distribution and its properties

RESEARCH ARTICLE

J.Natn.Sci.Foundation Sri Lanka 2018 46 (4): 505-518 DOI: http://dx.doi.org/10.4038/jnsfsr.v46i4.8626

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distribution and its properties

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Revised: 19 March 2018; Accepted: 27 April 2018

* Corresponding author ([email protected]; https://orcid.org/0000-0003-1713-2862)

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/).

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properly cited and is not changed in anyway.

$EVWUDFW This article suggests a new statistical distribution QDPHG ދ&XELF UDQN WUDQVPXWHG .XPDUDVZDP\ GLVWULEXWLRQ¶

using cubic rank transmutation map. Various statistical SURSHUWLHVRIWKLVQHZGLVWULEXWLRQVXFKDVKD]DUGIXQFWLRQDQG

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estimators of unknown parameters of this distribution are derived. A Monte Carlo simulation study based on bias and mean square error criteria of this estimator is conducted.

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INTRODUCTION

Various statistical distributions have been widely used for modelling lifetime data in many areas such DV PHGLFLQH HQJLQHHULQJ ¿QDQFH DQG K\GURORJ\ ,Q

OLWHUDWXUHPDQ\OLIHWLPHGLVWULEXWLRQVXVHGIRUVWDWLVWLFDO

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EDWKWXEHIDLOXUHUDWHVH[LVW5HFHQWO\PDQ\QHZOLIHWLPH

distributions which will be used for datasets having these kind of failure rates have been obtained by using methods developed for generating new distributions in lifetime analysis. Some of the studies on this subject were conducted by Elgarhy et al.*UDQ]RWWRet al.

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$O%DEWDLQ et al.  8VPDQ et al. 

$¿I\et al$NGR÷DQet al1RIDOet al.

.KDQet al.XO+DTet al.&RUGHLUR

et al.E.XPDUDVZDP\.Xú etc.

The Kumaraswamy distribution proposed by Poondi Kumaraswamy (1980) has been used to model hydrological phenomena. In the area of reliability DQDO\VLV*DQMLet al. (2006) have studied this distribution.

,Q OLWHUDWXUH PDQ\ OLIHWLPH GLVWULEXWLRQV ZKLFK

are obtained by means of this distribution exist. For H[DPSOH &RUGHLUR DQG &DVWUR  KDYH LQWURGXFHG

family of the Kumaraswamy generalised (Kw-G) distributions. Cordeiro et al. (2010; 2012b) have introduced Kumaraswamy-Gumbel and Kumaraswamy- Weibull distributions. Kumaraswamy distribution combined with other distributions has been also used in DUHDVVXFKDVUHOLDELOLW\DQDO\VLVPHFKDQLFDODQDO\VLVDV

well as hydrological phenomena.

Kumaraswamy distribution with a and b parameters is shown by Kw a b  . The cumulative distribution function (cdf) and probability density function (pdf) of this distribution are as follows;

     

^{a b}



G x a b   x

^...(1)

1 1

   

^a



^{a b}



g x a b abx

^

 x

^{ ...(2)}

where a

! 0

^{and b}!⁰ are the shape parameters and

>@

x ^.

December 2018 Journal of the National Science Foundation of Sri Lanka 46(4)

Shaw and Buckley (2009) have proposed a quadratic rank transmutation map (QRTM) to obtain new statistical distributions. This method has been used by many researchers to obtain new distributions called transmuted GLVWULEXWLRQV )RU H[DPSOH FGI DQG SGI RI WUDQVPXWHG

Kumaraswamy (TKw) distribution suggested by Khan et al. (2016) are as follows;

   

      

^{a b}

 

^{a b}

F x a b O  ^ª « ¬ O  x ^{º ª} » « ¼ ¬  O  x ^º » ¼

...(3)

 

1 1

    

^a



^{a b}

   

^{a b} f x a b

O

abx ^



x ^

^ª « ¬   O O 

x

^º » ¼

...(4)

 $¿I\et al. (2016) have studied the Kumaraswamy transmuted-G family of distributions. Nofal et al.

