Characterizations of the Exponentiated Pareto Distribution Based on Record Values
A. I. Shawky* and Hanaa H. Abu-Zinadah**
Girls College of Education, P. O. Box 32691, Jeddah 21438, Saudi Arabia.
Abstract
In this paper, the exact form of the probability density function (pdf) and moments of single, double, triple and quadruple of lower record values from Exponentiated Pareto distribution (EPD) are derived. We establish several recurrence relations between single, double, triple and quadruple moments of lower record values from EPD.
Key words:
Lower record values; Exponentiated Pareto distribution; Single, Double, Triple and Quadruple Moments; Recurrence relations.1. Introduction
Let us consider the Exponentiated Pareto distribution EP(θ ,λ) with probability density function (pdf)
, 0 , 0 , 0 ),
1 ) (
1 1 ( ] ) 1 ( 1 [ )
(x =θλ − +x −λ θ− +x − λ+ x> λ> θ >
f (1.1)
and cumulative distribution function (cdf)
, 0 ,
0 ,
0 ,
] ) 1 ( 1 [ )
( x = − + x − λ θ x > λ > θ >
F
(1.2)where θ and λ are two shape parameters. When θ = 1, the above distribution corresponds to the standard Pareto distribution of the second kind.
Let X1, X2, … be a sequence of i.i.d. random variables with cdf F(x) and pdf f(x). Set
Yn=max (min) {X1, …, Xn}, n≥1. We say that
X
jis an upper (lower) record value of thissequence if
( ) , 1
1
>
<
> Y
−j
Y
j j . By definition X1 is an upper as well as a lower record value. One can transform from upper records to lower records by replacing the original sequence of {X
j} by {− X , j ≥ 1
j } or (if P(
X
i> 0) = 1 for all i) by {1 / X , i ≥ 1
i }; the lower record values of this sequence will correspond to the upper record values of the original sequence. We will confine our attention to just lower record values. Many authors have studied characterization of record values from different kind of distributions; for example, Kamps (1992), Balakrishnan and Chan (1993), Balakrishnan and Ahsanullah (1994), Pawlas and Szynal (1999), Raqab and Awad (2000), Raqab (2002) and AlZaid and Ahsanullah (2003).
Permanent address: Fac. of Eng. at Shoubra, P.O. Box 1206, El Maadi 11728, Cairo, Egypt *
(E-mail address: aishawky @yahoo.com).
**E-mail address: [email protected].
In this paper, we derive the exact form of the pdf and moments of single, double, triple and quadruple lower record values from Exponentiated Pareto distribution (EPD) in Section 2. Section 3 gives several recurrence relations between moments of single, double, triple and quadruple lower record values from EPD.
2. Moments of lower record values
Let
) ( )
2 ( ) 1
(
, , ... ,
m L L
L
X X
X
be the first m lower record values from the EP distribution (1.1), then the single, double, triple and quadruple moments are derived as follows:The pdf of XL(m), m = 1, 2, …, can be written [see, Arnold et al. (1998)] as
.
0 ),
1 ( )]
( ln ) [ ( ) 1
( − − >
= Γ F x m f x x
x m
f m
(2.1)From (1.1), (1.2) and (2.1), the single moments of the lower record values
) (m
X
L are∫
∞ − −
= Γ
=
0
) 1 ( )]
( ln ) [
( ) 1 ) ( (
)
( x a F x m f x dx
m a
m X L a E
µ m
.By setting,
t = [ F ( x )] 1 / θ
, we find∑ ∫
∑ ∞ + − − −
=
=
− −
= Γ
1
0
1 , ] ln 1 [
0 !
) ) ( / ( 0
) 1 ) (
( )
( t k t m dt
k k
i k a
i
i a i
a m
a m
m λ θ
µ θ
where
=
>
− +
= +
. 0
; 1
0
; ) 1 (
...
) 1 ( )
( i
i i
r r
r r i
By putting, w = - ln t, we get
∑
∑ ∞
=
=
>
= +
− −
=
0
...
, 2 , 1 , 0 ,
) (
!
