General Recurrence Relations and Characterizations of Certain Distributions Based on Record Values

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Distributions 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, we establish general recurrence relations based on lower record values.

Then, we use these relations to obtain the corresponding recurrence relations for the moments, moment generating functions (MGF's) and factorial moment generating functions (FMGF's) of lower record values and to apply on exponentiated Pareto distribution (EPD). In addition, general classes of continuous distributions are characterized by considering the conditional expectation of a function of lower record values. We also introduce some examples of distributions from these general classes of continuous distributions.

Key words:

Lower record values; Exponentiated Pareto distribution; Single, Double Moments; Recurrence relations; Moment generating function; Factorial moment generating function; Conditional expectation.

1. Introduction

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

j is an upper (lower) record value of

this sequence 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 record values; for example, see Abu-Youssef (2003), AlZaid and Ahsanullah (2003), Arnold et al. (1998), Balakrishnan and Ahsanullah (1994), Kamps (1992), Mohie El-Din et al. (2000), Pawlas and Szynal (1999), Raqab (2002), Raqab and Awad (2000).

In this paper, we extend the previous results of establishing recurrence relations between the moments of lower record values by deriving general forms for the expected values of any measurable functions of single or double lower record values. Also, we

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].

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establish recurrence relations for the single and double moments, moment generating functions (MGF's) and factorial moment generating functions (FMGF's) of lower record values. Moreover, we use these relations to apply on EPD. In addition, general classes of continuous distributions have been characterized through conditional expectation of a function of lower record values. We also introduce some examples of distributions from these general classes of continuous distributions.

Let

) ( )

2 ( ) 1

(

, , ... ,

m L L

L

X X

X

be the first m lower record values from a population whose pdf f(x) and cdf F(x). Let H(x) = - ln [F(x)] and h(x)=f(x)/F(x). Then, the pdf of XL(m), m = 1, 2, …, is given by

) 1 ( )]

( ) [ ( ) 1

( H x m f x

x m

f m −

= Γ , − ∞ < x < ∞

, (1.1) and the joint pdf of two lower record values XL(m) and XL(n) is

), ( ) 1 ( )]

( ) ( 1 [ )]

( ) [ (

) ( ) 1 ,

, ( H x m H y H x n m h x f y

m n y m

n x

f m − − − −

− Γ

= Γ

1 ≤ m < n , − ∞ < y < x < ∞ ,

(1.2) where H(y) = - ln[F(y)] and h(y) = f(y) / F(y).

Let

) ] ) (

) 1 ( (

1 [ x

m X L m

X L m E

m =

= +

+ φ

µ

, (1.3) and

) ], 1 ) (

) ( (

1 [ y

m X L m X L m E

m =

= +

+ φ

µ

(1.4) where

φ ( . )

is a monotonic, continuous and differentiable function on

( α , β )

.

2. Main Results

From (1.1), the expected value for any measurable function g(x) is obtained as

...

, 2 , 1 ,

) 1 ( )]

( [ ) ) (

( )) 1 ( ( ) )) ( (

( ∞ − =

∞ Γ −

=

= ∫ g x H x m f x dx m

X m g m m E

X L g

E

(2.1)

and, the expected value for any measurable function

τ ( x , y )

is obtained from (1.2) as

, ) ( ) 1 ( )]

( ) ( 1 [ )]

( )[

, ) (

( ) (

1

)) , , ( ( ) )) , (

) ( (

(

dx dy y f x m h x n H y m H

x H y x x m

n m

Y n X E m

n X L m X L E

−

− −

∫

−

∞

−

∞

∫

∞

− − Γ

= Γ

=

τ τ τ

m, n =1,2, …, m < n. (2.2) The expected values given in (2.1) and (2.2), satisfy the following two theorems.

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Theorem 2.1

Let X be an absolutely continuous random variable with cdf F(x) and pdf f(x). For any measurable function g(x) of the m^th lower record value and m =1, 2, …,

).

( ) ( )]

( ) [

1 ( )) 1 ( )

1 (

( H x m F x dg x

X m g m m X

g

E ∞ ∫

∞ + − Γ

= −

+ −

(2.3)

Theorem 2.2

Let X be an absolutely continuous random variable with cdf F(x) and pdf f(x). For any measurable function

τ ( x , y )

of two lower record values and n, m = 1,2,…, m < n

, )

, ( ) ( ) ( )]

( ) ( 1 [ )]

( ) [

1 (

) (

1

)) , , ( )

, 1 ( ( ,

dx dy y y x y F x m h x n

H y m H

x H x m

n m

Y n X Y m

n X E m

τ τ

τ

− −

∫

−

∞

−

∞

∫

∞ + −

− Γ Γ

= − + −

(2.4) where

τ y ( x , y )

means that the partial differentiation is with respect to y.

