:

( ) [ (( )( ) ) ] ,0,,,;0;11 ;1;

>βα>−−=

β

⎥⎦⎤

⎢⎣⎡

ξ−ξα−

baxexF baxGxG

( ) ( ) ( )( )( ) [

( )( ) ]

( )( ) [ (( )( ) ) ]

,

11;1

1;;

;1;

1

1;1;

;1;

1

axGxG

axGxG

xGxG

exG

eexgxGabxh β

ββ

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−+β

−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−−β

−−ξ−

−ξξαβ=

( )

( ) ( )( )( ) [

( )( ) ]

[ (( )( ) ) ]

( )( ) { [ (( )( ) ) ] }

,

111;1

11

1;;

;1;

1

1;1;

1;1;

;1;

1

baxGxG

baxGxG

axGxG

xGxG

exG

e

eexgxGab

x β

β

ββ

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−+β

−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−

−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−−β

−−−ξ−

−−×

−ξξαβ

=τ

and

( ) ( ) [ (( )( ) ) ] .11ln1ln ;1;

baxGxG

exFxH

β

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−−−−=−−=

3. Statistical Properties

In this section, some properties of the GKwW - family will be obtained

as follows:

3.1. Quantile and median

Quantile functions are used in theoretical aspects of probability theory,


statistical applications and simulations. Simulation methods utilize quantile function to produce simulated random variables for classical and new

continuous distributions. The quantile function, say ( ) ( )uFuQ 1−= of X is

given by

[ (( )( )( )( ) ) ]bauQGuQG

eu

β

⎥⎦⎤

⎢⎣⎡−

α−−−−= 1111

after some simplifications, it reduces to the following form:

( ) { [ [ ( ) ] ] }

{ [ [ ( ) ] ] },

111ln1

111ln1111

11111

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−−−+

−−−=βα

−βα

−−

ab

ab

u

uGuQ (5)

where u is considered as a uniform random variable on the unit interval (0, 1)

and ( )⋅−1G is the inverse function of ( ).⋅G

In particular, the median can be derived from (5) by setting ,5.0=u i.e., the median is given by

{ [ [ ( ) ] ] }

{ [ [ ( ) ] ] }.

5.011ln1

5.011ln1111

11111

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−−+

−−=βα

−βα

−−

ab

abGMedian

3.2. Expansions for distribution and density functions

In this section, some representations of the cdf and pdf for Kumaraswamy Weibull-G family of distributions will be presented. The mathematical relation given below will be useful in this section.

It is well-known that, if 0>β is real non integer and ,1<z the

generalized binomial theorem is written as follows:

( ) ( )∑∞

=

−β ⎟⎠⎞

⎜⎝⎛ −β

−=−0

1 .1

11i

ii zi

z (6)

Then, by applying the binomial theorem (6) in (4), the probability density function of GKwW - family, where b is real non integer becomes


( ) ( ) ( )( )( )

( )( ) 1

;1;

1

;1;;

+β

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−−β

ξ−

ξβαβ=

β

xGexgxGabxf

xGxG

( ) [( )( ) ] ( )∑

∞

=

−+⎥⎦⎤

⎢⎣⎡

ξ−ξα−

β

−⎟⎠⎞

⎜⎝⎛ −

−×0

11;1;

.11

1i

iaxGxG

i ei

b

Then, using binomial expansion again in the previous equation, where a is real non integer, leads to:

( ) ( ) ( )( )( ) 1

1

;1;;

+β

−β

ξ−

ξβαβ=xG

xgxGabxf

( )( ) ( ) ( )

( )∑∞

=

⎥⎦⎤

⎢⎣⎡

ξ−ξ+α−

+

β

⎟⎠⎞

⎜⎝⎛ −+⎟⎠⎞

⎜⎝⎛ −

−×0,

;1;1

,111

1ji

xGxGjji e

jia

ib

(7)

by using the power series for the exponential function, we obtain

( ) ( )( ) ( ) ( ) ( )

( )∑∞

=

β⎥⎦⎤

⎢⎣⎡

ξ−ξ

+α−

⎥⎦⎤

⎢⎣⎡

ξ−ξ+α−=

β

0

;1;1

.;1;

!11

k

kkkkxG

xGj

xGxG

kje (8)

Inserting expansion (8) in (7) we have

( ) ( ) ( ) ( ) ( )∑∞

=

++

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+α−ξαβ=0,,

111!