(2016) have introduced a new lifetime model called Kumaraswamy transmuted exponentiated additive Weibull distribution. Al-Babtain et al. (2017) have suggested the Kumaraswamy-transmuted exponentiated PRGL¿HG:HLEXOOGLVWULEXWLRQ

 $ GDWDVHW FDQ ¿W WR VHYHUDO VWDWLVWLFDO GLVWULEXWLRQV

7KHUHIRUHLWLVLPSRUWDQWWRREWDLQÀH[LEOHGLVWULEXWLRQV

that will be used for various datasets and to determine WKHVWDWLVWLFDOGLVWULEXWLRQWKDWEHVW¿WVWRDGDWDVHW

The main purpose of this paper is to introduce a new distribution named cubic rank transmuted .XPDUDVZDP\ &57.Z GLVWULEXWLRQ ZKLFK ZLOO EH

used to model skewed datasets and datasets having increasing and bathtube failure rates by using the cubic UDQNWUDQVPXWDWLRQPDS&570SURSRVHGE\*UDQ]RWWR

et al. (2017). The new cdf and pdf obtained with CRTM are given as follows;

2 3

1 2 1 2

( ) ( ) ( ) ( ) (1 ) ( )

F x

O

G x

 O O 

G x

  O

G x

...(5)

₂

1 2 1 2

( ) ( )[ 2( ) ( ) 3(1 ) ( )]

f x g x

O  O O 

G x

  O

G x ...(6) where G x( ) is the cdf of any distribution and ( )

g x  LV WKH SGI RI WKH VDPH GLVWULEXWLRQ F x( ) denotes the cdf of cubic rank transmuted G (CRT-G) GLVWULEXWLRQ f x( ) is the pdf of CRT-G distribution and O1>@O2 > @ *UDQ]RWWR et al. (2017) have introduced cubic rank transmuted Weibull and cubic rank transmuted log-logistic distributions.

METHODOLOGY

&XELF UDQN WUDQVPXWHG .XPDUDVZDP\ &57.Z GLVWULEXWLRQ

Let X be a random variable having a CRTKw distribution with a b  

O O

_{1 2} parameters. The cdf and pdf RI WKLV GLVWULEXWLRQ DUH JLYHQ LQ HTXDWLRQV  DQG 

respectively.

2 3

1 2 1

( ) (1 (1 ^{a b}) ) ( )(1 (1 ^{a b}) ) (1 )(1 (1 ) ) F x O  x  O O  x  O  

2 3

( ) O(1 (1  ) ) ( O O )(1 (1  ) )  (1 O2)(1 (1 x^{a b}) ) _...(7)

1 1

1 2 1

( ) ^a (1 ^{a b}) [ 2( )(1 (1 ^{a b}) ) 3(1 )(1 (1 ) ) ] f x abx ^ x ^

O



O O

  x  

O

 

2

( ) ^(1 ) [^

O

2(

O O

 )(1 (1  ^{a b}) ) 3(1 

O

2)(1 (1 x^{a b}) ) ] _...(8) where a

! 0

^b!^0

O

1>@

O

2 > @ ^and x>@^{. If} it is taken as O 1 1 ^and O 2 1WKH.XPDUDVZDP\.Z distribution is obtained. The plots of pdf and cdf for various parameter values of CRTKw distribution are JLYHQLQ)LJXUHVDQGUHVSHFWLYHO\

7KH UHOLDELOLW\ IXQFWLRQ DQG KD]DUG IXQFWLRQ KI of the CRTKw

    

a b

O O

_{1 2} distribution are given in HTXDWLRQVDQGUHVSHFWLYHO\

2 3

1 2 1

( ) 1 ( )

1 ( (1 (1 ^{a b}) ) ( )(1 (1 ^{a b}) ) (1 )(1 (1 ) ) )

R x F x

x x x

O O O O



          

2 3

1 ( (1 (1O x ) ) (O O)(1 (1 x^{a b}) ) (1 O2)(1 (1 x^{a b}) ) ) 

           ...(9)

1 1

( ) ( )

1 ( )

(1 ) [ 2( )(1 (1 ) ) 3(1 )(1 (1 ) ) ]

a a b

h x f x

F x

abx x O O O O

O O O O

 



        

          

1 1 2

1 2 1 2

2 3

1 2 1 2

1 ( )

(1 ) [ 2( )(1 (1 ) ) 3(1 )(1 (1 ) ) ] 1 ( (1 (1 ) ) ( )(1 (1 ) ) (1 )(1 (1 ) ) )

a a b a b a b

a b a b a b

F x

abx x x x

x x x

O O O O

O O O O

 



        

          

...(10)

$FFRUGLQJ WR )LJXUH  WKH KD]DUG IXQFWLRQ RI

CRTKw     a b

O O

_{1 2} distribution is increasing for

1 2

  >@ DQG > @

a! b!