We consider here some special cases of Theorem 2.1 and Theorem 2.2 as follows:

1) By putting g(x) = x^a, a = 1,2,… in Theorem 2.1, we get

, ) ( )]

( 1 [ )

1 ( ) ( )

(

1 x a H x m F x dx

m a a

m a

m ∞ −

∞ + − Γ

= −

+ − µ ∫

µ

(2.5)

which represents the general form for establishing recurrence relations for the single moments of lower record values.

2) By putting g(x) = e^{t x}, in Theorem 2.1, we get

, ) ( )]

( ) [

1 ) (

( )

1 ( e t x H x m F x dx

m t t

M m m t

M ∞ ∫

∞ + − Γ

= −

+ −

(2.6)

which represents the general form for establishing recurrence relations for the single MGF's,

















= ( )

)

( tXL m

e E m t

M , of lower record values.

3) By putting g(x) = t^x, in Theorem 2.1, we get

, ) ( )]

( ) [

1 (

) ) (ln

( )

1 ( t x H x m F x dx

m t t

t m

m ∞ ∫

∞ + − Γ

= − Ψ

+ −

Ψ

(2.7)

which represents the general form for establishing recurrence relations for the single

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FMGF's,

















=

Ψ ( )

)

( XL m

t E

m t , of lower record values.

4) By putting

τ ( x , y ) = x a y b

, a, b = 1,2,… in Theorem 2.2, we get

, ) ( ) ( )]

( ) ( 1 [ )]

( 1 [

) 1 (

) ( ) , (

, )

, (

1 ,

dx dy y F x m h x n H y m H

x b H

a y x x

m n m b b

a n m b

a n m

− −

−

∫

−

∞

−

∞

∫

∞

−

×

+

− Γ Γ

= − + − µ

µ

(2.8) which represents the general form for establishing recurrence relations for the double moments of lower record values.

5) By putting

t x t y

e y

x , ) 1 2

( +

τ =

, in Theorem 2.2, we get

, ) ( ) ( )]

( ) ( 1 [ )]

( 2 [ 1

) 1 (

) ( ) 2 , 2 ( 1 ) ,

, 2 ( 1 1 ,

dx dy y F x m h x n

H y m H

x y H t x e t x

m n m

t t

n t M m t

n t M m

− − + −

∫

∞

−

∞

∫

∞

−

×

+

− Γ Γ

= − + −

(2.9) which represents the general form for establishing recurrence relations for the double MGFs of lower record values.

6) By putting

x y t x t y 2 ) 1

,

( =

τ

, in Theorem 2.2, we get

, ) ( ) ( )]

( ) ( 1 [ )]

( 2 [ 1

) 1 (

) (

2 ) (ln 2 )

1 , , ( 2 )

1 , 1 ( ,

dx dy y F x m h x n

H y m H

x y H x t x t

m n m

t t

n t t m

n t m

− −

∫

−

∞

−

∞

∫

∞

−

×

+

− Γ Γ

= − Ψ

+ − Ψ

(2.10) which represents the general form for establishing recurrence relations for the double FMGFs of lower record values.

Example

Let us consider the Exponentiated Pareto distribution EP(θ,λ) with pdf

f ( x ; θ , λ ) = θ λ [ 1 − ( 1 + x ) − λ ] θ − 1 ( 1 + x ) − ( λ + 1 ) , x > 0 , λ > 0 , θ > 0 ,

where λ is a positive integer. From (2.5) and (2.8), we can get

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...

, 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

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

,

respectively.

