on generalized order statistics from exponential distribution

2006, Vol. 40, No. 2, pp. 21–27 Bangladesh

ON GENERALIZED ORDER STATISTICS FROM EXPONENTIAL DISTRIBUTION

M. Ahsanullah

Department of Management Sciences

Rider University, Lawrenceville, NJ 08648-3099, USA

summary

Let X be a non-negative random variable (r.v.) with cumulative distribution function (cdf) F. SupposeX(r, n, m, k), (r= 1,2, . . . , n), m is a real number and k≥1, is the rth smallest generalized order statistic. Several distributional prop- erties of the generalized order statistics from exponential distribution are given.

Some characterizations of the exponential distribution based on the generalized order statistics are presented.

Keywords and phrases: Generalized Order Statistics, Exponential Distribution, Characterization.

1 Introduction

Let X be a random variable (r.v.) whose probability density function (pdf) f is given by

f(x) = 8

<

:

θe^−θx, x >0, θ >0, 0, otherwise.

(1.1) We denote Xi∈E(θ) if the pdf of Xiis as given in (1.1). SupposeX(1, n, m, k), . . . , X(n, n, m, k), (k≥1,m >−1 is a real number), are n generalized order statistics. Their joint pdff1,2,...,n(x1, x2, ..., xn) can be written as (see Kamps 1995, pp 50 51)

f1,2,...,n(x1, x2, ..., xn) = 8

>>

><

>>

>:

kQn−1

j=1γjQn−1

i=1(1−F(xi))^mf(xi)(1−F(xn))^k−1f(xn) F⁻¹(0)< x1< ... < xn< F⁻¹(1),

0, otherwise

(1.2)

whereγj=k+ (n−j)(m+ 1) andf(x) = _dx^dF(x). If m = 0 and k = 1, then X(r,n,m,k) reduces to the ordinary rth order statistic and (1.2) is the joint pdf of the n order statistics,X1,n< .. < Xn,n.If k = 1 and m = -1, then (1.2) is the joint pdf of the first n upper record values of the i.i.d. r.v.’s with distribution function F(x) and pdf f(x). For various distributional properties of order statistics and

c

Institute of Statistical Research and Training (ISRT), University of Dhaka, Dhaka 1000, Bangladesh.

upper record values, see David ( 1981) and Ahsanullah (2004) respectively. Galambos and Kotz (1978) discussed extensively the characterization of exponential distribution by order statistics.

Rossberg ( 1972) characterized the exponential distribution by considering the independence of Xr,n−Xr−1,n and Xr−1,n for 1 < r < n. Ahsanullah ( 1978 a) gave a characterization of the exponential distribution by the independence of X_U(n)- X_U(n−1)and X_U(n−1),n >1, where X_U(n) is the nth upper record value. In this paper some characterizations of the exponential distribution based on the generalized order statistics are presented.

2 Main Results

The marginal pdffr,n,m,k(x) of X(r,n,m,k) can be written as fr,n,m,k(x) = cr−1

(r−1)1(1−F(x))^γr−1f(x)g^r−1m (F(x)), (2.1) where

gm(x)) = 1

m+ 1(1−(1−x)^m+1, m >−1 =−ln(1−x), m=−1, x∈(0,1) andcr−1=γ1γ2...γr.

Since limm→−1 1

m+1(1−(1−x)^m+1=−ln(1−x), we will takegm(x)) = _m+1¹ (1−(1−x)^m+1) for all m with limm→−1gm(x)) =−ln(1−x).

The cdf ofFr,n,m,k(x) of X(r,nm,k) is Fr,n,m,k(x) =

Z x

−∞

cr−1

(r−1)1(1−F(u))^γr−1f(u)gm^r−1(F(u))du

= It(r, γr

m+ 1)

= Γr(α(x)), m=−1, wheret= 1−(1−F(x))^m+1,It(u, v) = _B(u,v)¹ Rt

ow^u−1(1−w)^v−1dw,is the incomplete Beta function, α(x) =−ln(1−F(x)) and Γr(y) = _Γr¹ Ry

0 u^r−1e^−udu.

Form >−1, we have

Fr,n,m,k(x)−Fr+1,n,m,k(x) = It(r, γr

m+ 1)−It(r+ 1, γr+1

m+ 1)

= It(r, γr

m+ 1)−It(r+ 1, γr

m+ 1−1)

= Γ(r+_m+1^γr )

Γ(r+ 1)Γ(_m+1^γr )t^r(1−t)^{m+1γr −1}

= γ1γ2...γr

r!(m+!)^r[1−(1−F(x))^m+1]^r(1−F(x))^m+1]

γr+1 m+1

= cr−1

r! (1−F(x))^γr+1[1−(1−F(x))^m+1

m+! ]^r

= fr+1,n,m,k(x)(1−F(x))

γr+1f(x) , if f(x)6= 0.

