RECURRENCE RELATIONS FOR MOMENTS OF RECORD VALUES FROM INVERTED WEIBULL DISTRIBUTION

RECURRENCE RELATIONS FOR MOMENTS OF RECORD VALUES FROM INVERTED WEIBULL DISTRIBUTION

By

SHAFYA A. AL-HIDAIRAH AND GANNAT R. AL-DAYIAN Department of statistics , Faculty of Science(Girls) , King Abdulaziz University, Jeddah KSA

ABSTRACT

Recurrence relations for the single and double moments of lower record values arising from inverted Weibull distribution (inverted exponential, and inverted Rayleigh as special cases) are derived. Numerical study of the recurrence relations for moments of record values arising from inverted Weibull and comments are given.

Keywords : Inverted Weibull distributions; record values; recurrence relations.

1.

INTRODUCTION

Record values and the associated statistics are of interest and important in many real life applications, such as: weather, educations, industry, economic, and sports data. The statistical study of record values started with Chandler (1952) and no spread in different directions. Foster and Stuart (1954) were pioneers in finding applications of record counting statistics in inference. They obtained the mean and variance for the total number of records in a sequence of length n. Feller (1968) presented some examples of record values in gambling problems. Rensick (1973) and Shorrok (1973) discussed the asymptotic theory of records.

A good elementary review concerning of record values is given by Glick (1978). Two more surveys are given by Nevzorov (1987) and Nagaraja (1988).

Properties of record values have been extensively studied in the literature see for example, Ahsanullah (1988), Arnold and Balakrishnan (1989), Arnold, Bbalakrishnan and Nagaraja (1992) and Arnold, Bbalakrishnan and Nagaraja (1998).

Because of the importance of the moments of record values in drawing inference, Balakrishnan, Ahsanullah and Chan (1992) established some recurrence relations for the single and double moments of lower record values from Gumble distribution.

Balakrishnan, Chan and Ahsanullah (1993) established some recurrence relations for single and double moments of record values from the generalized extreme value distribution.

Balakrishnan and Ahsanullah (1994) established some recurrence relations of single and double moments of record values from the Lomax distribution, and the generalized Pareto distribution. They noted that if the shape parameter tends to 0 in their results, simply become the results for moments from the standard exponential distribution, have been established in the previous paper in 1994.

Balakrishnan and Ahsanullah (1995) established some recurrence relations of single and double moments of upper record values from exponential distribution, they have extended their results to the case of independent nonidentical model, [for more details see Balakrishnan and Ahsanullah (1995)].

Pawlas and Szynal (1999) derived recurrence relations satisfied by single and double moments of the k^th upper record values from the Pareto, generalized Pareto and Burr distributions. They also gave some characterizations of the three distributions..

Pawlas and Szynal (2001) derived recurrence relations for single and double moments of GOSs under the concept of Kamps from Pareto, generalized Pareto and Burr distribution. The results include as particular cases the above relations for moments of k^th record values

Raqab (2001) derived the moments of record values from linear exponential distribution. He computed the means and double moments of record values from linear exponential distribution and the recurrence relations for both single and double moments of record values. He noted that these results can be used to establish similar results for exponential and Rayleigh distributions as special cases. Finally, he extended his results to include the moments of k^th record values.

Raqab (2002) derived exact expressions for single and product moments of record statistics of the generalized exponential distribution, and recurrence relations for single and product moments of record values are obtained. The means, variance and covariances of the record values are computed for various values of the shape parameter and for some record values. These values are used to compute the coefficients of the BLUE's of the location and scale parameters.

Sultan (2002) derived exact explicit expressions for the single, double, triple and quadruple moments of the upper and lower record values from uniform distribution.

Sultan, Moshref and Childs (2003) derived exact expressions for the single, double, triple and quadruple moments of lower record values from generalized power distribution. They used these expressions to compute the means, variances and the coefficients of skewness and kurtosis of certain linear functions of record values.

In this paper, the recurrence relations for moments of record values from inverted Weibull distribution are obtained. Numerical computations and a simulation study are presented to illustrate the procedures.

2.

INVERTED WEIBULL DISTRIBUTION

The Weibull distribution has been shown to be useful for modeling and analysis of life time data in medical, biological and engineering sciences. The use of the distribution in reliability and quality control work has been advocated by Kao in (1958), (1959). Some remarks on the history and also on the relation between extreme-value and Weibull distributions are in Mann (1968). In Figure (3.1), it can be use Weibull distribution as a father distribution.

The two ( shape and scale ) parameters Weibull distribution has the probability density function(pdf) given by







>

= ≥



 





−

otherwise ,

0

0) β , (θ 0, x , e x θ (x) β f

β θ - x 1 - β

β ,

then, the distribution of 1/X is referred as the inverse or inverted weibull distribution.

Inverted Weibull (IW) distribution have been recently derived as a suitable model to describe degradation phenomena of mechanical components Keller and Kanath (1982) such as the dynamic components (pistons, crankshaft, etc.) of diesel engines.