11;kji

kkkji

j

ia

i

bk

jxgabxf

( )[ ] ( ) ( )[ ] ( )[ ].;1; 1111 ++β−−+β ξ−ξ× kk xGxG (9)

Now, using the generalized binomial theorem, we can write

( )[ ] ( )[ ] ( )( )( )∑

∞

=

++β− ξ⎟⎠⎞

⎜⎝⎛ ++β

=ξ−0

11 .;1

;1 xGk

xG k (10)

Inserting (10) in (9), the probability density function of GKwW − can


be expressed as an infinite linear combination of exponentiated-G (exp-G for short) density functions, i.e.,

( ) ( ) ( ) ( )∑∞

=

−++βξξη=0,,,

11,,, ,;;

kji

kkji xGxgxf (11)

then

( ) ( ) ( )∑∞

=+ξ+β ξ=

0,,,,,, ,;

kjikkji xhWxf (12)

where

( ) ( )( )[ ]

( ) ( ),

1111!1

111,,, ⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++β⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛ −++β

+−α=+++ k

jia

ib

kkjabW

kkjikkji

and

( )[ ] ( ) ( ) ( ) .;;;,11,,,

,,,−ξξ=ξ

++βη

= aa

kjikji xGxagxhkW

If ba, and β are integers, the index i stops at jb ,1− stops at

( ) 11 −+ia and stops at ( ) .1 ++β k

Another form can be yielded when β is real non integer by adding and

subtracting 1 to ( ) ( ) 11; −++βξ kxG into (10) as follows:

( ) ( ) ( )[ ][ ] ( )∑∞

=

−++βξ−−ξη=0,,,,

11,,, .;11;

ukji

kkji xGxgxf

Using binomial expansion yields:

( ) ( )( )

( ) ( )[ ]∑∞

=

ξ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ ++βη−=

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

11

ukji

ukji

u xGxgu

kxf

Using binomial expansion more time yields:


( ) ( )( )

( ) ( )( )∑ ∑∞

= =

+ ⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ ++β

η−=0,,,, 0

,,, .1

1ukji

u

m

mkji

mu xGxgmu

uk

xf

So the expansion form of the pdf is as follows:

( ) ( ) ( )( )∑ ∑∞

= =

η=0,,,, 0

,,,,, ,ukji

u

m

mmukji xGxgxf (13)

where

( )( )

.1

1 ,,,,,,,, ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ ++βη−=η +

mu

uk

kjimu

mukji

Furthermore, an expansion for the cumulative function is derived as follows:

Using binomial expansion for ( )[ ] ,hxF where h is an integer, leads to:

( )[ ] ( ) [ (( )( ) ) ]∑

=

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−β

−−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

g

bgaxGxG

gh egh

xF0

;1;

.111

Using binomial expansion another time, where b is a real non integer, leads to

( )[ ] ( ) (( )( ) )∑∑

=

∞

=

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−+

β

−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

g p

apxGxG

pgh ep

bggh

xF0 0

;1;

.11

Using binomial expansion again, where a is a real non integer, leads to

( )[ ] ( )( )( )∑ ∑

=

∞

=

⎥⎦⎤

⎢⎣⎡

ξ−ξ

α−++

β

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

g qp

xGxGqqpgh e

q

ap

p

bg

g

hxF

0 0,

;1;

,1

by using the power series for the exponential function (8) in the previous equation, we obtain


( )[ ] ( ) ( ) ( )( )∑ ∑

=

∞

=

β+++

⎥⎦⎤

⎢⎣⎡

ξ−ξ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛α−=h

g tqp

tttqpgh

xGxG

q

ap

p

bg

g

ht

qxF0 0,,

,;1;

!1

by using the relation (10) in the previous equation where β is real non

integer, we obtain

( )[ ] ( ) ( ) ( )( )∑ ∑=

∞

=

+β+++

ξ⎟⎟⎠

⎞⎜⎜⎝

⎛β⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛α−=h

g tuqp

utttqpg

h xGut

qap

pbg

gh

tqxF

0 0,,,,;!

1

( )[ ] ( ) ( ) ( )[ ][ ]∑ ∑=

∞

=

+β+++

ξ−−⎟⎟⎠

⎞⎜⎜⎝

⎛β⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛α−=h

g tuqp

utttqpg

h xGut

qap

pbg

gh

tqxF

0 0,,,.;11!

1

Using binomial expansion again, leads to

( )[ ] ( ) ( ) ( )[ ]∑ ∑=

∞

=

++++ξ−⎟

⎠⎞

⎜⎝⎛ +β⎟⎠⎞

⎜⎝⎛β⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛α−=

h

g lutqp

ltltqpg

h xGl

utut

qap

pbg

gh

tqxF

0 0,,,,.;1!

1

Using binomial expansion again, where l is a real integer, leads to

( )[ ] ( ) ( )∑ ∑ ∑=

∞

= =

+++++ α−=ξh

g lutqp

l

z

tzltqpgh

tqxF

0 0,,,, 0!

1;

( ) .; zxGzl

lut

ut

qap

pbg

gh

ξ⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ +β⎟⎠⎞

⎜⎝⎛β⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛×

Replacing ∑ ∑∞

= =0 0l

l

zwith ∑ ∑

∞

=

∞

=zl z 0yielding

( )[ ] ( ) ( )∑ ∑ ∑∑=

∞

=

∞

=

∞

=

+++++ α−=h

g utqp zl z

tzltqpgh

tqxF

0 0,,, 0!