Form=−1,

Fr,n,m,k(x)−Fr+1,n,m,k(x) = fr+1,n,m,k(x)(1−F(x))

f(x) , f(x)6= 0

The joint pdffr,s,n,m,k(x, y) of X(s,n,m,k) and X(r,n,m,k),r < ss is given by fs,r,n,m,k(x, y) = c_s−1

(r−1)1(s−r−1)!(1−F(x))^mf(x)g^r−1m (F(x))

× [h(y)−h(x)]^s−r−1(1−F(y))^γs−1f(y), x < y, whereh(x) = _m+1¹ (1−x)^m+1form6=−1 andh(x) =−ln(1−x) form=−1, x∈(0,1).

The conditional pdf of X(s,n,m,k) given X(r,n,m,k)=x, 1≤r < s≤n,has the following form fs|r,n,m,k(y|x) = cs−1

cr−1(s−r−1)!(hm(F(y))−hm(F(y)))^s−r−1

× (1−F(y))^γs−1

(1−F(x))^γr+1f(y), y≥x.

The conditional pdf of X(r+1,n,m,k) given X(r,n,m,k) is fr+1|r,n,m,k(y|x) =γr+1

„1−F(y) 1−F(x)

«γ_r+1

f(y)

1−F(y), y≥x. (2.2)

Theorem 2.1: If Xk∈E(θ), k = 1,2,... and X’s are independent, thenγ1(X(1, n, m, k)∈E(θ) and X(r,n,m,k)=^d θ

r

P

j=1 W_j

γ_j, where W^′jsare i.i.d with cdfF(x) = 1−e^−x, x≥0.

Proof. See Ahsanullah (2000).

It follows from Theorem 2.1. that Dr+1 = γr+1(X(r+ 1, n, m, k)−X(r, n, m, k)) and Dr = γr(X(r, n, m, k)−X(r−1, n, m, k)), r≥ 1, with X(0,n,m,k) =0, are identically distributed.The following Theorem gives a characterization of the exponential distribution using this property.

An absolutely continuous ( with respect to Lebesgue measure) cumulative distribution function F with probability density function f, the ratio _1−F^f(x)_(x) for 1−F(x) >0 is called hazard rate. We will say F blongs to class C if r(x) is either monotone increasing or decreasing.We write Dr = γr{X(r, n, m, k)−X(r−1, n, m, k)}, r >1.

Theorem 2.2: Let X be a non-negative r.v. having an absolutely continuous (with respect to Lebesgue measure) strictly increasing distribution function F(x) with F(0) = 0 and F(x)<1 for allx >0. Then the following properties are equivalent

(a) X has an exponential distribution with density as given in (1.1);

(b) For some fixed r, (1≤r < n), the statistics Dr+1and Dr, r≥1, are identically distributed and F belongs to class C.

Proof: From Theorem 2.1, it is easy to show that (a) =⇒(b). We will prove here that (b) =⇒(a).

P(Dr+1≥x) = 1−FD_r+1(x)

= cr−1

(r−1)!

Z ∞ x

Z∞ 0

(1−F(y))^mf(y)g_m^r−1(F(y))

× (1−F(y+ z γr+1

))^γr+1−1f(y+ z γr+1

))dzdy

= cr−1

(r−1)!

Z ∞ 0

(1−F(y))^mf(y)g^r−1m (F(y))

× int^∞x (1−F(y+ z γr+1

))^γr+1−1.f(y+ z γr+1

))dzdy

= cr−1

r!

Z ∞ 0

(1−F(y))^mf(y)g^r−1m (F(y))

× (1−F(z+ x γr+1

))^γr+1dy (2.3)

Similarly, for r≥2,

P(Dr≥x) = 1−FD_r(z)

= cr−2

(r−2)!

Z ∞ 0

(1−F(y))^mf(y)g^r−2m (F(y))

× (1−F(y+ z γr+1

))^γr+1−1.f(y+ z γr

))dzdy

= cr−2

(r−1)!

Z ∞ 0

(1−F(y))^mf(y)g^r−2_m (F(y))

× Z ∞

x

(1−F(y+ z γr+1

))^γr+1−1.f(y+ z γr

))dzdy

= cr−2

(r−2)!

Z ∞ 0

(1−F(y))^mf(y)g^r−2m (F(y))(1−F(y+ x γr

))^γrdy

= cr−1

(r−1)!

Z ∞ 0

g^r−1m (F(y))(1−F(y+ x γr

))^γr−1f(y+ x γr

))dy

= cr−1

(r−1)!

Z ∞ 0

g^r−1m (F(y))(1−F(y+ x γr

))^γrr(y+ x γr

))dy (2.4)

The relation (2.4) is also true for r=1. Since Dr+1and Drare identically distributed, we must have from (2.3) and (2.4).

0 = cr−1

(r−1)!