Erto (1989) discussed the properties of this distribution and its potential use as lifetime model. A lot of work has been done on IW distribution; for example, Calabria and Pulcini (1990) studied the maximum likelihood and least-squares estimation of IW parameters. Calabria and Pulcini (1994) studied Bayes 2-sample prediction for IW distribution.

Mahmoud, Sultan, and Amer (2003) considered the order statistics arising from IW distribution, and derived the exact expression for the single moments of order statistics. The variances and covariances. The obtained based on the moments of order statistics, the best linear unbiased estimates BLUE's for the location and scale parameters of IW distribution.

Sultan, Ismail, and AL-Moisheer (2006) investigated the mixture model of two inverted Weibull distributions, and discussed some properties of this model.

Let X be a random variable distributed as IW (θ,β) with scale parameter 0

θ> and shape parameter β > 0 , denoted by X ≈IW (θ,β), has pdf , cumulative distribution function (cdf), reliability function (SF), and hazard rate function(HRF), given respectively by







>

= ^ ≥





 +

−

−

otherwise ,

0

0) β , (θ 0, x , e x θ (x) β f

β x θ - 1 1) (β

β .

(2.1)







>

= ^ >





−

othewise ,

0

0) β , (θ 0, x ,

(x) e F

β x θ

1

. (2.2)







>

= ^ >





−

othewise ,

0

0) β , (θ 0, x ,

e - S(x) 1

β x θ

1

. (2.3)

and

0).

β , (θ 0, x , e

- 1

e x θ

h(x) β β

β

x θ

1 x θ

1 1) (β β

>

>

=







−







− +

−

−

(2.4)

It may be observed that if 0<β<1, the HRF is a monotone decreasing, on the other hand, if β>1 the HRF is increasing and then decreasing. The limit of the HRF, as βtends to infinity, equals zero.

The curves of four IW (θ,β) populations and the corresponding HRF's are plotted in Figure (2.1). The first population, in Fig. (2.1.a), is when 0<β≤θ, (θ=0.5,1.0,β=0.5), the second population, in Fig. (2.1.b), is whenβ=1, (θ=0.1,1.5,β=1.0), the third and fourth populations, in Fig. (2.1.c) and Fig. (2.1.d), are whenβ>θ, (θ=0.1,1.0,β=2.0) and (θ=0.1,0.5,β=3.0), respectively.

Figure (2.1): The pdf and HRF's of IW (θ,β) Distributions

Fig. (2.1.a) θ=0.5,1.0,β=0.5. Fig. (2.1.b) θ=0.1,1.5,β=1.0.

Fig. (2.1.c) θ=0.1,0.5,β=2.0. Fig. (2.1.d) θ=0.1,0.5,β=3.0.

Figure (2.1) shows that

When 0<β≤θ, the curve of the pdf and HRF are monotone decreasing.

When β=1 the curve of the pdf takes the shape of the inverted exponential distribution. When β=2 the curve of the pdf takes the shape of the inverted Rayleigh distribution. When β>θ the curves of the pdf and HRF are increasing and then decreasing.

The r^th moment of the IW (θ,β) distribution is given by The mean of the IW (θ,β) distribution is given by

. 1 β β , 1 1 θ Γ

E(x) 1  >







 −

= (2.5) The variance of the IW (θ,β) is given by

, 2 β β , 1 1 θΓ - 1 β 1 2 θ Γ V(x) 1

2

2  >



 











 −







 −

= (2.6)

0 10 20 30

0 0.1 0.2 0.3

f1 x( ) f2 x( ) h1 x( ) h2 x( )

x

0 10 20 30

0 0.5 1

f1 x( ) f2 x( ) h1 x( ) h2 x( )

x

0 10 20 30

0 0.2 0.4 0.6

f1 x( ) f2 x( ) h1 x( ) h2 x( )

x

0 10 20 30

0 0.5 1

f1 x( ) f2 x( ) h1 x( ) h2 x( )

x

Inverted

Exponential Distribution

Exponential Distribution

Extreme Value Distribution

Rayleigh

Distribution Inverted

Rayleigh

Distribution Inverted

Weibull Distribution

1 β=

β=2

1/X 1/X

1 β=

1/X

2 β= 1

β= β=2

(

x θ

)

log β -

Weibull Distribution where Γ(.)is the gamma function.

The quantile of the IW (θ,β) distribution is given by

( )

[

^{θ logq}

]

^{, 0 q 1,}

x_q = − ^{β−1 −1} < < (2.7) and the special cases may be obtained by using (1.3.7) such as first and third quartiles, whenq=0.25 andq=0.75, respectively. Also, ifq=0.5, we obtain the median of x, which is given by

( )

[

^β

]

¹

median

2 1

log θ x

median≡ = ^{− −} . (2.8) It may be observed, from (2.1), that the pdf is a monotone decreasing when 1,

β

0< < in which case the mode is zero. On the other hand, the pdf is a monotone increasing then decreasing in the case of β>1, in which case the mode is obtained by maximizing the pdf. It is given by

( )

[ ]







>

+

<

= <

≡ θ 1 β− + − , β 1

1 β ,0 0

x

mode 1

2) β 1 (

mode β . (2.9) Figure(2.2): Relationship Between Weibull Distribution and Other Distributions

3.