1

( ) .; zxGzl

lut

ut

qap

pbg

gh

ξ⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ +β⎟⎠⎞

⎜⎝⎛β⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛×

Finally,


( )[ ] ( )∑∞

=

ξ=0

,;z

zz

h xGsxF (14)

where

( ) ( )∑ ∑ ∑=

∞

=

∞

=

+++++

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +β⎟⎟⎠

⎞⎜⎜⎝

⎛β⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛α−=h

g utqp zl

tzltqpgz

z

l

l

ut

u

t

q

ap

p

bg

g

ht

qs0 0,,,

.!1

3.3. Moments

If X has the pdf (11), then its rth moment can be obtained through the following relation:

( ) ( )∫∞

∞−==μ′ .dxxfxXE rr

r (15)

First, when β is an integer, then substituting (11) into (15) yields:

( ) ( ) ( )( ) ( )∑ ∫∞

=

∞

∞−−++βξξη==μ′

0,,,

11,,, .;;

kji

krkji

rr dxxGxgxXE

Then

( )∑∞

=−++βτη=μ′

0,,,11,,, ,

kjikrkir

where ( ) 11, −++βτ kr is the probability weighted moments (PWMs) of the

baseline distribution. Second, when β is a real non integer, then substituting

(13) into (15) yields:

( ) ( ) ( )( )∑ ∑ ∫∞

= =

∞

∞−ξξη==μ′

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

ukji

u

m

mrmukji

rr dxxGxgxXE

Then, ∑ ∑∞

= =τη=μ′

0,,,, 0,,,,, .

ukji

u

mmrmukir


Another form based on the parent quantile function:

First, when β is an integer yields

( )( ) ( )∑ ∫∞

=

−++βη=μ′0,,,

1

011

,,, .kji

krGkjir duuuQ

Second, when β is a real non integer yields

( )( )∑ ∑ ∫∞

= =

η=μ′0,,,, 0

1

0,,,,, .ukji

u

m

mrGmukjir duuuQ

3.4. The probability weighted moments

A general theory for PWMs covers the summarization and description of the theoretical probability distributions, estimation of parameters, quantiles of probability distributions and hypothesis testing for probability distributions. The PWMs method can generally be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly. The probability weighted moments can be obtained from the following relation:

[ ( ) ] ( ) ( )( )∫∞

∞−==τ ., dxxFxfxxFXE srsr

sr (16)

First, when β is an integer, by substituting (11) and (14) into (16), replacing

h with s, leads to:

( ) ( )( ) ( )∑ ∑ ∫∞

=

∞

=

∞

∞−−++β+ξξη=τ

0,,,, 0

11,,,, .;;

ukji z

kzrkjizsr dxxGxgxs

Then,

( )∑ ∑∞

=

∞

=−++β+τη=τ

0,,, 011,,,,, .

kji zkzrkjizsr s


Second, when β is real non integer, by substituting (13) and (14) into (16),

replacing h with s, leads to:

( ) ( )∑ ∑∑ ∫∞

= =

∞

=

∞

∞−+ξξη=τ

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

ukji

u

m z

zmrmukjizsr dxxGxgxs

Then

∑ ∑∑∞

= =

∞

=+τη=τ

0,,,, 0 0,,,,,,, .

ukji

u

m zmzrmukjizsr s



( )( ) ( )∑ ∑ ∫∞

=

∞

=

−++β+η=τ0,,, 0

1

011

,,,, .kji z

kzrGkjizsr duuuQs

Second, when β is real non integer yields

( )( )∑ ∑∑ ∫∞

= =

∞

=

+η=τ0,,,, 0 0

1

0,,,,,, .ukji

u

m z

mzrGmukjizsr duuuQs

3.5. The moment generating function

Generally, the moment generating function is ( ) ( ),tXX eEtM = by

using the exponential expansion, it can written as ( ) ( )∑∞

==

0.!r

rr

X XErttM

Another form based on the parent quantile function: first, when β is an

integer yields

( ) ( )( ) ( )∑ ∫∞

=

−++βη=0,,,

1

011

,,, .kji

kutQkjiu duuetM G



( ) ( )( )∑ ∑ ∫∞

= =

η=0,,,, 0

1

0,,,,, .ukji

u

m

mutQmukjiu duuetM G

3.6. The mean deviation

Basically, the mean deviation is a measure for the amount of scatter in X. Generally, the mean deviation is expressed by:

( ) ( ) ( )μ−μμ=δ TFX 221 and ( ) ( ),22 MTX −μ=δ

where ( ) ( )∫ ∞−=

qdxxxfqT which is the first incomplete moment.