O



O

  . On the other

KDQG WKH KD]DUG IXQFWLRQ KDV D EDWKWXE VKDSH IRU

 a   b 

O

1>@ ^and

O

2

  > @

^. 0RPHQWVRI&57.ZGLVWULEXWLRQ

Let X be a random variable having a CRTKw distribution with a b  

O O

_{1 2} parameters. r^th moment of this random variable is obtained as follows;

CRTKw distribution and its properties 507

Journal of the National Science Foundation of Sri Lanka 46(4) December 2018

0 0.2 0.4 0.6 0.8 1 1.2

0 1 2 3 4 5 6

x

f(x)

b=1.3,λ(1)=0.2,λ(2)=0.5

a=1 a=2 a=3 a=4

 

1.3, 0.2, 0.5, 1, 2, 3, 4

b a  

 

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9  



1, 0.5, 0.5, 0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

 

1 2

1.3, 0.2, 0.5, 1, 2, 3, 4

b a  

0 0.2 0.4 0.6 0.8 1 1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

x

f(x)

a=2,b=1,λ(2)=0.7

λ(1)=0.3 λ(1)=0.5 λ(1)=0.7 λ(1)=0.9

 

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9

a b  



1, 0.5, 0.5, 0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

 

 

 

2 1

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9

a= b= λ = λ =  

Figure 1. The plots of pdf for different parameter values of

 1, 0.5, 0.5, 0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9



^{00 0.2 0.4 0.6 0.8 1 1.2}

0.5 1 1.5 2 2.5 3 3.5 4 4.5

x

f(x)

a=5,λ(1)=0.7,λ(2)=0.3

b=2 b=3 b=5 b=8



1.3, 0.2, 0.5, 1, 2, 3, 4  a 5, 0.7, 0.3, b 2, 3, 5,8 

 

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9  1, 4, 0.5, 0.5, 0.3, 0.5, 0.8 



1, 0.5, 0.5, 0.6,1,1.3, 2

 1.2, 0.6, 0.6, 1,1.2,1.5,1.7 

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

1.3, 0.2, 0.5, 1, 2, 3, 4  a 5,

₁

0.7,

₂

0.3, b 2, 3, 5,8 



^{00 0.2 0.4 0.6 0.8 1 1.2}

1 2 3 4 5 6 7 8 9

x

f(x)

a=1,b=4,λ(1)=0.5

λ(2)=-0.5 λ(2)=0.3 λ(2)=0.5 λ(2)=0.8

 2, 1, 0.7, 0.3, 0.5, 0.7, 0.9  a 1, b 4, 0.5, 0.5, 0.3, 0.5, 0.8 



1, 0.5, 0.5, 0.6,1,1.3, 2

 1.2, 0.6, 0.6, 1,1.2,1.5,1.7 

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

 

 

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9 a=1,b=4,λ1=0.5,λ2= −0.5, 0.3, 0.5, 0.8 The plots of pdf for different parameter values of CTKw distribution



1, 0.5, 0.5, 0.6,1,1.3, 2

 1.2, 0.6, 0.6, 1,1.2,1.5,1.7

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

1.3, 0.2, 0.5, 1, 2, 3, 4 a=5,λ1=0.7,λ2=0.3,b=2, 3, 5,8



a=1,b=4, (1)=0.5

 2, 1, 0.7, 0.3, 0.5, 0.7, 0.9 1, 4, 0.5, 0.5, 0.3, 0.5, 0.8



1, 0.5, 0.5, 0.6,1,1.3, 2

 1.2, 0.6, 0.6, 1,1.2,1.5,1.7

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

1 2

1.3, 0.2, 0.5, 1, 2, 3, 4

b= λ = λ = a=  

a=2,b=1, (2)=0.7

 

2, 1, 0.7, 0.3, 0.5, 0.7, 0.9 

 1, 0.5, 0.5, 0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

)LJXUH The plots of pdf for different parameter values of CRTKw distribution

)LJXUH The plots of cdf for various parameter values for CRTKw distribution

 

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

F(x)

b=1,λ(1)=0.5,λ(2)=-0.5

a=0.6 a=1 a=1.3 a=2



1, 0.5, 0.5, 0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

 