RECURRENCE RELATION FOR MOMENTS OF RECORD VALUES FROM INVERTED WEIBULL DISTRIBUTION

3.1 Record values

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of the observations.

Let X1 , X2 , … , Xn be a sequence of i.i.d random variables with cumulative distribution function ( cdf ) F(x) and pdf f(x). Set Y_n = max(min){ X₁ , X₂ , … , X_n},

1

n ≥ . We say Xj is an upper ( lower ) record of this sequence if Yj > ( < ) Yj-1 , j > 1.

By definition, X1 is an upper as well as lower record values. One can transform from upper record values to lower records by replacing the original sequence of random variables by {-Xj , j≥1}or by {1/Xi , i≥1} ( if P( Xi > 0) =1 for all i ); the lower record values of this sequence will correspond to the upper record values of the original sequence.

Since the study will involve both lower and upper record values ( depending on the population under consideration), we shall use the following notations for convenience: XU(n) for the n^th upper record and XL(n) for the n^th lower record.

The indices at which the upper record values occur are given by the record times {U(n),n≥1}, where ^U(n)=^min

{

^{j j}>^U(n-1),Xj >^XU(n-1)

}

,n≥1, withU(1)=1. Also, the indices at which the lower record values occur are given by the record times {L(n), n≥1 }, where L(n)=min

{

j j>L(n-1),Xj <XL(n-1)

}

, n>1, with L(1)=1. We will confine our attention to record values from continuous random variables.

For more details on both continuous and discrete record values, [see Ahsanullah (1995)], and [Arnold, et al.(1998)].

3.2 Probability density functionof record values

The joint pdf of the first n upper record values XU(1) , XU(2) , … , XU(n) is given by, [see Arnold, et al.(1998)]

( ) ( ) ∏

⁻

= −

=

1 n

1

i U(i)

U(i) U(n)

U(n) U(2)

U(1) n , ...

1,2, ,

) F(x 1

) x f(x

f x , ...

, x , x f

, x

...

x

x_U(1) < _U(2) < < _U(n) <∞

<

∞

− (3.1)

and the pdf of the n^th upper record value XU(n) is obtained to be ), x ( f ]}

) x ( F 1 [ log Γ(n){ ) 1 x (

f_n = − − ⁿ⁻¹

, ...

1,2, n , x<∞ =

<

∞

−

(3.2)

and the joint pdf of XU(m) and XU(n) (m < n) is obtained to be

f(y), ]}

F(x) 1 log[

] F(y) 1 log[

{

F(x) 1 ]} f(x) F(x) 1 log[

m){ Γ(m)Γ(n y) 1

(x, f

1 m n

1 m n

m,

−

−

−

− +

−

−

×

− −

− −

=

n, m , ...

1,2, m ,

y

x < <∞ = <

<

∞

− (3.3)

where x = XU(m) and y = XU(n).

The joint pdf of the first n lower record values XL(1) , XL(2) , ... , XL(n) is given by

( ) ( ) ∏

⁻

=

=

1 n

1

i L(i)

L(i) L(n)

L(n) L(2)

L(1) n , ...

1,2, ,

) F(x

) x f(x

f x , ...

, x , x

f

, x

...

x

x_L(n) < _{L(n 1)} < < _L(1) <∞

<

∞

− ₋ (3.4)

and the pdf of the n^th lower record value XL(n) is obtained to be:

, ) x ( f } ) x ( F log Γ(n){ ) 1 x (

f_n = − ⁿ⁻¹

, ...

1,2, n , x<∞ =

<

∞

− (3.5)

and the joint pdf of the lower record values XL(m) and XL(n) (m < n) is obtained to be

, f(y) ]}

log[F(x) ]

F(y) log[

{

F(x) ]} f(x) F(x) log[

m){ Γ(m)Γ(n y) 1

(x, f

1 m n

1 m n

m,

−

−

−

+

−

×

− −

=

n, m , ...

1,2, m , x

y< <∞ = <

<

∞

− (3.6)

where x = XL(m) and y = XL(n).

3.3 Moments of record values

Let us denote the single moments of the n^th upper record values E(X^k_U(n)) by

k

µ′n , the double moments of the upper record valuesE(X^k_U(m)X^s_U(n)) by µ′_m,^k,_n^s, the single moments of the n^th lower record value E(X^k_L(n)) by µ′_(n)^k, and the double moments of lower record values E(X^k_L(m)X^s_L(n)) by µ′_(m),^k,s_(n). The single and double moments of both upper and lower record values are given as follows:

(i) Single moments of the upper record value XU(n) are given by ...