) ) ( / ( 0
) 1 ) (
(
k
a and m a
k k
i k a
i
i a i
m a a
m λ
θ λ θ
µ
. (2.2)The joint pdf of two lower record values XL(m), XL(n), m, n = 1, 2, …, m < n, can be written [see Arnold et al. (1998)] as
), ) ( (
) 1 ( )]
( ln ) ( ln 1 [ )]
( ln ) [ (
) ( ) 1 ,
, ( f y
x F
x m f
x n F y
m F x m F
n y m
n x
f m − − − + − −
− Γ
= Γ
0 < y < x < ∞ .
(2.3) From (1.1), (1.2) and (2.3), the double moments of the lower record values) (m
X
L and) (n
X
L , m < n, are given bydx dy y m f
x n F y
F
x F
x m f
x b F
a y x x m n m b
n X L a
m X L b E
a n m
) 1 ( )]
( ln ) ( ln [
0 ( )
) 1 ( )]
( ln [ ) 0
( ) ( ) 1 ) ( ) ( (
) , (
,
− + −
−
×
∞ − −
− Γ
= Γ
= ∫ ∫
µ
, ) (
0 ( )
) 1 ( )]
( ln ) [
( ) (
1 I x dx
x F
x m f
x a F
m x n
m ∞ ∫ − −
− Γ
= Γ
(2.4)where
. ) 1 ( )]
( ln ) ( ln [ 0 )
( y b F y F x n m f y dy
x x
I = ∫ − + − −
Setting,
w = − ln F ( y ) + ln F ( x )
, we find that1. )] /
( [ 0 !( )
) )( / ( 0
) 1 ( )
( )
( ∞ +
=
+ −
=
− −
−
− Γ
=
∑
∑
θ
θ λ
θ F x l
l
m l n
l j l b
j
j b j
m b m n
x n I
Put I(x) in (2.4), we obtain
. ) 1 ( )]
( ln / [ )]
( [ 0 !( ) 0
) )( / ( 0
) 1 ) (
( ) , (
, xa F x l F x m f x dx
l
m l n
l j l b
j
j b j
b m
m b n
a n
m − −
∞ ∞
=
+ −
=
− − Γ
−
=
∑
∑ ∫
θ
θ θ λ
µ
By setting,
1 ln[ ( )]
x F t θ
= −
, we have) , (
) (
!
!
) )( / )( )( / ( 0
0 )
1 ( 0
0 )
, (
, k l l n m k l m
j l i k
k l i j b a i
a j b a i b j b n
a n
m θ θ
λ λ
θ
µ ∞∑ + − + +
=
∑∞
=
−
−
− +
∑
=
∑
=
=
λ >max(a,b) and a,b=0,1,2,... . (2.5)
The joint pdf of three lower record values XL(m), XL(n), XL(ℓ), m, n, ℓ = 1, 2, …, m< n<
ℓ, can be written [see Arnold et al. (1998)] as
. 0
), ) ( ( ) (
) ( ) 1 ( )]
( ln ) ( ln [
)] 1 ( ln ) ( ln 1 [ )]
( ln ) [ ( ) (
) ( ) 1 , , , ( ,
∞
<
<
<
− <
+ −
−
×
− + −
− −
− − Γ
− Γ
= Γ
x y z z
y f F x F
y f x n f
y F z
F
m x n F y
m F x n F
m n z m
y n x
f m
l l l
(2.6) By the same way, we can get the triple moments of the lower record values
) (m
X
L ,) (n
X
Land
X
L(l), m < n < ℓ as. ...
, 2 , 1 , 0 , , )
, , max(
, ) (
) (
) (
!
!
!