Moreover, we can find from (2.6) that

)

,

1 (

! 1 1 0 ) 1

(

! 1 1 1 1 2 0 ) 1

( )

1(

+ + +

∑∞

= + −

+ + +

∑+

=

∑∞

=

= −

+ − ^^_







 j

j m t j j j

i j m t j i i

j m t

M m t

M µ

µ θ λ λ

θ λ

m = 1,2,…

Also, from (2.9), we have

. 1

)

,

1 , (

1

! ,

! 1 2 1 0 0 1

) , (

1

! ,

! 1 2 1 1

1 2 0 0 ) 1

, 2 (1 ) ,

, 2 (1 1 ,

n

k

m

j n k m

j t k t j j k

i k j

n k m

j t k t j i i

j k t

n t Mm t

n t Mm

<

+

≤

+ +

∑∞

=

∑∞

=

−

+ + + +

∑+

=

∞∑

=

∞∑

=

= −

+ − ^^_









θ µ

λ µ λ θ

λ

From (2.7), we get

), 1 (

! 1 ) 1 (ln 0 ) 1

(

! 1 ) 1 1 (ln 1

2 0 ) 1

( )

1(

+ + +

∑∞

= + −

+ + +

∑+

=

∞∑

=

= − Ψ + −

Ψ _^









 j

j m t j

j j

i j m

t j i i

j m t

m t µ

µ θ λ λ

θ λ

m = 1,2,…

From(2.10), we obtain

. 1

), 1 , (

1

! ,

! ) 1 (ln 2 1) (ln 0 0 1

) , (

1

! ,

! ) 1 (ln 2 1) 1 (ln 1

2 0 0 ) 1

, 2 (1 ) ,

, 2 (1 1 ,

n k m

j n k m

j t k t j

j k

i k j

n k m

j t k t j

i i j k t

n t t m

n t m

<

+ ≤ + +

∑∞

=

∑∞

=

−

+ + + +

∑+

=

∑∞

=

∑∞

=

= − Ψ

+ −

Ψ _^











θ µ

λ µ λ θ

λ

Theorem 2.3

Let X be an absolutely continuous random variable with cdf F(x) and pdf f(x).

Suppose F(x) < 1 for

x ∈ ( α , β ), F ( α ) = 0

and

F ( β ) = 1

. Then

x c

a b x

F ( ) = [ − φ ( )]

(2.11) if and only if

], ) ( 1 [ 1

1 a

x b c c

m

m +

= +

+ φ

µ

(2.12) where

a ≠ 0 , c , b

are finite constants.

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Proof

It is clear that

).

( ) ) (

( 1

1 y dF y

x x m F

m φ

α

µ + = ∫ (2.13) Integrating (2.13) by parts, we get

).

( ) ) (

( ) 1

1 ( F y d y

x x x F

m

m φ

α φ

µ + = − ∫ (2.14) In view of (2.11) and (2.14), we obtain

) . 1 (

) ) (

1 ( +

+ −

+ = a c

x a x b

m m

φ φ

µ

Then, the necessary condition is proved. To prove the sufficient condition, from (2.12) and (2.13), we obtain

].

) ( 1 [ ) 1

( ) ) (

( 1

a x b c c

dy y f y x x

F +

= +

∫ φ φ

α

Taking the derivative we, have

), ( ) ( ]

) ( [ ) ( ) ( ) ( ) 1

( c F x x

a x b c x f x f x

c + φ = φ + + φ ′

which gives

) ( ) ( ) ]

[ ( )

( cF x x

a x a x b

f φ φ ′

− =

−

,

and then

) . (

) ( )

( ) (

x a b

x a c x

F x f

φ φ

−

− ′

=

Thus, the theorem is proved.

Theorem 2.4

x ∈ ( α , β ), F ( α ) = 0

and

F ( β ) = 1

, then

F ( x ) = [ b − a φ ( x )] c

if and only if

1 , ) ) (

( ) 1 ( ) 1

( ) 1 ( 2

1 + + ≥

+ −

= + +

+ m

a y b y

H m c m

y m H

m c m

m µ φ

µ

(2.15)

and

,

) (

)]

( ) ( [ 2

1 a

b y

H

y

c − +

= φ β φ

µ

(2.16) where

a ≠ 0 , c , b

are finite constants and

φ (x )

is monotonic, continuous and

differentiable in x in the interval

[ α , β ]

.

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Proof

It is clear that

[ ( )] ( ) .

)]

( [ ) 1

1 ( H x m x dx

y y m H m y

m φ

β φ

µ = + ′

+ ∫ (2.17)

In view of (2.11), we have

( ) ( )[ ( )] .

c a

x a b x x h

−

= −

′ φ

φ

(2.18) Using (2.18) in (2.17), we obtain

,

) 1 (

) ) (

2 ( ) 1

1 (

) (

1 = − +

+ + +

+ − a c m

y H y b

m m m

c y H m

m µ φ

µ

thus, the necessary part is proved.