, 2 , 1 n , dx f(x) F(x)]}

log[1 { Γ(n) x

µ′_n^k = 1 ^{∞ k} − − ⁿ⁻¹ =

∞

∫

− (3.7)

(ii) The double moments of the upper record values X_U(m) and X_U(n) (m < n) are given by

[ ]

{ }

dx, dy f(y) F(x)]}

1 log[

F(y)]

log[1 {

F(x) 1 F(x) f(x) 1 log y

m) x Γ(m)Γ(n µ 1

1 m n x

1 s m

k s

k, n m,

−

−

∞

∞

−

∞ −

− +

−

−

×

− −

− −

′ =

∫ ∫

n.

m , ...

, 2 1, m , y

x< <∞ = <

<

∞

− (3.8)

(iii) The single moments of lower record values XL(n) are given by ...

, 2 , 1 n dx, f(x) } F(x) log { Γ(n) x

µ′_(n)^k= 1 ^{∞ k} − ⁿ⁻¹ =

∞

∫

− (3.9)

(iv) The double moments of the lower record values XL(m) and XL(n) (m < n) are given by

{ }

dx, dy f(y) }

F(x) log F(y) log {

F(x) F(x) f(x)

log y m) x

- (n Γ(m) µ 1

1 m n

x k s m1

s k,

(n) (m),

−

−

∞

∞

− −∞

−

+

−

× Γ −

′ =

∫ ∫

. n m , ...

, 2 , 1 m , x

y< <∞ = <

<

∞

− (3.10)

3.4 Probability density functionof the lower record values from inverted Weibull distribution

Let X_L(1), X_L(2), … , X_L(n) be the first n lower record values arising from IW(θ,β) in (2.1), and (2.2). By using the pdf of the n^th lower record values XL(n) given in (3.5), the pdf of the n^th lower record value from IW (θ,β) as

{ ( )

^θx

}

^{e ,}

Γ(n)x θ (x) β f

β x θ

1 1 β n 1)

(β β n







−

− − +

−

−

=

1 n 0), β , (θ 0,

x> > ≥ . (3.11) Similarly by using the joint pdf of the lower record values X_L(m) , X_L(n) in (3.6) and f(x)and F(x) as given, respectively, by (2.1) and (2.2), the joint pdf of XL(m) and XL(n)

from lower record values from IW(θ,β) is given by

{ ( ) } { }

, e y

x y x

θ m)x

Γ(n Γ(n)

θ y) β (x, f

β y θ

1 (β1)

1 m β n - β 1 - β m 1) (β 1) m - n ( β 2 n

m,







− +

−

−

− − +

− +

−

×

− −

=

n.

m 1 0), β , (θ , x y

0< < <∞ > ≤ < (3.12)

3.5 Recurrence relation for moments of record values

In this section, we use the relation between pdf and cdf of IW (θ,β) to establish some recurrence relation for the single and double moments of lower record values.

From (2.1) and (2.2), we have

{

^logF(x)

}

^F(x)

x β

f(x)= ⁻¹ − .

(3.13)

This relation will be used in the following result to establish recurrence relation.

(i) Relation for the single moments

Result 3.1

For n = 1, 2 … and k = 1, 2 … the relation for single moments of lower record values from the IW (θ,β) distribution is given by:

.

),

( )

1

( n k

n k

n _k

n k

n  ′ >



 



 −

′₊ = µ β

β

µ β

(3.14)

Proof

For n ≥1 and k =1, 2 … the single moments of the lower record values from IW can be written as

{

^logF(x)

}

^f(x)dx.

Γ(n) x

µ 1 ^n-1

0 k k

(n) = −

′

∫

^∞

By using (3.13) in the above relation, we obtain

{

^logF(x)

}

^F(x)dx.

Γ(n) x

µ β ⁿ

0 1 - k k

(n) = −

′

∫

^∞

Integrating by parts treating x^k−1 for integration and the rest for differentiation, then the integral can be written as follows

µ , nβ k.

nβ k

µ_(n^k₁₎ nβ  ′_(n)^k >







 −

′₊ = (ii) Relation for double moments

Once again the relation (3.13) will be used to derive recurrence relation for the double moments of the lower record values from the IW (θ,β) distribution assuming their existence

.

Result 3.2

For m≥1 and k,s=1,2, ... the relation for double moments of lower record values from the IW(θ,β) distribution is given by

k.

β m , kµ mβ

β

µ_(m),(^k,s_{m )} m _(m′^{k s}₎ >

= −

′ _{+1 +}⁺₁ (3.15) and for1≤m≤n−2, we have

1 µ , mβ k.

β m

β k

µ_(m^k,s_),(n) m − _(m),(n)′^k,s >

′ ₊ = (3.16) Proof

The double moments of the lower record values from IW (θ,β) distribution may be written as

, x y 0 , dy I(y) g(y) m) y

Γ(m)Γ(n µ 1

0 s s

k, (n)

(m), ≤ < <∞

= −

′

∫

^∞ (3.17)

where

{ } {

^{logF(y) logF(x)}

}

^dx.

