Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.5, 2013

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( ) ( )

,

<

<

(1)

is said to have weighted distribution, corresponding to the distribution of . The definition in the discrete case is analogous. One of the basic problems when one use weighted distributions as a tool in the selection of suitable models for observed data is the choice of the weight function that fits the data. Depending upon the choice of weight function( ), we have different weighted models. For example, when the weight function depends on the lengths of units of interest, i.e. ( ) = , the resulting distribution is called length-biased. In this case, the pdf of a length-biased (rv) is defined as

40

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( )=

( )

( )=

! ( )

,

<

<

(2)

Where = ( ) < ∞ .More generally, when the sampling mechanism selects units with probability proportional to some measure of the unit size, i.e., when ( ) = ; > 0, then the resulting distribution is called size-biased. This type of sampling is a generalization of length-biased sampling and majority of the literature is centered on this weight function. Denoting = ( ) < ∞ , distribution of the size-biased (rv) of order is specified by the pdf !

,

<

<

(3)

= 1, (3) reduces to the pdf of a length-biased (rv).

Clearly, when

We present the Even-Power Weighted distribution (EPWD), take two types of weighted functions, and \$ ( ) = % , we derive the pdf, cdf, and some other useful distributional properties. 2.

#(

)=

Even-Power Weighted Distribution

2.1.

Definition: The Even-Power Weighted Distribution (EPWD) is given by &'

where

( )=

&'

( ( ))

*

( )

, −∞ <

\$2

< ∞ , , = 1,2,3, …

1( ( )) 3 = 46∞( ( ))

0=

\$2

(4)

( )5

And ∈ 8, the weight function ( ) raised to the power of 2,, where , ∈ 9 : . Therefore we can use any other distribution such as the normal distribution or any other distributions. 2.2.1. Even-Power Weighted Normal Distribution #(

Consider the weight function

Now let , = 1 and

)=

( ; , ; \$) = 0=

1(

#

√\$= >

#(

, −∞<

< ∞ and Normal distribution given by

6

−∞<

%

(?@A)& &B&

\$2

)) 3 =

#

√\$=

,

< ∞ C5 ; \$ > 0

(?@A)&

\$ 6 &B& %

4 > 6∞ ∞

5 = ;\$ +

\$

And according to equation (1) the pdf of EPWND is as follows E

&

( ; , ; \$) =

And the cdf of EPWND is:

E

Where K( ) = Now let And Now let

\$(

#

√\$=

)=%

0=

&

4 L\$ % > 6∞

( ; , ; \$) = 6

(H@A)& &B&

, −∞ <

1(

\$(

=M ⟹

?

4@∞ F & G

#

√\$= >(> & : & )

@

(H@A)& &B& IF

√\$= >(> & : & )

=

(?@A)&

\$ 6 &B& %

# √\$= >(> & : & )

(5)

× K( )

5L

< ∞ with Normal pdf, ( ; , ; \$ )

\$2

)) 3 =

=M+

#

√\$=

4 % \$2 % > 6∞ ∞

6

(?@A)& &B&

, 5 = 5M

41

5

, , = 1,2,3, …

(6)

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.5, 2013

0=

& & G &'AO&' B

√\$= >

Then the pdf of (EPWND) is

46∞ % ∞

&

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E Q@&'B& R & B

6 P

&'

5M = % \$2

( ; , ; \$ , ,) =

And the cdf of (EPWND) is

&

Where 1) Let L − 2)

S6\$2> & >

&'

?