 

 

 

 



0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

F(x)

a=0.5,b=0.5,λ(2)=0.3

λ(1)=0.2 λ(1)=0.3 λ(1)=0.5 λ(1)=0.9

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

a b

 

 

 

 



b=1,λ₁=0.5,λ₂= −0.5,a=0.6,1,1.3, 2





0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

 

 

 

 



1, 0.5, 0.5, 0.6,1,1.3, 2

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

F(x)

a=1.2,λ(1)=0.6,λ(2)=0.6

b=1 b=1.2 b=1.5 b=1.7

 1.2, 0.6, 0.6, 1,1.2,1.5,1.7 

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

 

 

 



1, 0.5, 0.5, 0.6,1,1.3, 2



1 2

1.2, 0.6, 0.6, 1,1.2,1.5,1.7

a= λ = λ = b= 

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9

1 2

( , , , )

 

 

 

 

 



0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

F(x)

a=1,b=1.5,λ(1)=0.6

λ(2)=-0.6 λ(2)=0.1 λ(2)=0.6 λ(2)=0.9

1, 1.5, 0.6, 0.6, 0.1, 0.6, 0.9 a b

1 2

( , , , )

 

 

 

 

 



2 2

0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 a= b= λ = λ =

a b

 

 

 

 

 



0.5, 0.5, 0.3, 0.2, 0.3, 0.5, 0.9 a=1,b=1.5,λ₁=0.6,λ₂= −0.6, 0.1, 0.6, 0.9

1 2

( , , , )

December 2018 Journal of the National Science Foundation of Sri Lanka 46(4)

0 0.2 0.4 0.6 0.8 1 1.2

0 5 10 15 20 25 30 35 40 45 50

x

h(x)

b=5,λ(1)=0.3,λ(2)=0.5

a=2 a=5 a=7 a=10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 1 2 3 4 5 6

x

h(x)

a=5,b=2,λ(2)=0.3

λ(1)=0.1 λ(1)=0.3 λ(1)=0.5 λ(1)=0.7

Figure 3. The plots of hf for different parameter values for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

h(x)

a=3,b=5,λ(1)=0.6,λ(2)=0.3 a=0.3,b=0.7,λ(1)=0.4,λ(2)=0.9 a=1.6,b=1.2,λ(1)=0.5,λ(2)=0.2 a=1.5,b=1,λ(1)=1,λ(2)=-1 a=1,b=1.5,λ(1)=1,λ(2)=-1 a=1,b=1.5,λ(1)=0,λ(2)=1

0 0.2 0.4 0.6 0.8 1 1.2

0 2 4 6 8 10 12

x

h(x)

a=0.2,λ(1)=0.8,λ(2)=-0.8

b=0.1 b=0.2 b=0.5 b=1

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.5 1 1.5

x

h(x)

a=1.3,b=1.5,λ(1)=0.6

λ(2)=-0.3 λ(2)=0.3 λ(2)=0.6 λ(2)=0.9

)LJXUH The plots of hf for different parameter values for CRTKw distribution

1

2

1 2 1 2

0

( ) ( )[ 2( ) ( ) 3(1 ) ( )]

         

     

r r

E X x g x G x G x dx

r r

b Beta b b Beta b

O O O O

O O O O

O

   

§ ·

  ¨©  ¸¹   

  

³

0

1 2 2 1

         

     

r r

b Beta b b Beta b

a a

b Beta r b

O O O O

O O O O

O

   

§ ·

  ¨©  ¸¹   

  

³

2 1

         

     

r r

Beta b

a a

O O O O

O O O O

O

   

§ ·

  ¨©  ¸¹   

  

³

2

         

    r 

b Beta b

a

O O O O

O O O O

O

   

§ ·

  ¨©  ¸¹   

  

³

...(11) where ^Beta

 

^ LVEHWDIXQFWLRQ7KLVIXQFWLRQLVGH¿QHG

as follows for D E ƒ^.