F(x) F(x) f(x)

log x I(y) y

1 m n 1

k m

∫

^{∞ − − × − + − −}

= (i) Forn =m+1, we get

, x y 0 dy, (y) I f(y) Γ(m) y

µ 1

0 0

s s

k, 1) (m

(m), = ≤ < <∞

′ ⁺

∫

^∞ (3.18)

where

{ }

^dx.

F(x) F(x) f(x)

log x (y)

I y

1 k m

o ⁼

∫

^{∞ − −}

Upon using the relation in (3.13) in the above expression, yields

{

^logF(x)

}

^dx.

x β (x)

I y

1 m k

o ⁼

∫

^{∞ − −}

Integrating by parts treating x^k−1 for integration and the rest of the integrand for differentiation in each part of the right hand side, the integral, Io(y), can be written as follows

{ } { }

dx.

F(x) F(x) f(x)

log k x

m F(x) β

log ky

(x) β

I y

1 k m

k m

o ^{=− − +}

∫

^{∞ − −}

Upon substituting the above expression of Io (y) instead of I(y) in (2.2.7), we obtain

{ }

{ }

^f(y)dxdy.

F(x) F(x) f(x)

log y Γ(m) x

k m β

dy f(y) F(x) log Γ(m) y

k µ β

0 y

1 s m

k

m 0

s k s

k, 1) (m (m),

∫ ∫

∫

∞ ∞ −

∞ +

+

− +

−

−

′ =

The relation (3.15) is proved simply upon using the definition of the single and double moments of lower record values in the above equation and simplifying the resulting expression to get

k β m , kµ β m

β

µ_(m),^k,s_(m1) m ′_(m^ks₁₎ >

= −

′ ₊ ⁺₊ .

(ii) For 1≤m≤n−2

{ } { }

∫

^{∞ − − × − + − −}

= y

1 m n 1

k m logF(y) logF(x) dx.

F(x) F(x) f(x)

log x I(y)

Upon using the relation in (2.2.4) in the above expression, yields

{

^logF(x)

} {

^{logF(y) logF(x)}

}

^dx.

x β I(y) y

1 m n 1 m

∫

^∞ k^{− − − + − −}

= (3.19)

For the first part I(y), upon integrating by parts treating x^k-1 for integration and the rest of the integrand for differentiation, we obtain

{ } { }

( ) { } { }

F(x) dx.

f(x)

F(x) log F(y) log F(x)

log k x

1 m n β F(x) dx

f(x)

F(x) log F(y) log F(x)

log k x

m I(y) β

y

2 m n k m

y

1 m n 1

k m

×

+

−

− −

− −

×

+

−

−

=

∫

∫

∞ − −

∞ − − −

Upon substituting the above expression of I(y) into (2.2.7), we obtain

{ } { }

( ) { } { }

dy.

dx F(x) f(y)

f(x)

F(x) log F(x) log F(x)

log y m) x

Γ(m)Γ(n k

1 m n β

dy dx F(x) f(y)

f(x)

F(x) log F(y) log F(x)

log y m) x

Γ(n Γ(m) k

m µ β

0 y

2 m n s m

k

1 m n 1

m

0 y

s k s

k, (n) (m),

×

+

−

− −

−

− −

×

+

−

− −

′ =

∫ ∫

∫ ∫

∞ ∞ − −

−

−

∞ ∞ −

The relation in (3.16) is proved simply upon using the definition of the single and double moments of lower record values in the above equation and simplifying the resulting expression to get

k.

β m , β µ

m β k

µ_(m^k,s_1),(n) m − ′_(m),^k,s_(n) >

′ ₊ = Remarks

(a) If β=1, we get the recurrence relations for single and double moments of lower record value from the inverted exponential distribution.

k.

n , n µ

k

µ_(n^k₁₎ n  ′_(n)^k >







 −

′₊ = (3.20)

k.

m , kµ m

µ_(m),^k,s_(m1) m ′_(m^{k s}₁₎ >

= −

′ ₊ ⁺₊ (3.21) k.

m , m µ

k

µ_(m^k,s_1),(n) m− _(m),′^k,s_(n) >

′ ₊ = (3.22)

(b) If β=2, we get the recurrence relations of single and double moments of lower record value from the inverted Rayleigh distributions.

k.

n , n µ

2 k n

µ_(n^k₁₎ 2  ′_(n)^k >







 −

′ ₊ = (3.23)

k.

m , kµ m 2

m

µ_(m),^k,s_(m1) 2 _(m′^{k s}₁₎ >

= −

′ ₊ ⁺₊ (3.24) k.

m , m µ

2 k m

µ_(m^k,s_1),(n) 2 − ′_(m),^k,s_(n) >

′ ₊ = (3.25)

12

4.