( ) ( ) ( )∑ ∫∞

=

+++βη=0,,,

011

,,, .kji

q kGkji duuuQqT


( ) ( )∑ ∑ ∫∞

= =

η=0,,,, 0

1

0,,,,, .ukji

u

m

mGmukji duuuQqT

3.7. Order statistics

Let ( ) ( ) ( )nXXX <<< 21 be the order statistics of a random sample

of size n following the Kumaraswamy Weibull-G family of distributions, with parameters ,,, αba and .β The pdf of the kth order statistic, as

mentioned in David [7], can be written as follows:

( ) ( ( ) )( ( ) )

( ) ( ) ( ( ) )∑−

=ν

−+νν⎟⎟⎠

⎞⎜⎜⎝

⎛ν

−−

+−=

knk

kk

kX xFkn

knkBxf

xf k0

1.11, (17)


First, when β is an integer substituting (11) and (14) in (17), replacing h

with ,1−+ν k leads to

( ) ( ( ) )( ( ) )

( )1,;+−ξ

= knkBxg

xf kkX k

( ( ) ) ( )∑∑ ∑−

=ν

∞

=

∞

=

−++β+ξη×kn

z kji

kzkvzkji xGp

0 0 0,,,

11,,,, ,; (18)

where

( ) .1, zz skn

p ⎟⎟⎠

⎞⎜⎜⎝

⎛ν

−−= ν

ν

Second, when β is real non integer substituting (13) and (14) in (17),

replacing h with ,1−+ν k leads to

( ) ( ( ) )( ( ) )

( ) ∑∑−

=ν

∞

=−−ξ

=kn

z

kkX knkB

xgxf k

0 01,

;

( ( ) )∑ ∑∞

= =

+ξη×0,,,, 0

,,,,,, ,;ukji

u

m

zmkvzmukji xGp (19)

where ( )⋅g and ( )⋅G are the density and cumulative functions of the

GKwW - distributions, respectively.

Moreover, rth moment of kth order statistics is obtained as follows:

( ( ) ) ( ) ( ( ) ) ( )∫∞

∞−= .kk

rk

rk dxxfxXE (20)

First, when β is an integer substituting (18) in (20), leads to

( ( ) ) ( ) ( )∑∑ ∑−

=ν

∞

=

∞

=−++β+τη

+−=

kn

z kjikzrvzkji

rk pknkBXE

0 0 0,,,11,,,,, .1,

1


Second, when β is real non integer substituting (19) in (20), leads to

( ( ) ) ( ) ∑∑ ∑ ∑−

=ν

∞

=

∞

= =+τη

+−=

kn

z kji

u

mmzrvzmukji

rk pknkBXE

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

1

4. Some Special Sub-models

Here, we discuss a few examples of GKwW - family of distributions.

4.1. KwW-uniform distribution

As a first example, suppose that the parent distribution is uniform in

the interval ∞<θ<< x0 and ( ) .1;θ

=θxg Therefore, the KwW-uniform

(KwWU) distribution has the following cdf, pdf by direct substituting

( ) ,;θ

=θ xxG in (3) and (4) as follows:

( ) [ ( ) ] ,0,0,,,,111 θ<<>βα−−−=

β⎟⎠⎞⎜

⎝⎛

−θα−

xbaexF baxx

( )( )

[ ] [ [ ] ] ,111 111

1−⎟

⎠⎞⎜

⎝⎛

−θα−−⎟

⎠⎞⎜

⎝⎛

−θα−⎟

⎠⎞⎜

⎝⎛

−θα−

+β

−ββββ

−−−−θαβθ= bax

xax

xx

x

eeex

xabxf

.0,0,,, θ<<>βα xba

Furthermore, the survival, hazard rate and reversed hazard rate functions are obtained, respectively, as follows:

( ) [ ( ) ] ,11 baxx

exF

β⎟⎠⎞⎜

⎝⎛

−θα−

−−=

( ) [ ]

( ) [ [ ] ]

,

11

1

1

11

axx

axx

xx

ex

eexabxh β

ββ

⎟⎠⎞⎜

⎝⎛

−θα−+β

−⎟⎠⎞⎜

⎝⎛

−θα−⎟

⎠⎞⎜

⎝⎛

−θα−

−β

−−−θ

−αβθ=


and

( ) [ ] [ [ ] ]

( ) [ [ [ ] ] ]

.

111

111

1

111

baxx

baxx

axx

xx

ex

eeexabx β

βββ

⎟⎠⎞⎜

⎝⎛

−θα−+β

−⎟⎠⎞⎜

⎝⎛

−θα−−⎟

⎠⎞⎜

⎝⎛

−θα−⎟

⎠⎞⎜

⎝⎛

−θα−

−β

−−−−θ

−−−αβθ=τ

4.2. KwW-BurrXII distribution

Let us consider the parent BurrXII distribution with pdf and cdf given by

( ) 0,,1,,;1

1 >σμ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞⎜

⎝⎛μ

+σμ=μσ−σ−

−− cxxccxgc

cc and ( ) =μσ,,; cxG

,11σ−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞⎜

⎝⎛μ

+−cx respectively. Then the KwW-BurrXII (KwWBurrXII)

distribution, has the following cdf, pdf, survival, hazard rate and reversed hazard rate functions:

( ) [ [ ] ] ,0,0,,,,,,,11111

>>σμβα−−−=

βσ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−

xcbaexF ba

x c

( ) [ ] [ [ ] ] ,

1

111

11

1

111

111

1111

σ−

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−−⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−

−βσ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+

−−−×

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+αβσμ

=

βσβσ

βσ

c

ba

x

a

x

cx

cc

x

ee

xexabc

xf

cc

c

( ) [ [ ] ] ,1111

ba

x c

exF

βσ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−

−−=


( ) [ ]

[ [ ] ]

,

111

1

11

111

111

1111

a

xc

a

x

cx

cc

c

c

c

ex

e

xexabc

xh βσ

βσ

βσ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−σ−

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−

−βσ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−−−

−−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+

−×

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+αβσμ

=

and

( )

[ ]

[ [ ] ]

[ [ [ ] ] ]

,

1111

11

111

111

111

111111

1

ba

xc

ba

x

a

xc

x

cc

c

c

cc

ex

e

exexabc

x βσ

βσ

βσβσ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−σ−

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−−βσ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+α−−−

−−−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+

−−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞⎜

⎝⎛μ

+αβσμ

=τ

where 0,0 >μ>x is scale parameter, 0,,,,, >σβα cba are shape

parameters.

4.3. KwW-Weibull distribution

If the random variable X follows the Weibull distribution with cdf given

by ( ) ,0,,,1,; >γλ−=γλγλ− xexG x where λ and γ are scale and shape

parameters. The pdf and cdf of KwW-Weibull (KwWW) is derived from the KwW-G family of distributions as follows:

( ) [ ( ( ) ) ] ,0,0,,,,,111 1 >>γλβα−−−=βγλ −α− xbaexF bae x

( ) [ ] { ( ) }( ( ) ) 111111 1 −−α−λ−−α−−β−λ−γ βγλγβγλγ−αβλγ= aexex xx

eeexabxf

[ ( ( ) ) ] .11 11 −−α− βγλ−−× bae x

e


Additionally, survival, hazard rate and reversed hazard rate functions of KwWW take the following forms:

( ) [ ( ( ) ) ] ,11 1 bae xexF

βγλ −α−−−=

( ) [ ] { ( ) }( ( ) )

( ( ) ),

11

111

11111

ae

aexex

x

xx

e

eeexabxhβγλ

βγλγβγλγ

−α−

−−α−λ−−α−−βλ−γ

−−

−−αβλγ=

and

( )

[ ] { ( ) }

( ( ) ) [ ( ( ) ) ]

[ ( ( ) ) ],

111

111

1

1

1111

111

bae

baeae

xex

x

xx

x

e

ee

eexab

xβγλ

βγλβγλ

γβγλγ

−α−

−−α−−−α−

λ−−α−−βλ−γ

−−−

−−−

−αβλγ

=τ

respectively. Note that for 1=γ the KwW-Weibull distribution reduces to

KwW-exponential distribution.

4.4. KwW-quasi Lindley distribution

Quasi Lindley distribution is introduced by Shanker and Mishra [12]. They defined the cdf and pdf of the quasi Lindley with two parameters as follows:

( ) ,111,; ⎥⎦⎤

⎢⎣⎡

+θ+−=θ θ−

pxepxG x

( ) ( ) .1,; xexpppxg θ−θ++θ=θ

The cdf and pdf for Kumaraswamy Weibull quasi Lindley (KwWQL) distribution are derived from (3) and (4), respectively, as follows:

( ) [ ( ) ] ,0,1,0,,,,,111

1

11>−>>θβα−−−=

β−θ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛

+θ+α−

xpbaexF bapxe x


( )

( )

( ) [ ([ ]

) ]

( )2

111111

11111

111

1

111

111

111

1

⎟⎠⎞⎜

⎝⎛

+θ++

−×

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+×θ+αβθ

=−

β−−⎟⎠⎞⎜

⎝⎛

+θ+θα−

β−θ

β−θ

−−−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+α−

−β−θ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡θ−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+α−

pxp

e

pxeexpab

xfbap

xxex

x

eapxe

xxp

xe

[ ( ) ] .11 1111

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛

+θ+α−

β−θ

−−× bapxe x

e Furthermore, the survival, hazard rate and reversed hazard rate functions

are obtained as follows:

( ) [ ( [ ] ) ] ,111

111

bae pxx

exFβ

−−⎟⎠⎞⎜

⎝⎛

+θ+θ

α−−−=

( )

( ) ( [ ] )

( ) [ ( [ ] ) ]

,

11111

1111

11

11

1

11

11

2

111

111

ae

xpxe

aex

pxx

x

pxx

epxp

e

epxexpab

xhβ

−−⎟⎠⎞⎜

⎝⎛

+θ+θ

β−θ

β−

−⎟⎠⎞⎜

⎝⎛

+θ+θ

α−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡θ−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+α−