       

 

1 1 1

 0 

Beta D E x^D x ^E dx * DD E* E

³

* 

...(12)

where ^*

 

^. LVJDPPDIXQFWLRQDQGLWLVGH¿QHGDV

 

0 ¹

x^D e dxx

D

^{f  }

*

³

...(13) *UDGVWHLQ 5\]KLN

CRTKw distribution and its properties 509

Journal of the National Science Foundation of Sri Lanka 46(4) December 2018

Parameter values _{  } a bO O_{1 2} E X( ) E X( ²) Var X( ) ^{CK CS}

 0.8544 0.8002 0.0702 6.1734 -2.0869

 0.3143 0.1427 0.0439 2.4260 0.4757

 0.6408 0.4313 0.0206 5.1081 -1.2126

 0.9525 0.9075 0.0002 7.3080 -1.6164

7DEOH 0HDQ YDULDQFH FRHI¿FLHQWV RI VNHZQHVV DQG NXUWRVLV IRU YDULRXV SDUDPHWHU YDOXHV RI &57.Z

distribution

§ ·

¨ ¸

© ¹

0 10 20 30 40 50 60 70 80 90 100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b=1λ(1)=0.6,λ(2)=0.5

a

E(X)

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

a b

_{ }

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

^{ }

§ ·

¨ ¸

© ¹

 

[

0,100 ,

]

1, 1 0.6, 2 0.5

a∈ b= λ= λ =  

Figure 4. Expected values for different parameter values of

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5} 

§ ·

¨ ¸

© ¹

 ^{0 10 20 30 40 50 60 70 80 90 100}

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a=3,λ(1)=0.5,λ(2)=-0.5

b

E(X)

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}



^{a 3, b} [ ^0,100 ] ^{, 0.5, 0.5}

 

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

^

^3, [ ] ^{, 0.5, 0.5}

^

§ ·

¨ ¸

© ¹

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5} a=3,b∈

[

0,100

]

,λ1=0.5,λ2= −0.5 Expected values for different parameter values of CTKw distribution

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5 3,}

[ ]

^{, 0.5, 0.5}

)LJXUH Expected values for different parameter values of CRTKw distribution

§ ·

¨ ¸

© ¹

 

 

Figure 4. Expected values for different parameter values of

^{0 10 20 30 40 50 60 70 80 90 100}

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

b=1,λ(1)=0.6,λ(2)=0.5

a

Var(X)

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

a b _{ }

§ ·

¨ ¸

© ¹

 

 

 

[

0,100 ,

]

1, 1 0.6, 2 0.5

a∈ b= λ = λ =  

Figure 5. Variance values for different parameter values of

§ ·

¨ ¸

© ¹

 

 

Expected values for different parameter values of CTKw distribution

_

^{0 10 20 30 40 50 60 70 80 90 100}

0 0.005 0.01 0.015 0.02 0.025 0.03

a=3,λ(1)=0.5,λ(2)=-0.5

b

Var(X)

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}  ^{a 3, b} [ ^0,100 ] ^{, 0.5, 0.5}  

§ ·

¨ ¸

© ¹

 

 

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5} a=3,b∈

[

0,100

]

,λ1=0.5,λ2 = −0.5 Variance values for different parameter values of CTKw distribution

)LJXUH Variance values for different parameter values of CRTKw distribution

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&RHI¿FLHQWV RI VNHZQHVV DQG NXUWRVLV IRU &57.Z

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respectively.

 

3

3 2 1

3 2 2

2 1

3 2

CS *  *  *

*  * ...(14)

December 2018 Journal of the National Science Foundation of Sri Lanka 46(4) )LJXUH &RHI¿FLHQWRINXUWRVLVYDOXHVIRUGLIIHUHQWSDUDPHWHUYDOXHVRI&57.ZGLVWULEXWLRQ

0 10 20 30 40 50 60 70 80 90 100

0 2 4 6 8 10 12

14 b=1λ(1)=0.6,λ(2)=0.5

a

CK

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

a b

_{ }

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

^{ }

a → ∞ ^{E X} ( ) ^b

variance

X ( , ,

₁

,

₂

)

( ) M t

(3 ) b (1 , ) 2(2 b 3) (1 , 2 )

³ ¦ ³

¦ ¦

¦

 

[

0,100 ,

]

1, 1 0.6, 2 0.5

a∈ b= λ = λ =  

Figure 6. Coefficient of Kurtosis values for different param

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5 }

a→ ∞ ^{E X}

( )

^b

variance

X ( , ,

λ λ

₁, ₂)

M t( )

(3 ) b (1 , ) 2(2b 3) (1 , 2 )

³ ¦ ³

¦ ¦

¦

 ^{-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1}

-5 -4 -3 -2 -1 0 1 2 3 4 5

a=1,b=0.9,λ(1)=0.1

λ(2)