SOME NUMERCAL RESULT

some numerical studs with the comments of the recurrence relations for single and double moments of lower record values from inverted Weibull distribution (3.14), and (4.58), in Table (4.1), and (4.2).

Table (4.1): Single Moments of the Lower Record Values from Doubly Truncated Inverted Weibull Distribution

7 6

5 4

3 2

1 k n

β

,

θ

22.57 23.938

25.647 27.878

30.975 35.74

44.117 2.5, 0.02

1

28.265 29.468

30.941 32.817

35.341 39.061

45.543 3.5, 0.02

33.501 34.487

35.676 37.162

39.118 41.912

46.569 5, 0.02

9.028 9.575

10.259 11.151

12.39 14.295

17.814 2.5, 0.05

11.306 11.787

12.377 13.127

14.136 15.624

18.227 3.5, 0.05

13.4 13.795

14.27 14.865

15.647 16.765

18.628 5, 0.05

4.514 4.788

5.129 5.576

6.195 7.148

8.925 2.5, 0.1

5.653 5.894

6.188 6.563

7.068 7.812

9.114 3.5, 0.1

6.7 6.897

7.135 7.432

7.824 8.382

9.314 5, 0.1

7 6

5 4

3 2

1 k n

β

,

θ

520.99 588.22

678.71 807.99

1.01(10³) 1.38(10³)

1.74(10³) 2.5, 0.02

2

807.998 879.82

972.433 1.098(10³)

1.281(10³) 1.582(10³)

2.187(10³) 3.5, 0.02

1.128(10³) 1.197(10³)

1.282(10³) 1.394(10³)

1.549(10³) 1.787(10³)

2.233(10³) 5, 0.02

83.359 49.114

108.594 129.278

161.598 220.359

310.698 2.5, 0.05

129.28 140.771

155.589 175.665

204.943 253.165

353.299 3.5, 0.05

180.558 191.501

205.18 223.021

247.802 285.925

357.403 5, 0.05

20.84 23.529

27.148 32.32

40.399 55.09

81.816 2.5, 0.1

32.32 35.193

38.897 43.916

51.236 63.291

88.508 3.5, 0.1

45.14 47.875

51.295 55.755

61.95 71.481

89.351 5, 0.1

Comment: It is noted that, from Table (4.1), as β increases the single moments increases. While, as n increases the single moment's decreases. And, as θ increases the single moments decreases. If k increases the single moment's increases.

13

Table(4.2): Double Moments of the Lower Record Values from Inverted Weibull Distribution