√\$= >G

?@&'B&

4@∞ B

=

√\$= >G

#

% &'AO&'& B&

(H@A)& &B& IF &'AO&'& B&

4@∞ G &'H G

( ; , ; \$ , ,) =

:\$2 & > &

@

G

T& @& IU

√\$=

= M ⟹ L = M + , 5L = 5M

=

\$2 6

@

(?@A)& &B&

(7)

Q&

?@A 4@∞ G &'Q G &B& IS & & √\$= >G &' B

#

= 11 + %, V \$

6 6\$2> & √\$>

W3

(8)

= X ⟹ M = ;X + 2,; \$ , 5M = ;5X

2.2.1.1. The shape The shapes of the density functions given in (5) and (7) can be clarified by studying those functions defined over the real line (-∞, ∞) and the behavior of its derivative as follows: 2.2.1.1. 1. Limit and Mode of the function Note that the limits of the Density functions given in (5) and (7) are as follow:lim Also

E

⟶]

lim

&

⟶]

&

( ; , ; \$ ) = ?⟶a

&'

lim

Therefore

⟶∞

lim

⟶∞

lim

And

⟶6∞

lim

⟶6∞

Since lim % ⟶±∞

6

(?@A)&

@ &B& ^_` & G

√\$= >(> & : & )

( ; , ; , ,) = \$

E

&

&

&

(?@A)& &B&

&'?@

(?@A)& &B&

(?@A)&

&

( ; , ; \$ , ,) =

( ; ,;

&'

\$)

=

?⟶∞

^_`

√\$= >G

(?@A)& &B&

&G @

(?@A)& &B&

√\$= >(> & : & )

( ; , ; \$ , ,) =

= 0 and

=

& & √\$= >G &'AO&' B

?⟶@∞

A& &B&

% &'AO&'& B&

6

A& &B&

#

\$(

(12)

=0

#

(13) lim %

\$2 6

(?@A)& &B&

2.2.1.1. 2. The mode (modes) of EPWND when )=

=0

) = , , = 1 have two maximum values, modes say ) = % has one mode say .

Then we must verify for that by finding the mode (modes) of them as: #(

, , = 1 is given by

c ^de

c

(10)

=0

lim % \$2 = 0 .From these limits, we conclude that #(

(9)

(11)

& & √\$= >G &'AO&' B ⟶6∞

⟶6∞

1) the pdf of EPWND when 2) the pdf of EPWND when

1)

=0

=0

√\$= >(> & : & ) &'?@

6

√\$= >(> & : & )

@ &B& ^_` & G

^_` G

×0×%

#

& & √\$= >G &'AO&' B

( ; , ; \$ ) = ?⟶∞

&'

E

^_` G

?⟶a

=

E

&( )

=0

42

(14)

#

and

\$,

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.5, 2013

log

Now

E

I ^de

&

E

I

( ) = log V

&( )

\$

= −

# W √\$=>(> & : & )

#

>&

\$(

)=%

I ^de

I

&

\$

log

6

= 2, −

&

&'

=0

>&

#

\$> &

\$> & 6 ( 6 ) >&

h & :i> &

= +

#

we get

&' ( )

+ 2 log −

( − )=0⟹

That is the two modes are as 2)

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\$

( ) = log V

=0

= −

\$

and

( − )\$

\$

h & :i> & \$

# & &W √\$= >G &'AO&' B

(15)

+ 2, −

( 6 )& \$> &

= 2,; \$ +

Then the mode of EPWND is

(16)

The Figure 1 (part -1- and -2-) and Figure 2 shows the shapes of pdf and cdf of EPWND where # ( ) = , , = 1 depending on different values of the parameters, and Figure 3(part -1-, -2- and -3-) shows the shapes of pdf of EPWND where \$ ( ) = % depending on different values of the parameters. 2.2.1.2. Reliability function The reliability functions of EPWND are given by 8

E

8

&

=1−

& & ( ; ,> )

E

=1−

& &' ( ; ,> )

&

&

( ) = 1−

(&')

#

# √\$=>(> & : & )

( ) = 11 − %, V \$

× K( )

6 6\$2> & √\$>

(17)

W3

(18)

2.2.1.3. Hazard function The hazard functions of EPWND are given by ℎ

E

&

(?@A)& & G @ &B&

=

& & ( ; ,> )