) )( / )( )( / )( )( / ( 0
0 0 )
1 ( 0
0 0 ))
( ) ( ) ( (
) , , (
, ,
=
>
+ +
− + +
− + +
∑∞
=
×
∑∞
=
∑∞
=
−
−
− +
− +
∑
=
∑
=
∑
=
=
=
c b a and c b m a
e v m k e n
n v e
k v e
i k j v
s e k
v e i j s a b c i
a j b s c a i b j c s c
XL b
n XL a
m XL c E
b a
n m
θ λ θ
θ
λ λ
λ
θ µ
l
l l l
(2.7) The joint pdf of four lower record values XL(m), XL(n), XL(ℓ), XL(v), m, n, ℓ, v = 1, 2,
…, m < n < ℓ < v, can be written [see Arnold et al. (1998)] as
. 0
), ) (
( ) ( ) (
) ( ) ( ) (
)] 1 ( ln ) ( ln 1 [ )]
( ln ) ( ln 1 [ )]
( ln [
)] 1 ( ln ) ( ln ) [ ( ) ( ) (
) ( ) 1 , , , , ( , ,
∞
<
<
<
<
<
×
− + −
− − + −
− −
−
×
− + −
− − Γ
− Γ
− Γ
= Γ
x y z w w
z f F y F x F
z f y f x f
z v F w
n F y F z
m F x F
m x n F y
l F v n m
n w m
z y v x n f m
l l
l l
(2.8)
Similarly, we can obtain the quadruple moments of the lower record values
) (m
X
L ,) (n
X
L ,) (l
X
L and) (v
X
L , m < n < ℓ < v as, ) (
) (
) (
) (
!
!
!
!
) )( / )( )( / )( )( / )( )( / (
0 0 0 0 )
1 (
0 0 0 0 ))
( ) ( ) ( ) ( (
) , , , (
, , ,
e m r p m k e n
r n p e
v r r k p e r
i k j p
s e q r
k p e r i j s q a b c d
i a j b s c q a d i b j c s d q v d
v XL c XL b
n XL a
m XL d E
c b a
v n m
θ θ
θ θ
λ λ
λ λ
θ µ
+ + +
− + + +
− + +
− + +
×
∑∞
=
∑∞
=
∑∞
=
∞∑
=
−
−
−
− + +
− +
×
∑
=
∑
=
∑
=
∑
=
=
=
l l
l l
λ > max( a , b , c , d ) and a , b , c , d = 0 , 1 , 2 , ...
. (2.9)3. Recurrence relations for moments of record values
We have some results when λ is a positive integer, as follows:
1- For the single moments of lower record values
) (m
X
L from EP distribution (1.1)...
, 2 , 1 , 0 ) ,
( ) 1 1 ) (
1 (
1 1 1
2 )
( =
+ +
− + + + + +
=
=
∑
a a
m i a
a i m i
a a
m µ
µ θ λ λ
θ
µ λ
. (3.1)Proof Since
) . (
) 1 ( )]
( ln [ ) ( ) 0
( 1
) 1 ( )]
( ln [ ) 0
( ) 1 (
x dx F
x m f
x F x
a F m x
dx x m f
x a F
m x a
m
− −
∞
= Γ
− −
∞
= Γ
∫
∫
µ
From (1.1) and (1.2), we find that
1 ], 1 2 [ ) 1 ( )
( x i x
i i x f x
F λ λ λ
θ
λ +
+ +
=
=
∑ λ is a positive integer. (3.2) Integrating by parts, we get
}.
)]
( ln [ ) ( 0
) ( )]
( ln 1[ 0
){ ( ) 1
( a xa F x mF x dx xa f x F x mdx
m m a
m ∫ −
∞ +
− −
∫
∞
= Γ µ
From (3.2), we have
} )]
( ln [ ) ( 0
1 ] 1 2 [ ) ( )]
( ln 1[ 0
{ ) ( ) 1
( xi x dx xa f x F x mdx
i i x m f x a F
a x m m a
m ∞∫ −
+ + +
∑+
=
− −
∫
∞
= Γ
λ λ
λ θ
µ λ
) . ( } 1 ) (
1 )
1 (
1 1 1
2
{ a
m a
m i
a i m i
a
+ + + +
− + + + +
=
=
∑ λ µ λµ µ
λ θ λ Then,
...
, 2 , 1 , 0 ) ,
( ) 1 1 ) (
1 (
1 1 1
2 )
( =
+ +
− + + + + +
=
=
∑
a a
m i a
a i m i
a a
m µ
µ θ λ λ
θ
µ λ
.2- For the double moments of lower record values
) (m
X
L and) (n
X
L , m < n from EP distribution (1.1)...