CK

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}



^{a 1, b 0.9, 0.1, ∈ −} [ ^1,1 ]

 

 

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}



[ ] ¹



a → ∞ ^{E X} ( ) ^{b → ∞} ( ) ^{0 a}

variance a

X ( , ,

₁

,

₂

)

( ) M t

(3 ) (1 , ) 2(2 3) b (1 , 2 ) b

³ ¦ ³

¦ ¦

¦

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5} a=1,b=0.9,λ1=0.1,λ2∈ −

[

1,1

]

 Coefficient of Kurtosis values for different parameters of CTKw distribution

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5}

[ ]

^1

a→ ∞ ^{E X}

( )

^{b → ∞}

( )

^{0 a}

variance a

X ( , , ₁, ₂)

( ) M t

(3 ) (1 , ) 2(2 3) b (1 , 2 )b

³ ¦ ³

¦ ¦

¦

 

 

0 10 20 30 40 50 60 70 80 90 100

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

2 b=1λ(1)=0.6,λ(2)=0.5

a

CS

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5}

a b  

a → ∞ ^{E X} ( ) ^b

variance

X ( , ,

₁

,

₂

)

( ) M t

(3 ) b (1 , ) 2(2 b 3) (1 , 2 )

³ ¦ ³

¦ ¦

¦

 

 

 

[

0,100 ,

]

1, 1 0.6, 2 0.5

a∈ b= λ = λ =  

Figure 7. Coefficient of Skewness values for different param

a→ ∞ ^{E X}

( )

^b

variance

X ( , ,λ λ₁, ₂)

( ) M t

(3 ) b (1 , ) 2(2b 3) (1 , 2 )

³ ¦ ³

¦ ¦

¦

 

 

 ^{0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1}

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

a=1,b=2,λ(2)=-1

λ(1)

CS

[ ^{0,100 ,} ] ^{1, 0.6, 0.5}

^

^{a b} [ ] ^{0,1 , 1}

^ 

a → ∞ ^{E X} ( ) ^{b → ∞} ( ) ^{0 a}

variance a

X ( , ,

₁

,

₂

)

( ) M t

(3 ) (1 , ) 2(2 3) b (1 , 2 ) b

³ ¦ ³

¦ ¦

¦

 

 

 

[

^{0,100 ,}

]

^{1, 0.6, 0.5} a=1,b=2,λ1∈

[ ]

0,1 ,λ2= −1 Coefficient of Skewness values for different parameters of CTKw distribution

a→ ∞ E X

( )

b → ∞

( )

⁰ a

variance a

X ( , , ₁, ₂)

( ) M t

(3 ) (1 , ) 2(2 3) b (1 , 2 )b

³ ¦ ³

¦ ¦

¦

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2 4

4 1 3 1 2 1

2 2

2 1

4 6 3

=

CK *  * *  * *  * ª*  * º

¬ ¼ ...(15)

where

1 2

         

        

j

j j

b Beta b b

a a

b Beta j b j

O O O O

O

§ ·

*   ¨©  ¸¹   

  

2 1

         

        

j j

b Beta b b Beta b

a a

b Beta b j

O O O O

O

§ ·

*   ¨©  ¸¹   

  

2

         

     j   

b Beta b j

a

O O O O

O

§ ·

*   ¨©  ¸¹   

  

7KH ¿UVW PRPHQW WKH VHFRQG PRPHQW YDULDQFH

FRHI¿FLHQWV RI NXUWRVLV DQG VNHZQHVV IRU YDULRXV

parameter values of CRTKw distribution with     a bO O1 2

parameters are given in Table 1.

It is seen that ao f ^{then E X}

 

^{ob and bo f then}

 

⁰

E X o $OVRDVaLQFUHDVHVvarianceDSSURDFKHV]HUR

It can be seen that increasing the value of

a

parameter OHDGVWRGHFUHDVLQJWKHYDOXHRIFRHI¿FLHQWRIVNHZQHVV

in Figure 7.

0RPHQWJHQHUDWLQJIXQFWLRQRI&57.ZGLVWULEXWLRQ Let

X

has a CRTKw     a b

O O

_{1 2} distribution. Moment JHQHUDWLQJ IXQFWLRQ RI WKLV GLVWULEXWLRQ M t_x( ) LV

obtained as follows;