√\$=> (> & : & )6k( )

=

& &' ( ; ,> )

\$G &'? G

@

(19)

(?@A)& &B&

&

?@A@&'B & & Rm √\$=>G &'AO&' B l#6G2 P √&B

Figure4 shows the hazard functions of EPWND where 2.2.1.4. Moment Generating Function Theorem (1): If 1) no(

E

2) no( Proof:

) (L)

=

no(

− ) E&

=M⟹

(L) =

no(

W %F

>&: &

E

\$(

, if

p

H & B& &

:

(L) = (% F ) = &)

=M+

, 5 = 5M, so

&&'

) (L)

=

4@∞ G (HO&')? G ∞

2) t

E &

&

( )=

&'

, , = 1 and

\$(

)=% .

( )=

Proof: We know that

(u> & : & ) >&: &

, where

+ 2,; , where vn(2) (L)w o

Fx]

#(

\$(

(21)

= V1 + =

)=

(?@A)&

\$ 6 &B& %

F & ) 46∞ % ∞

√\$=>(> & :

& & √\$=>G &'AO&' B

\$

)= , ,=1

#

(?@A)& @ &B& I

Result (1): The mean of EPWND is given by: 1) t

)=

(22)

(Q& @&B& HQ) &B& IS

√\$=>(> & : & )

#(

, if

)=%

@ ∞ G HA 4@∞(S& :\$ S: & )G

no(

2) We have

> & (F & > & :\$F

B& (qO&r) qµO & p &r& B&

1) We have Let

#(

is distributed EPWND, then its moment generating function is:

(L) = V1 + &)

&&'

(20)

> & F(F> & :\$ >&: &

p

W %F

B& (HO&')& qµO & & & p&r B

, ,=1

)=%

= ( )2 , , = 1,2, …. 43

5

H & B& &

:

∎ (23) (24) (25)

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So that 1) n′o(

) E&

(L) = {V1 + n′oV

2) n′o(

) &&'

W E&

(L) =

F & > { :\$ F> & >&: &

(0) =

E

2) • , Proof: 1) M ′′ ‚(ƒ

„E&

&

) (t)

n′′o(

&

& & G &' B

%F

:

\$

= †V1 +

E&

>&: &

( ) = ; , if

&'

vV\$F>

) (0)

\$(

‡& σ{ :\$µ‡σ& σ& :µ&

{ :\$ > & >&: &

2) n′′o(

E

&

n′′o(

( )=

) &&' &&'

(L) =

) (0)

E

, if

)=%

(

\$)

}> & ~

& & G &' B

%

W ( + L; \$ )Š % F =

#(

−1

E

&

)= +

= ( + 2,; \$ )\$ + ; \$ = \$(

E

&

\$

:

(

, ,=1

(26) (27) &

> :

H & B& &

\$> { W >&: &

+

\$)

, , = 1, is given by

( )3 =

B& (HO&')& F : &

Then the variance of EPWND if

)=

{

>&: &

&

#(

\$F> :\$ > Wv ˆ(µ + tσ\$ )\$ v + vσ\$ ‰ + ( + L; \$ ) V & & W + V

Then the variance of EPWND, if • ,

H & B& &

:

B& (HO&')& &

u> { :€> & & : {

=

W}% F

+ 2,; \$

u> { :€> & & : {

( )=

>&: &

>&: &

Result (2): The variance of EPWND is given by: 1) • ,

\$F> { :\$ > &

(u> & : & )

} :> & (F:\$2)~

n′o (0) =

W ( + L; \$ ) + V

> & }u> { : & (‹> & : & 6‹)~ (> & : & )&

} :> & (F:\$2)~ & & G &' B &

&'

(

&

\$)

) = % , is • ,

%F

&

:

(&')

B& (HO&')& &

( )=

&

&'

(

\$)

\$

&

(&')

( ) = ;\$

Result (3): The coefficient of variation, skewness and kurtosis of EPWND are, respectively as follows: 1) When