, 2 , 1 , 0 , ) , , (
, 1 )
, (
1 , 1 )
, 1 (
1 , 1 1 1
2 )
, (
, + − + + + + + =
+ +
∑+ +
=
=
a b a b
n m b
a n m b
i a
n i m
i b a
a n m
a
µ µµ θ λ λ
θ
µ λ , (3.3) and
. ...
, 2 , 1 , 0 , ) , , (
1 ) ,
1 ) (
1 ,
( 1 , 1 1
2 )
, (
, =
+ +
− + + +
∑+ +
=
=
a b a b
n m i b
b a
n i m
i b b
a n
m
µ
µ θ λ λ
θ
µ λ
(3.4) ProofWe prove (3.3), at first:
dy m dx
x n F y
F
x F
x y f m f
x b F
a y x m y
n m b
a n m
)] 1 ( ln ) ( ln [
) (
) ) ( 1 ( )]
( ln [ ) 0
( ) ( ) 1 , (
,
− + −
−
×
− −
∞
∞
− Γ
= Γ ∫ ∫
µ
( ) ( ) ,
) 0 (
) (
1 y b f y I y dy
m n
m ∞ ∫
− Γ
= Γ
(3.5) where) . (
) 1 ( )]
( ln ) ( ln 1 [ )]
( ln [ )
( dx
x F
x m f
x n F y
m F x a F
x y y
I − − − + − −
∞
= ∫
Integrating by parts, we get
) }.
( ) 2 ( )]
( ln ) ( ln [ )]
( ln [ )
1 (
) (
) 1 ( )]
( ln ) ( ln [ )]
( ln )[ (
) 1 ( 1{
) (
x dx F
x m f
x n F y
m F x a F
x y m n
x dx F
x m f
x n F y
m F x x F
f x a F
x y m a y I
− + −
−
−
∞
−
− +
− + −
−
− −
∞
=
∫
∫
From (3.2), we have
}.
) (
) 2 ( )]
( ln ) ( ln [ )]
( ln [ ) 1 (
) (
) 1 ( )]
( ln ) ( ln [ )]
( ln [ 1 ]
1 1 2 [ 1{
) (
dx x F
x m f
x n F y m F
x a F
x y m n
dx x F
x m f
x n F y m F
x a F
i x xa i i
y a m y I
− + −
−
∫ −
∞
−
− +
− + −
−
−
− +
+
+
∑+
=
∫
∞
= λ λ λ
θ λ
Putting I(y) in (3.5), we find that
) . , (
, 1 )
, (
1 , 1 )
, 1 (
1 , 1 1 1
2 )
, (
, a b
n m b
a n m b a
i a
n i m
i b a
a n
m + +
+ + +
− +
+ + + +
=
=
∑
µ µ
µ θ λ λ
θ µ λ
Now we prove (3.4),
dx m dy
x n F y
F
x F
x y f m f
x b F
a y x x m n m b
a n m
)] 1 ( ln ) ( ln [
) (
) ) ( 1 ( )]
( ln [ 0
) 0 (
) ( ) 1 , (
,
− + −
−
×
− −
∞
− Γ
= Γ ∫ ∫
µ
( ) ,
) (
) 1 ( )]
( ln [ ) 0 (
) (
1 J x dx
x F
x m f
x a F
m x n m
− −
∞
− Γ
= Γ ∫ (3.6)
where
) . (
) 1 ( )]
( ln ) ( ln [ ) ( 0
)] 1 ( ln ) ( ln [ ) ( 0
) (
y dy F
y m f
x n F y
F y
b F y x
m dy x n
F y
F y
b f y x x J
− + −
−
=
− + −
−
=
∫
∫
Integrating by parts, we get
}.
)]
( ln ) ( ln [ ) ( 0
)]
( ln ) ( ln [ ) 1 ( 0
1 { ) (
m dy x n
F y
F y
b f y x
m dy x n
F y
F y
b F y x m b x n
J
+ −
− +
+ −
− −
= −
∫
∫
From (3.2), we have
}.