K =1 and S =1

β

,

θ

m 1 2 3 4 5 6

n

2.5, 0.02

2 1.23×10³

3 501.897 1.798×10³

4 145.537 471.078 2.026×10³

5 32.771 100.513 389.489 1.991×10³

6 6.03 17.908 65.793 302.577 1.714×10³

7 0.938 2.726 9.711 42.368 215.711 1.303×10³

3.5, 0.02

2 1.705×10³

3 731.166 3.041×10³

4 220.119 834.413 5.652×10³

5 51.024 184.865 1.137×10³ 1.024×10⁴

6 9.611 33.924 199.695 1.631×10³ 1.693×10⁴

7 ^{1.524 5.29 30.391 237.647} 2.235×10³ 2.512×10⁴

5, 0.02

2 2.058×10³

3 921.261 5.012×10³

4 285.797 1.432×10³ 2.045×10⁴

5 67.762 327.246 4.287×10³ 9.637×10⁴

6 12.996 61.485 777.806 1.604×10⁴ 4.352×10⁵

7 ^{2.092 9.77 121.346} 2.42×10³ 6.018×10⁴ 1.793×10⁶

2.5, 0.05

2 316.645

3 121.912 1.738×10³

4 ^{34.619 403.11} 1.433×10⁴

5 ^{7.716 82.645} 2.393×10³ 1.263×10⁵

6 ^{1.411 14.459 387.593} 10665×10⁴ 1.032×10⁶

7 ^{0.219 2.179 56.204} 2.241×10³ 1.128×10⁵ 7.585×10⁶

3.5, 0.05

2 345.39

3 ^144.425 2.638×10³

4 ^{43.115 665.138} 8.013×10⁴

5 ^{9.954 143.943} 1.468×10⁴ 3.279×10⁶

6 ^{1.871 26.147} 2.523×10³ 4.779×10⁵ 1.299×10⁸

7 0.296 4.055 380.717 6.837×10⁷ 1.575×10⁷ 4.694×10⁹

5, 0.05

2 ^356.662

3 ^158.549 3.902×10³

4 ^49.078 1.062×10³ 9.386×10⁵

5 ^{11.624 239.91} 1.883×10⁵ 4.08×10⁸

6 ^{2.228 44.854} 3.39×10⁴ 6.541×10⁷ 1.772×10¹¹

7 ^{0.359 7.108} 5.273×10³ 9.813×10⁶ 2.367×10¹⁰ 7.083×10¹³

2.5, 0.1

2 ^87.191

3 ^33.021 1.135×10³

4 ^{9.335 247.664} 4.562×10⁴

5 ^{2.077 50.16} 7.177×10³ 2.217×10⁶

6 ^{0.38 8.732} 1.152×10³ 2.768×10⁵ 1.016×10⁸

7 0.059 1.313 166.529 3.698×10⁴ 1.054×10⁷ 4.208×10⁹

3.5,0.1

2 88.205

3 36.776 1.608×10³

4 10.97 393.767 4.696×10⁵

5 2.531 84.745 8.436×10⁴ 2.149×10⁸

6 0.476 15.361 1.447×10⁴ 3.083×10⁷ 1.148×10¹⁰

7 0.075 2.379 2.181×10³ 4.405×10⁶ 9.606×10¹⁰ 3.922×10¹³

5, 0.1

2 89.34

3 39.707 2.285×10³

4 12.29 614.773 1.509×10⁷

5 2.911 138.557 3.018×10⁶ 2.094×10¹¹

6 0.558 25.882 5.433×10⁵ 3.354×10¹⁰ 2.91×10¹⁵

7 0.09 4.099 8.451×10⁴ 5.027×10⁹ 3.881×10¹⁴ 3.721×10¹⁰

14

Comment: It is noted that, from Table (4.2), as βand m increases the double moments increases. While, as n and θ increases the double moment's decreases.

CONCLUSIONS

Some contributions based on record values have been made. The contributions included work on the single, and double moments of record values.

Some properties for inverted Weibull distributions were discussed. Several recurrence relations were established for the moments of record values from inverted Weibull distribution.

General conclusions from this study can be summarized as follows:

pdf's and joint pdf's are defined for lower record values arising from inverted Weibull distribution.

The connection between the cdf and the pdf of the inverted Weibull distribution has been established to institute the recurrence relations for moments of record values.

Several recurrence relations are established for the single and double moments of lower record values arising from inverted Weibull distribution

( inverted exponential, and inverted Rayleigh distributions as special cases).

The verification from legitimacy of the recurrence relations for single and double moments of lower record values from inverted Weibull distribution, by using Mathcad program was completed. And we checked the recurrence relations for moments, then we made sure were accepted .

15

REFERENCES

Ahsanullah, M. (1995). Record Statistics, Nova Science Publishers, Inc., New York.

Ahsanullah, M., and Bhoj, D.S. (1996). Record values of extreme value distributions and a test for domain of attraction of Type I extreme value distribution, Ind. J. Statist., Series B, 58(2),151-158.

AL-Saleh, J.A., and Agarwal, S.K. (2006). Extended Weibull Type distribution and finite mixture of distributions, Statist. Metho.,3, 224-233.

Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1998). Records, John Wiley

& Sons, New York.

Bain, L.J. (1978). Statistical Analysis of Reliability and Life-Testing Models ( Theory and Method), Marcel Dekker, Inc., New York.

Balakrishnan, N., and Ahsanullah, M. (1994). Recurrence relations for single and product moments of record values from the Lomax distribution, and generalized Pareto distribution, Commun. Statist.-Theor. Meth.,23(10)2841-2852.

Balakrishnan, N., and Ahsanullah, M. (1995). Relations for single and product moments of record values from exponential distribution, J. Appl. Statist. Scie. 2(1), 73-87.

Balakrishnan, N., and Chan, P.S. (1998). On the normal record values and associated inference, Statist. Probab. Lett., 39, 73-80.

Balakrishnan, N., Ahsanullah, M., and Chan, P.S. (1992). Relations for single and product moments of record values from Gumbel distribution, J. Appl. Statist. Scie., 15, 223-227.

Balakrishnan, N., Chan, P.S. and Ahsanullah, M., (1993). Recurrence relations of record values from generalized extreme value distribution, Commun. Statist.-Theor.

Meth.,22(5), 1471-1482.

Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Continuous univariate distributions,2, 2^nd ed., JohnWiley & Sons, Inc., New York.

Kundu, D., and Raqab, M.Z. (2005). Generalized Rayleigh distribution: different methods of estimation, Comput. Statist. Da. Anal.,49, 187-200.

Mohammad, H.H. (2003). A study on the moments of record values and associated inference, M. D. Thesis, AL-Azhar University Girls Branch, Cairo, Egypt.

16

Mahmoud, M.A.W., Sultan, K.S., and Amer, S.M. (2003). Order statistics from inverse Weibull distribution and associated inference, Comput. Statist. Da. Anal.,42, 149-163.

Nadarajah, S. (2005). On the moments of the modified Weibull distribution, Reliab.

Engine. Syst.,90, 114-117.

Nassar, M.M, and Eissa, F.H. (2003). On the exponentiated Weibull distribution, Commun. Statist.-Theor. Meth.,32(7),1317-1336.

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.-Theor. Meth.,28(7),1699-1709.