#(

Œ•

E&

=

Œ’

E&

=

Œ—

E&

)=

Ž•2• ‘•

E

E

(o) &

=

> & }u> { : & (‹> & : & 6‹)~ (> & : & )(u> & : & )

(28)

& (> & : & ) (#“> { :‹> & & : ” )6u (> & : & )(u> & : & )(u> { :€> & &: { ):\$ • (u> & : & )•

= ˆ(v; \$ +

4(; \$ +

\$ u (15; €

)

)

\$ \$

• > • ˆu> { : & (‹> & : & 6‹)‰ –&

+ 21; ™

(3; \$ +

+ 30; “ + 15; \$

\$ )(15; ™ \$ \$ (3; ™

6(; + ) (3; + ) 3 ™ v(3; \$ + \$ )™ ‰⁄; ™ (3; ™ + \$

Proof

, ,=1

( ) &

\$

\$

\$

+ 9; \$

+ 6;

\$

\$ \$

(9; + \$

2) When

\$(

Œ•

E&'

=

Œ

E&'

=

)=%

>& ž

where Ÿ =

E&

, Œ’

+

\$

− 9))\$

+

and Œ—

E&

™)

+

u

+

€)

of EPWND where

(30) #(

)=

, , = 1.

+ 2,; \$

(31)

=0

(32)

ž• :u> & ž6už(ž& :> & ):\$ž• >&

E&

“)

+ 6; u

\$

Using (25), (26) we can prove this result.

Figure 5 shows the plots of Œ•

(29)

44

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.5, 2013

Œ—

=

E&'

ž{ :€> & ž& :u> { 6™ž(ž• :u> & ž):€ž& (ž& :> & )6už{

Œ’ Œ—

E&'

E&'

E&'

where and

>&

=

:\$2> & ‘•

= =

‘•

& &

&'

&

&

(

(&')

=

ž

(

&

&'

(o)‘•

PŽ•2• (o){ 6™‘•

u)

&

(&')

&

&

(o & ):\$¡‘• &'

&' (o)R

(o)‘•

= Ÿ u + 3; \$ Ÿ

™)

(33)

>&

(o • )6u‘• &'

(&')

=3

>{

Proof Using (25), then Œ•

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•– &

&

&

E

(o • ):€¡‘• ¡Ž•2•

&

(&')

&

(o)¢

E

(&')

=0 &

(o)¢ ‘•

&

(o)¢

&

&

(o & )6u¡‘•

&

(&')

{

(o)¢

(34)

= Ÿ ™ + 6; \$ Ÿ \$ + 3; ™

(35) ∎

Since Œ = 0 then the shape of the pdf of EPWND where \$ ( ) = % is symmetric and bell-shaped. From E&' equation (34) we get Œ— &' = 3, then the shape of the pdf of EPWND where \$ ( ) = % is peaked like the E normal distribution. Figure 6 shows the plots of Œ• 3.

Conclusions

E&'

of EPWND where

\$(

)=% .