−α−−β−

θ

−−⎟⎠⎞⎜

⎝⎛

+θ++

−⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+θ+αβθ

=

and

( )

( )

[ [ [ ] ) ] ( [ ] )

( ) [ [ ( [ ] ) ] ]bae

aebae

pxexp

xe

pxx

pxxp

xx

xx

epxp

ee

expab

x

β

−⎟⎠⎞⎜

⎝⎛

+θ+θ

β

−⎟⎠⎞⎜

⎝⎛

+θ+θ

β

−⎟⎠⎞⎜

⎝⎛

+θ+θ

×β−θ

β−θ

−α−

−−α−−−α−

×⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡θ−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+α−

−−−⎟⎠⎞⎜

⎝⎛

+θ++

−−−

θ+αβθ

=τ

12

1111

111111

1

11

1

111

11

111

111111

111

[ ( ) ] ,11 1111

1

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞⎜

⎝⎛

+θ+α−

β−θ

−−× bapxe x

e respectively. For θ=p the KwW-Lindley distribution will be obtained.


Plots of the pdf and hazard rate function for the new four special different distributions deduced from KwW-G family for some parameter values are displayed in Figures 1 and 2, respectively.

(a)

(b)

(c)

(d)

Figure 1. Density plots (a) KwWU, (b) KwWBurrXII, (c) KwWW (d) KwWQL.


(a)

(b)

(c)

(d)

Figure 2. Hazard plots (a) KwWU, (b) KwWBurrXII, (c) KwWW, (d) KwWQL.


5. Maximum Likelihood Estimation

The maximum likelihood estimators of the unknown parameters of the KwW-G family from complete samples are determined. Let nXX ...,,1 be

observed values from the KwW-G distribution with parameters βα,,, ba

and .ξ Let ( )Tba ξβα=Θ ,,,, be the 1×p parameter vector. The total

log-likelihood function for the vector of parameters Θ can be expressed as

( )ΘLln

( ) ( )[ ]∑=

ξ−β+β+α++=n

iixGnnbnan

1,ln1lnlnlnln

( ) ( )[ ] ( )[ ] ( )[ ]∑ ∑ ∑= = =

βξα−ξ+ξ−+β−n

i

n

i

n

iiii xHxgxG

1 1 1,,ln,1ln1

( ) [ ( )[ ] ] ( ) [ ( ( )[ ] ) ],11ln11ln11

,

1

, ∑∑=

ξα−

=

ξα− ββ−−−+−−+

n

i

axHn

i

xH ii ebea

where ( ) ( )( ) .,1,,

ξ−ξ

=ξi

ii xG

xGxH The elements of the score function ( ) =ΘU

( )ξβα UUUUU ba ,,,, are given by

[ ( )[ ] ] ( ) ( ( )[ ] ) ( ( )[ ] )

( ( )[ ] )∑ ∑= =

ξα−

ξα−ξα−ξα−

β

βββ

−−

−−−−−+=n

i

n

i axH

xHaxHxH

ai

iii

e

eebeanU

1 1,

,,, ,

11

1ln111ln

[ ( ( )[ ] ) ]∑=

ξα− β−−+=

n

i

axHb ieb

nU1

, ,11ln

( )[ ] ( ) ( )[ ] ( )[ ]

( )[ ]∑ ∑= =

ξα−

ξα−ββ

α β

β

−

ξ−+ξ−

α=

n

i

n

ixH

xHi

ii

i

e

exHaxHnU1 1

,

,

1

,1,

( ) ( )[ ] ( )[ ] [ ( )[ ] ]

[ ( )[ ] ]∑=

ξα−

−ξα−ξα−β

β

ββ

−−

−ξ−−

n

i axH

axHxHi

i

ii

e

eexHba1

,

1,,,

11

1,1


( )[ ] ( )[ ] ( )[ ]∑ ∑= =

ββ ξξα−ξ+

β=

n

i

n

iiii xHxHxHnU

1 1,ln,,ln

( )( )[ ]

( )[ ] ( )[ ]( )[ ]∑

=ξα−

β

β

βξα−

−

ξξ−α+

n

ixH

ii

i

ixH

e

xHxHea1

,1

,ln,1,

( ) ( )[ ] ( )[ ] ( )[ ] [ ( )[ ] ]

[ ( )[ ] ]∑=

ξα−

−ξα−ξα−β

β

ββ

−−

−ξξ−α−

n

i axH

axHxHii

i

ii

e

eexHxHba1 ,

1,,,

11

1,ln,1

and

kUξ ( ) ( )( ) ( ) ( )