)]
( ln ) ( ln [ ) ( 0
)]
( ln ) ( ln [ 1 ]
1 1 2 )[
( 0 1 {
) (
mdy x n
F y
F y
bf y x
mdy x n
F y
b F i y
yb i i
y f b x m x n
J
+ −
− +
+ −
−
− + + +
+
− =
=
∫
∫
∑ λ λ
λ θ
λ
Putting J(x) in (3.6), we obtain
) . , (
1 ) ,
1 ) (
1 ,
(
1 , 1 1
2
) , (
1 } ,
) , (
1 , )
1 ,
(
1 , 1 1
2 ) {
, (
,
b a
n m i b
b a
n i m
i b
b a
n m b
a n m i
b a
n i m
i b b
a n m
+ +
− + + + + +
=
=
+ + + +
− + + + +
=
=
∑
∑
θ µ λ µ
λ θ λ
µ µ
λ λ µ
λ θ µ λ
3- For the triple moments of lower record values
) (m
X
L ,) (n
X
L and) (l
X
L ,m < n < l
, from EP distribution (1.1)) , , , (
, , 1 )
, , (
1 , 1 , 1 )
, , 1 (
1 , 1 , 1 1 1
2 )
, , (
, ,
c b a
l n m c
b a
l n m c
b i a
l n i m
i c
b a
l n m
a a
+ + + + + +
− +
+ + +
∑+ +
=
=
µ µ
µ θ λ λ
θ
µ λ
a,b,c=0,1,2,... (3.7) and
( , , ) , , , 0,1,2,...
1 , ) , 1 ) (
1 ,
, (
1 , , 1 1
2 )
, , (
,
, =
+ +
− + + +
∑+ +
=
=
a b c a b c
l n m i c
c b a
l n i m
i c c
b a
l n
m
µ
µ θ λ λ
θ
µ λ
(3.8)4- For the quadruple moments of lower record values
) (m
X
L ,) (n
X
L ,) (l
X
L and) (v
X
L ,v
l n
m < < <
, from EP distribution (1.1)) , , , , (
, , , 1 )
, , , (
1 , 1 , 1 , 1 )
, , , 1 (
1 , 1 , 1 , 1 1 1
2 )
, , , (
, , ,
d c b a
v l n m d
c b a
v l n m d
c b i a
v l n i m
i d
c b a
v l n m
a a
+ + + + + + +
− +
+ + + +
∑+ +
=
=
µ µ
µ θ λ λ
θ µ λ
a,b,c,d =0,1,2,... (3.9) and
) , , , , (
1 , , ) , 1 ) (
1 ,
, , (
1 , , , 1 1
2 )
, , , (
, , ,
d c b a
v l n m i d
d c b a
v l n i m
i d d
c b a
v l n
m + − + + +
+
∑+ +
=
=
µ
µ θ λ λ
θ µ λ
a,b,c,d =0,1,2,... . (3.10) Results 3 and 4 can be proved in the same way as that for result 2.
References
[1] AlZaid, A. A. and Ahsanullah, M. (2003). A characterization of the Gumbel distribution based on record values. Commun. Statist. - Theory Meth., 32(11), 2101-2108.
[2] Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York.
[3] Balakrishnan, N. and Ahsanullah, M. (1994). Relations for single and product moments of record values from Lomax distribution. Sankhya, 56 B(2), 140-146.
[4] Balakrishnan, N. and Chan, P. S. (1993). Record values from Rayleigh and Weibull distributions and associated inference. National Institute of Standards and
Technology Journal of Research, Special Publications 866, 41-51.
[5] Kamps, U. (1992). Identities for the difference of moments of successive order statistics and record values. Metron, 50, 179-187.
[6] Pawlas, P. and Szynal, D. (1999). Recurrence relations for single and product moments of k-th record values from Pareto, generalized Pareto and Burr distributions.
Commun. Statist.-Theory Meth., 28(7), 1699-1709.
[7] Raqab, M. Z. (2002). Inferences for generalized exponential distribution based on record statistics. Journal of Statistical Planning and Inference, 104, 339-350.
[8] Raqab, M. Z. and Awad, A. M. (2000). Characterizations of the Pareto and related distributions. Metrika, 52, 63-67.