For the sufficient part, from (2.15), we have

) . 1 (

) ) (

(

) ( )]

( [ ) 1 (

)]

( [

) ) (

1 ( )]

( [ ) ( )]

( [

− +

=

− +

− ∫

∫

m a c

b y y H

dx x m h x H x m y

y H c

y dx H

x m h x H x y y m H

m

φ

φ β φ

β

Taking the derivative and after some calculations, we obtain

c

a y a b y

y h ( )[ ( )]

)

( φ

φ ′ = −

−

which gives

) . (

) ( )

( ) (

y a b

y a c y

F y f

φ φ

−

− ′

=

Thus, we obtain the required result.

For the necessary condition of (2.16), it is easy to obtain

.

) ( ) ) (

( 1 2

1 x h x dx

y y

H φ

β

µ = ∫ But, from (2.11)

) .

( ) ) (

( b a x

x c x a

h φ

φ

−

− ′

=

So,

) ( ) ] 1 (

) [ 2 (

1 d x

x a b

b y y

H

c φ

φ β

µ = ∫ − −

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which gives

) , (

)]

( ln[

) (

)]

( ) ( [ 2

1 a H y

y a b b c y

H

y

c φ β φ φ

µ − − −

=

(2.19)

where

)]

( ln[

)

( y c b a y

H = − − φ

,

thus (2.19) gives the necessary part. Using the same argument in the above theorem, we obtain the sufficient condition and the theorem is proved.

By repeatedly appealing the recurrence relation in (2.15), we simply derive the following relation in terms of

2 µ 1

:

. 2 ,

1 2 ] 1

) [ ( 2 1

1 ) ]

[ ( 1 1 ) 2 (

1 )] 1

( [

1 1 !

≥

∏ + +

=

−

∑

− −

= +

+

=

− −

=

− −

− + =

















∑ ∏

m m j

i j i m y H m c

a i b

m j i j i m y H m c

i m y

y H

c m m m

m µ φ

µ

(2.20)

Theorem 2.5

Let X be an absolutely continuous random variable with cdf F(x) and

x ∈ ( α , β )

, then

) ) (

( x a b e c x

F = + − φ

, (2.21) if and only if

, 0 ) ,

(

)]

( ) ( [ ) 1

1 ( − ≠

− +

+ = c

x F x a x c

m m

α φ φ φ

µ

(2.22)

where a, b, c are finite constants and

φ ( . )

is a monotonic, continuous and differentiable function on

( α , β )

.

Proof

In view of (2.21) and (2.14), we have

[ ( ) ( ) ].

) ( )

(

)]

( ) ( ) [

1 (

α φ α φ

φ φ φ

µ e c x e c

x F c

b x

F x x a

m m

− − + −

− −

+ =

(2.23)

But

. ) ) (

( )

( and b e c x F x a

b c a

e − φ α = − − φ = −

Thus, substituting in (2.23), the necessary condition is proved. To prove the sufficient condition, from (2.22), we have

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1 ] ( ) ( ) )

( )[

( )

( φ φ φ α

φ α

a x c a

x x F dy y f y x

+

− +

∫ = , (2.24)

differentiating both sides with respect to x, (2.24) follows:

), 1 ( ] ) ( )[

( f x

a c x F

x − = −

φ ′

hence the theorem is proved.

Table 1: Examples of (2.11) distributions

A b c φ(x) F(x) Name

1 1 λ _e⁻⁽^x^/^θ⁾^p [1−e⁻⁽^x^/^θ⁾^p]^λ; x>0 Exponentiated Weibull 1 1 θ (1+x)⁻^λ [1−(1+x)⁻^λ]^θ;x>0 Exponentiated

Pareto -1 0 θ ^p e⁻^x⁻^p e⁻⁽^θ^/^x⁾^p; x>0 Inverse-Weibull

−1

−λ ⁰ ^p x λ⁻^px^p;0≤x≤λ Power function

1 1 1 (1+θx)⁻^λ 1−(1+θx)⁻^λ;x>0 Lomax 1 1 α _e⁻⁽^λ^x⁾² [1−e⁻⁽^λ^x⁾²]^α; x>0 Generalized

Rayleigh ) 1

( − ⁻

− λ β ⁰ ¹ ⁽^x⁻^β⁾ ^β ^λ

β λ

β ≤ <

−

− x

x );

( Rectangular

1 1 1 (1+θx^p)⁻^λ 1−(1+θx^p)⁻^λ;x>0 ^{Burr XII}

1 1 α _e⁻^x² (1−e⁻^x²)^α Burr X

−p

− )