( )∑ ∑= =

ξ−ξ∂ξ∂

+β+ξ

ξ∂ξ∂−β=

n

i

n

i iki

iki

xGxG

xGxG

1 1,1

,1,,1

( )( ) ( )[ ] ( )∑ ∑

= =

−β ξ∂ξ∂ξαβ−ξξ∂ξ∂

+n

i

n

ikii

iki xHxHxg

xg

1 1

1 ,,,,

( ) ( )[ ] ( )[ ] ( )( )[ ]∑

=ξα−

ξα−−β

β

β

−

ξ∂ξ∂ξ−αβ+

n

ixH

kixH

i

i

i

e

xHexHa1

,

,1

1

,,1

( )

( )[ ] ( )[ ]

[ ( )[ ] ] ( )

[ ( )[ ] ]∑=

ξα−

−ξα−

ξα−−β

β

β

β

−−

ξ∂ξ∂−×

ξ

−βα−n

i axHki

axH

xHi

i

i

i

e

xHe

exH

ba1

,

1,

,1

.11

,1

,

1

Setting βα UUUU ba ,,, and ξU equal to zero and solving the

equations simultaneously yields the maximum likelihood estimator =Θ

( )ξβα ,,,, ba of ( ) .,,,, Tba ξβα=Θ These equations cannot be solved

analytically and statistical software can be used to solve them numerically.

For interval estimation of the parameters, the 55 × observed information matrix ( ) { }ν=Θ ,uII for ( ),,,,,, ξβα= bavu whose elements are the


second derivatives of the log-likelihood ( )Lln function. Under the regularity

conditions, the known asymptotic properties of the maximum likelihood

method ensure that: ( ) ( ( ))Θ→Θ−Θ −1,0 INnd

as ,∞→n where d→

means the convergence in distribution, with mean zero and covariance matrix

( )Θ−1I then, the ( )%1100 ω− confidence interval for ( )ξβα=Θ ,,,, ba

is given as follows:

( ),var2 Θ±Θ ωZ

where 2ωZ is the upper ( )2100 ω percentile of the standard normal

distribution and ( )⋅var denote the diagonal elements of ( )Θ−1I

corresponding to the model parameters.

6. Applications

In this section, we explain the flexibility and superiority of new KwWE as special distribution from the Kumaraswamy Weibull-G family proposed here. We examine three real data sets to compare the fits of KwWE with exponentiated Weibull exponential (EWE) distribution and Weibull exponential (WE) distribution with the following corresponding densities:

( ) [ ] { ( ) }( ( ) ) 1111 11 −−α−λ−−α−−βλ βλβλ−−αβλ= aexex

KwWExx

eeeabxf

[ ( ( ) ) ] ,0,0,,,,,11 11 >>λβα−−× −−α− βλxbae bae x

( ) [ ] { ( ) }( ( ) ) ,11 1111 −−α−λ−−α−−βλ βλβλ−−αβλ= aexex

EWExx

eeeaxf

,0,0,,, >>λβα xa

and

( ) [ ] { ( ) } .0,0,,,1 11 >>λβα−αβλ= λ−−α−−βλ βλxeexf xex

WEx


In each real data set, the parameters are estimated by maximum likelihood method. Akaike information criterion (AIC), the correct Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan information criterion (HQIC), –2 ln, L, the Kolmogorov-Smirnov ( )SK - and

p-value statistics are considered to compare the three models. The formula for these criteria is as follows:

( ) ,112,ln22−−++=−= kn

kkAICCAICLkAIC

( ) ( )[ ] ,ln2lnln2andln2ln LnkHQICLnkBIC −=−=

where k is the number of parameters in the statistical model, n is the sample size and ln L is the maximized value of the log-likelihood function under the

considered model. Also, ( ) ( )[ ],sup yFyFsk ny −=− where ( ) nyFn1=

(number of observation ,)y≤ and ( )yF denotes the cdf.

In general, the best distribution corresponds to the smallest values of SKHQICCAICBICAICL -,,,,,ln2− and the biggest value of p-value

criteria.

6.1. The first real data set

The first data set was taken from Hinkley [9], which consists of thirty successive values of March precipitation (in inches) in Minneapolis/St Paul. The data set is as follows:

0.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43, 3.37, 2.20, 3.00, 3.09, 1.51, 2.10, 0.52, 1.62, 1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.90, 2.05.

Table 1 provides the maximum likelihood estimates (MLEs) of the model parameters. Also, the numerical values of statistics, AIC, BIC, CAIC, HQIC and Kolmogorov-Smirnov values as well as the p-value statistics are listed in Table 1. Figures 3 and 4 provide the plots of estimated cumulative and estimated densities of the fitted KwWE, EWE and WE models for the first data set.


Table 1. The S-KHQICBICCAICAICMLEs ,,,,, and p-value of the

models based on the first data set

Model

KwWE EWE WE

901.3ˆ =α 402.12ˆ =α -

001.32ˆ =b - -

583.8ˆ =α 016.39ˆ =α 708.168ˆ =α

578.0ˆ =β 581.0ˆ =β 572.1ˆ =β

Parameter Estimates

310328.4ˆ −×=λ 310066.8ˆ −×=λ 02.0ˆ =λ

–2 ln L 108.217 115.109 117.798

AIC 118.217 123.109 123.798

CAIC 120.717 124.709 124.721

BIC 115.603 121.017 122.229

HQIC 120.458 124.902 125.142

K-S 0.06 0.089 0.0796

p-value 0.9999 0.971 0.991

Based on the values of AIC, CAIC, BIC, HQIC, K-S and p-value in Table 1, the KwWE model provides better fit than the EWE and WE models.