We can develop the weighted distribution into Derivation Even-power weighted distribution (EPWD), like EvenPower Weighted distribution, Even-power Weighted Normal distribution. Therefore, we can discuss some of statistical properties on them. References Ahmed A. ,Reshi J. A. and Mir K. A. (2013). Structural Properties of size biased Gamma distribution, IOSR Jornal of Mathematics, Vol. 5, Issue 2, pp 55-61. Broderick X. S., Oluyede B. O. and Pararai M., (2012). Theoretical Properties of Weighted Generalized Rayleigh and Related Distributions, Theortical Mathematics & Applications, Vol. 2, no.2, 2012, 45-62. Castillo, J.D. and Casany, M. P., (1998). Weighted Poisson Distributions and Underdispersion Situations, Ann. Inst. Statist. Math. Vol. 50, No. 3. 567-585. Cox, D.R. (1962). Renewal Theory, Barnes & Noble, New York. Das K. K. and Roy T. D. (2011). On Some Length-Biased Weighted Weibull Distribution, Pelagia Research Library,Advances in Applied Science Research, , 2 (5):465-475. ISSN: 0976-8610, USA. Das, K. K. and Roy, T. D. (2011). Applicability of Length Biased Weighted Generalized Rayleigh distribution, Advances in Applied Science Research, , 2 (4): 320-327. El-Shaarawi, Abdel .and Walter W. Piegorsch, (2002). Weighted distribution, G. P. Patil, Vol. 4, pp 23692377. Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies, The Annals of Eugenics 6, 13–25. Ghitany, M. E. and Al-Mutairi, D. K. (2008). Size-biased Poisson-Lindley distribution and its application, METRON-International Journal of Statistics. Vol. LXVI, n. 3, pp. 299-311. Gupta, R. C., and Keating, J. P. (1985). Relations for Reliability Measures under Length Biased Sampling, Scan. J. Statist., 13, 49-56, Gupta, R. C., and Kirmani, S. N. U. A. (1990). The Role of Weighted Distributions in Stochastic Modeling, Commun. Statist., 19(9), 3147-3162. G.P. Patil and J.K. Ord, (1997). Weighted distributions, in Encyclopedia of Bio-statistics, P. Armitage and T. Colton, eds, 6, Wiley, Chichester, 4735-4738. JING, X. K. (2010). Weighted Inverse Weibull and Beta-Inverse Weibull distribution, 2009 Mathematics Subject Classification: 62N05, 62B10 45

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.5, 2013

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M. Zelen and M. Feinleib, On the Theory of Chronic Diseases, Biometrika, 56, (1969), 601-614. Oluyede, B. O. (1999). On Inequalities and Selection of Experiments for Length-Biased Distributions, Probability in the Engineering and Informational Sciences, 13, 169-185. Oluyede, B. O. and George, E. O. (2000). On Stochastic Inequalities and Comparisons of Reliability Measures for Weighted Distributions. Mathematical problems in Engineering, Vol. 8, pp. 1-13. Oluyede B. O. and Terbeche M.,(2007). On Energy and Expected Uncertainty Measures in Weighted Distributions, International Mathematical Forum, 2, no. 20, 947-956. Patil, G.P. &Rao, C.R. (1978). Weighted distributions and size-biased sampling with applications to wildlife populations and human families, Biometrics34, 179–184. Patil, G. P. and Ord,J. K.,(1976). On size-Biased Samling and Related Form-Invariant Weighted distribution, Sankhya: The Indian Journal Of Statistics, Vol. 38, Series B, Pt. 1, pp. 48-61. Priyadarshani, H. A. (2011). Statistical properties of weighted Generalized Gamma distribution, Mathematics Subject Classification: 62N05, 62B10. Rao, C.R. (1965). On discrete distributions arising out of methods of ascertainment, in Classical and Contagious Discrete Distributions, G.P. Patil, ed., Pergamon Press and Statistical Publishing Society, Calcutta, pp. 320–332. Rao, C.R. (1997).Statistics And Truth Putting Chance to Work, World Scientific Publishing Co. Pte. Ltd.Singapore. Shaban, S. A. and Boudrissa, N. A. (2000). The Weibull Length Biased distribution properties and Estimation, MSC Primary: 60E05, secondary: 62E15, 62F10 and 62F15 Zelen, M. (1974). Problems in cell kinetics and the early detection of disease, in Reliability and Biometry, F. Proschan& R.J. Serfling, eds, SIAM, Philadelphia, pp. 701–706. Dr. Kareema A. k,, assistant professor, Babylon University for Pure Mathematics Dept.. Her general specialization is statistics and specific specialization is Mathematics Statistics. Her interesting researches are : Mathematics Statistics, Time Series, and Sampling that are poblished in international and local Journals. 1) the Doctor degree in Mathematical Statistics: 30/12/2000 2) the Master degree in Statistics:1989/1990 3) the Bachelor's degree in Statistics: 1980/1981 4) the Bachelor's degree in English Language: 2007/2008

-1-

Figure 1 the pdf of EPWND where # ( ) = , , = 1. (-1-) the parameter ; = 2. (-2-) the parameter ; take the values (1,2,4) with = 0.