(λ β ¹ ¹ (λ−x)^p ^β ^λ

β λ

λ ≤ ≤

−

− −x _p x

; ) (

1 Beta of the first

kind

1 1 θ e⁻^λ^x (1−e⁻^λ^x)^θ;x>0 Exponentiated

exponential 1 1 θ e⁻^x(1+x) [1−e⁻^x(1+x)]^θ;x>0 Exponentiated

Gamma

λp ¹ ¹ x⁻^p 1−λ^p x⁻^p; x≥λ Pareto of the first

kind

Theorem 2.6

x ∈ ( α , β ), F ( α ) = 0

and

F ( β ) = 1

, then

F ( x ) = a + b e − c φ ( x )

if and only if

,

) 1 (

) ( )

2 ( ) ( )

1 (

) ) (

1 ( = − +

+ +

−

+ =













m c

y y H

m X L X e c m E

b c

y H y a

m m

φ φ

µ

(2.25)

where

c ≠ 0 , b ≠ 0 , a

are finite constants,

x > y and 0 < m < n .

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Proof

To prove the necessary condition, (2.21) gives

.

) 1 (

) (

= + −

x F

x e c

b

a φ

(2.26) In view of (2.17) and (2.26), we obtain

. ) ( )]

( [ )]

( [

1

) ( )]

( ) [ ( )]

( [ ) 1 (

dx x m h x H y y m H c

dx x m h x x H e c y y m H b c y a m

m

∫

−

− + =

β β φ φ

µ

After integrating the last term, we obtain (2.25).

For the sufficient condition, using the same argument in the above theorem, it is easy to obtain

) . (

) ( )

( ) ) (

(

x F

x f x

e c b a

x e c

x c

b =

+ −

′ −

−

φ φ φ

Thus the theorem is proved.

Table 2: Examples of (2.21) distributions

a b c φ(x) F(x) Name

1 -1 θ x^p 1−e⁻^θ^x^p;x>0 ^Weibull

0 1 θ^p x⁻^p e⁻⁽^θ^/^x⁾^p; x>0 Inverse-Weibull

1 -1 λ² x² 1−e⁻⁽^λ^x⁾²;x>0 ^Rayleigh

1 -1 λ x 1−e⁻^λ^x;x>0 Exponential

1 -1 λ ln(1+x) 1−e⁻^λ^ln(¹⁺^x⁾;x>0 Standard Pareto of the second kind

0 λ⁻^p − p ln x λ⁻^pe^p^ln^x;0≤x≤λ Power function

1 -1 λ ^ln(¹+θx⁾ 1−e⁻^λ^ln(¹⁺^θ^x⁾;x>0 Lomax

0 (λ−β)⁻¹ ^-1 ^ln(^x⁻^β⁾ ^β ^λ

β λ

β

<

− ≤

−

e ^x x );

(

) ln(

Rectangular 1 -1 λ ln(1⁺θx^p) 1−e⁻^λ^ln(¹⁺^θ^x^p⁾;x>0 Burr XII

1 −(λ−β)⁻^p ⁻ ^p ^ln(^λ⁻^x⁾ ^β ^λ

β λ

λ

≤

− ≤

−

e x

p x p

) ; 1 (

)

ln( Beta of the first

kind 1 −λ^p p ln x 1−λ^pe⁻^p^ln^x;λ≤x<∞ Pareto of the first

kind 1 -1 1 x−ln(1+x) 1−e⁻^x⁺^ln(¹⁺^x⁾;x>0 Gamma

(11)

References

[1] Abu-Youssef, S. E. (2003). On characterization of certain distributions of record values. Applied Mathematics and Computation, 145, 443-450.

[2] 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.

[3] Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York.

[4] Balakrishnan, N. and Ahsanullah, M. (1994). Relations for single and product moments of record values from Lomax distribution. Sankhya, 56 B(2), 140- 146.

[5] Kamps, U. (1992). Identities for the difference of moments of successive order statistics and record values. Metron, 50, 179-187.

[6] Mohie El-Din, M. M., Abu-Youssef, S. E. and Sultan, K. S. (2000). General recurrence relations based on upper record values. The Egyptian Statistical Journal, 44(1), 68-80.

[7] 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.

[8] Raqab, M. Z. (2002). Inferences for generalized exponential distribution based on record statistics. Journal of Statistical Planning and Inference, 104, 339-350.

[9] Raqab, M. Z. and Awad, A. M. (2000). Characterizations of the Pareto and related distributions. Metrika, 52, 63-67.