Figure 3. Estimated cumulative densities for the first data.

Figure 4. Estimated densities of models for the first data.


It is clear from Figure 4 that the fitted density for KwWE model is closer to the empirical histogram than EWE and WE models.

6.2. The second real data set

The second data set was taken from Murthy et al. [10]. The corresponding data is referring to the time between failures for a repairable item. The data set is as follows:

1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86, 1.17.

The following Table 2 provides MLEs, the AIC, BIC, CAIC, HQIC and K-S values as well as the p-value statistics for the second data set.

Table 2. The MLEs, AIC, CAIC, BIC, HQIC, K-S and p-value of the models based on second data set

Model

KwWE EWE WE

797.2ˆ =α 355.6ˆ =α -

55.29ˆ =b - -

521.6ˆ =α 227.30ˆ =α 933.127ˆ =α

593.0ˆ =β 58.0ˆ =β 318.1ˆ =β

Parameter estimates

310326.4ˆ −×=λ 310347.9ˆ −×=λ 015.0ˆ =λ

–2 ln L 106.964 113.344 111.901

AIC 116.964 121.344 117.901

CAIC 119.464 122.944 118.824

BIC 114.35 119.252 116.332

HQIC 119.205 123.137 119.246

K-S 0.062 0.078 0.083

p-value 0.9998 0.993 0.987


In order to assess whether the KwWE model is appropriate, Figures 5 and 6 provide some plots of the estimated cdf as well as the estimated probability densities of the fitted KwWE, EWE and WE models for the second data set.

Figure 5. Estimated cumulative densities for the second data.

Figure 6. Estimated densities of models for the second data.


From Figure 6, it is concluded that the KwWE distribution is quite suitable for this data set.

6.3. The third real data set

The following data represent the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [2]. The data is as follows:

0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, .08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.

The following Table 3 gives the MLEs, AIC, BIC, CAIC, HQIC and K-S values as well as the p-value statistics for the third data set.

Table 3. The MLEs, AIC, CAIC, BIC, HQIC, K-S and p-value of the models based on third data set

Model

KwWE EWE WE

494.4ˆ =α 973.10ˆ =α -

374.22ˆ =b - -

739.9ˆ =α 229.32ˆ =α 738.163ˆ =α

556.0ˆ =β 601.0ˆ =β 584.1ˆ =β

Parameter Estimates

31037.4ˆ −×=λ 011.0ˆ =λ 02.0ˆ =λ

–2 ln L 265.787 283.242 292.659

AIC 275.787 291.242 298.659


CAIC 278.287 292.842 299.582

BIC 275.074 290.671 298.036

HQIC 280.319 294.867 301.378

K-S 0.105 0.111 0.1394

p-value 0.4015 0.338 0.122

The values in Table 3, indicate that the KwWE model is a strong competitor among other distributions that was used here to fit this data set. Figures 7 and 8 provide some plots of the estimated cdf and pdf of the fitted KwWE, EWE and WE models for the third data set.

Figure 7. Estimated cumulative densities for the third data.


Figure 8. Estimated densities of models for the third data.

Based on the plots of Figure 8 one can notice that the fitted density of the KwWE model is closer to the empirical histogram than the corresponding densities of the EWE and WE models.

7. Conclusion

In view of an idea of the family of Weibull-G distributions and Kumaraswamy distribution, we introduce a new family of distributions, called Kumaraswamy Weibull-G. The Kumaraswamy Weibull-G family extends several widely known distributions. An expansion for the density function and explicit expressions for the moments, probability weighted moments, generating function, mean deviation, quantile function and order statistics are obtained. The estimation of parameters is approached by the method of maximum likelihood. Some applications of the new family to three real data are given to demonstrate the importance and usefulness of the suggested family. In this study, we wish that this new generalization will


attract a broadly applications in several areas such as engineering, survival and lifetime data, hydrology, etc.

Acknowledgment

The authors are thankful to the editor and a reviewer whose comments led to improvement of the article.

References

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[9] D. Hinkley, On quick choice of power transformations, Journal of the Royal Statistical Society, Series (c), Applied Statistics 26 (1977), 67-69.

[10] D. N. P. Murthy, M. Xie and R. Jiang, Weibull Models, John Wiley, New Jersey, 2004.

[11] R. R. Pescim, G. M. Cordeiro, C. G. B. Demetrio, E. M. M. Ortega and S. Nadarajah, The new class of Kummer beta generalized distributions, Statistics and Operations Research Transactions 36 (2012), 153-180.


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