-2take the values (0,1,-1) with

From Figure 1(-1-), (-2-), we note that behaves as a shape parameter, while it is location parameter in the original (Normal ) distribution. Also we note if = 0 then there are two equal peaks in height, while if = 1 then the peak at the right of zero to is more high than other, and if = −1 then the peak at the left of zero is more high peak than other. From this we conclude that the more the value of the parameter increased peak height on the positive side of zero and decreased peak height on the negative side of zero. Conversely, the

46

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smaller the value of the parameter increased peak height in the negative direction of the zero and reduced peak height in the positive direction of the zero. In Figure 1(-2-) we can see that the parameter ; behaves as scale and location parameter, the impact parameter ; on the spread of shape the bigger ;, the more spread out the graph is. While in Figure 2 below, we see the impact parameter in simple bow in shape of the cdf , the less value of , increased curvature simple bow, therefore stay behaves as a shape parameter. Figure 2 the cdf of EPWND where ;.

#(

)=

, , = 1 and the parameter

take the values (2, 1,-1) with fixed

Figure 3 the pdf of EPWND where \$ ( ) = % , the parameters , ; and , take the values (0,4, −4), (1,2,3) and (1,2,5), with ; = 2, , = 2, = 2, , = 2 and = 2, ; = 2 respectively.

From Figure 3 we note that the parameter behaves as location parameter as in the Normal distribution. In Figure 2 -2- the parameter ; behaves as location and scale parameter, while in the original distribution it is behaves as scale parameter. Also in Figure 2 -3- the parameter , behaves as location parameter

Figure 4: The hazard function of EPWND where = −2, , = 2, ; = 2 respectively.

#(

)=

47

, , = 1,

= −2, ; = 2 and where

\$(

)=% ,

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Figure 4 -1- shows us that the curve of hazard function is changed from increasing to decreasing into the zero, and then it is strictly increasing, that because the effect of the variable \$ in the hazard function, while Figure 3 -2- shows us that the hazard function is strictly increasing.

Figure 5 The coefficient of (Variation, Skewness and Kurtosis) of EPWND when

#(

)=

, ,=1

Figure 5 -1-, shows us the relationship between Œ• & and at fixed ; such that the Œ• & behaves a E E constantly decreasing when ; increases and then Œ• & monotonically decreases when approaches to zero, at E which Œ• & increases to maximum value. After that Œ• & decreases and then constantly increases with E E increasing . Figure 5 -2-, shows us the relationship between Œ• & and ; at fixed such that the Œ• & E E monotonically decreases and then increases when ; increases. The relationship between Œ’ & and at fixed E . While in Figure 5 -4; is shown in Figure 5 -3- such that Œ’ & monotonically decreases with increasing E Œ’ & behaves as an exponential distribution with increasing ; at = −2, and behaves as natural logarithm E function at = 2.

Figure 6 The coefficient of variation of EPWND where \$ ( ) = % with fixed (;, ,), ( , ,), ( , ,) and ( , ;) in parts -1-, -2-, -3-, -4- respectively. Figures 6, the relationship between , ;, , and Œ• &' in which Œ• &' monotonically decreasing at increasing E E ( from -5 to 5 in part -1-), ; (from 1 to 5 in part -2-), , (from 0 to 5 in part -4-) respectively. While in part -3- , the relationship between ; and Œ• &' shows that Œ• &' monotonically increasing at increasing ; (from 1 to E E 5 in ).

48

Even power weighted distribution
Even power weighted distribution