Pawlas, P., and Szynal, D. (2000). Recurrence relations for single and product moments of k^th record values from Weibull distribution, and a characterization, J. Appl. Statist. Scie.10(1), 17-26.

Pawlas, P., and Szynal, D. (2001). Recurrence relations for single and product moments of generalized order statistics from Pareto, generalized Pareto, and Burr distributions, Commun. Statist.-Theor. Meth.,30(40),739-746.

Soliman, A.A., Abd Ellah, A.H., and Sultan, K.S. (2005).Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches, Comput. Statist. Da. Anal.(in press).

Stewart, J. (1994). Calculus, 3^rd edition, Brooks/Cole, New York.

Sultan, K.S. (2002). Moments of record values from uniform distribution and associated inference, Egypt. Statist. J. ISSR, UNIV.,44(2),137-149.

Sultan, K.S., Ismail, M.A., and AL-Moisheer, A.S. (2006). Mixture of two inverse Weibull distributions: properties and estimation, Compu. Statist. Da. Anal.( in press).

Sultan, K.F., Mosherf, M.E., and Chils, A. (2003). Record values from generalized power function distribution and associated inference, J. Appl. Statist. Scie. Journal of Faculty of Commerce AL-Azhar University Girls Branch,18, 451-480.

Raqab, M.Z., and Ahsanullah, M. (2003). On moment generating function of record from extreme value distribution, Pak. J. Statist.,19(1), 1-13.

17 Appendix

Result 4.1

For real m,k with m≥−1,k ≥1, and for integers r, j≥1, the relation for single moments of GOSs within a class of DT distribution(1.2.30) is given by

[ ] [ ]

{

^{( ) (} ; , , ⁾

}

^,

)) ( 1 ( , ,

; ,

,

; 1 , ,

;

, ,

;

k m n r X

k m n r r

j k m n r j

k m n

r j PE X e _rnmk E X

η γ η

µ

µ′ − ′₋ = ^λ − (4.37)

where

) (

) ) (

(

2 1 ,

,

; x

x x X_rnmk ^j

λ η λ

= ⁻ ′ is a continuous function.

Proof

Let X has the pdf (4.16), then from (4.31) in the double truncated case, we obtain

[

^{F (x)}

]

^{f (x)dx.}

(x)) (F g Γ(r) x

µ C ^P _dt ^{γ 1} _dt

Q dt

1 r m 1 j

j r k m, n, r;

1 r 1

− −

−

∫

′ = (i)

Integrating by parts treating

[

^{F (x)}

]

^fdt^(x) 1 γ dt

r−

for integration and the rest for differentiation, then the integral can be written as follows

[ ]

[

^{F (x)}

]

^{f (x)dx.}

(x)) (F g r) x

( γ

C 1) (r

dx (x) F (x)) (F g ) x

r ( γ

C µ j

1

1

r 1

1

r

P

Q dt

γ dt dt 2 r m j r

1 r

P Q

γ dt dt 1 r m 1 j r

1 j r

k m, n, r;

∫

∫

− −

−

− −

Γ + −

= Γ

′

which can be written as

[ ] [ ]

[

^{F (x)}

]

^{f (x)dx.}

(x)) (F g ) x 1 r ( γ

C

dx (x) F (x) F (x)) (F g ) x

r ( γ

C µ j

1

1

1 - r 1

1

r

P

Q dt

1 γ dt dt 2 r m j r

1 r

P

Q dt

1 γ dt dt 1 r m 1 j r

1 j r

k m, n, r;

∫

∫

− −

−

− −

− −

− + Γ

= Γ

′

(ii)

Substituting from (4.20), in (ii), we get

[ ]

dx.

(x) (x) f λ

(x) P λ

(x) F (x)) (F g ) x

r ( γ

C µ j

µ

dt 2 2

P Q

γ 1 dt dt 1 r m 1 j r

1 j r

k m, n, 1;

- r j

k m, n, r;

1

1

r









− ′

×

= Γ

− ′

′ ⁻

∫

^{− − −}

(iii)

From (4.17), we can write P2 equal to e f (x) (x)

λ (x) Pλ ^(x) _dt

2 1 λ

′ .

Substituting the above result in (iii), we obtain

[ ]

[

^{F (x)}

]

^{f (x)dx.}

(x)) (F (x) g

λ (x) λ x ) r ( γ

C j

dx (x) f (x) F (x)) (F g (x) e

λ (x) λ x ) r ( γ

PC µ j

µ

1

1

r 1

1

r

P

Q dt

1 γ dt dt 1 r m 2 1 j

r 1 r

P

Q dt

1 γ dt dt 1 r m λ(x) 2 1

1 j

r 1 r j

k m, n, 1;

- r j

k m, n, r;

∫

∫

− −

−

−

− −

−

−

Γ ′

− Γ ′

′ =

′ −

Hence,

18

(x) . λ

(x) λ E x (x) e

λ (x) λ PE x γ µ j

µ

2 1 j λ(x)

2 1 1 j

r j

k m, n, 1;

-