Z ∞ 0

gm^r−1(F(y))[(1−F(y))^mf(y) (1−F(z+ x γr+1

))^γr+1

− (1−F(y+ x γr

))^γrr(y+ x γr

)]dy (2.5)

If F has increasing hazard rate, then ln (1-F) is concave and we have for m≥1, ln(1−F

„ z+ x

γr

«

) = ln{1−F

„(m+ 1)z γr

+γr+1

γr

(z+ x γr+1

)

« }

≥ m+ 1 γr

ln(1−F(z)) +γr+1

γr

ln(1−F

„ z+ x

γr+1

« ) i.e. (1−F

„ z+ x

γr

«

)^γr ≥(1−F(z))^m+1(1−F

„ z+ x

γr+1

« )^γr+1

(2.6) Substituting (2,6) in (2.5), we obtain

0 = cr−1

(r−1)!

Z ∞ 0

g^r−1m (F(y))[(1−F(y))^mf(y)

× (1−F(z+ x γr+1

))^γr+1−(1−F(y+ x γr

))^γrr(y+ x γr

)]dy

≤ cr−1

(r−1)!

Z ∞ 0

g^r−1m (F(z))(1−F(z))^m+1(1−F(z+ x γr+1

))^γr+1[r(z)−r(z+ x γr

)]dx

≤ 0.

Thus (2.5) to be true we must haver(z) =r(z+_γ^x

r) for all z and almost all x.Hence the hazard rate is constant and F(x) = 1-e^−θx,for all x≥0 and any realθ >0. If F has decreasing hazard rate, then we obtain similarly the equalityr(z) =r(z+_γ^x

r) for all z and almost all x.

If k = 1 and m = -1, then from the Theorem 2.2, we obtain the result based on r upper records ( see Ahsanullah (1978 a)).. If k = 1 and m = 0, then we obtain the characterization result of the exponential distribution given by Ahsanullah ( 1978 b) based on the identical distribution of XU(n)

- X_U(n−1)and X_U(n+1)−X_U(n).

It can be that under the condition of the existence of the first moment, the condition (b) of Theorem 2.2 can be replaced by the equality of E(Dr+1) and E(Dr). However we need the condition that F belongs to class C.For an alternative proof of the characterization of the exponential distribution using the equality property of E(Dr+1) and E(Dr),see Cramer, Kamps and Raqab(2000).

Let F be the distribution function of a nonnegative random variables. We will call F ”new better than used” (NBU) if 1−F(x+y)≤(1−F(x))(1−F(y)), x, y≥0,and F is ”new worse than used”

(NWU) if 1−F(x+y) ≥(1−F(x))(1−F(y)), x, y≥0.We will say that F belongs to the class C1 if F is either NBU or NWU. The following theorem gives a characterization of the exponential distribution if F belongs to class C1.

Theorem 2.3.

Let X be a non-negative r.v. having an absolutely continuous (with respect to Lebesgue mea- sure) strictly increasing distribution function F(x) with F(0) = 0 andF(x)<1 for allx >0. Then the following properties are equivalent

(a) X has an exponential distribution with density as given in (1.1);

(b) For some fixed r, (1≤r < n), the statisticsDr+1/γr+1and X(1,n-r,m,k ) are identically distributed and F belongs to class C1.

Proof.

From (2.3) , we have P(Dr+1≥xγr+1) = cr−1

r!

Z ∞ 0

(1−F(y))^mf(y)gm^r−1(F(y))(1−F(x+x))^γr+1dy (2.7) Substituting F(x) =1- e^−θxin (2.7), we obtain

P(Dr+1≥xγr+1) = cr−1

r!

Z ∞ 0

e^−mθyθe^−θy [ 1

m+ 1(1−e^−θ(m+1)y]^r−1e^{−θγr+1(x+y)}dy

= cr−1

r! e^−θγr+1x Z∞

0

θe^−yθγr[ 1

m+ 1(1−e^−θ(m+1)y]^r−1dy

= e^−θγr+1x. We now prove that (b) =⇒(a).

From (2.3) , we have

P(Dr+1≥xγr+1) = cr−1

r!

Z∞ 0

(1−F(y))^mf(y)g_m^r−1(F(y))(1−F(y+x))^γr+1dy

= Z ∞

0

fr,n,m,k(y)(1−F(y+x))^γr+1/(1−F(y))^γr+1dy (2.8) The pdf of X(1,n-r,m,k) is

f1,n−r,m,k(x) =γr+1(1−F(x))^γr+1f(x) (2.9) Thus

P(X(1, n−r, m, k)≥x) = (1−F(x))^γr+1 (2.10) Since P(Dr+1≥xγr+1) =P(X(1,n-r,m,k)≥x),we obtain from (2.6) and (2.10)

0 = Z ∞

0

fr,n,m,k(y)[(1−F(y+x))^γr+1/(1−F(y))^γr+1−(1−F(x))^γr+1]dy (2.11) If F belongs to class C1,then for (2.11) to be true , we must have

(1−F(y+x))^γr+1/(1−F(y))^γr+1 = (1−F(x))^γr+1∀x≥0 and almost all y≥0. (2.12) The solution of (2.12) ( see Acz´el, J. (1966)) is

F(x) = 1−e^−θx, x≥0 and any arbitrary, θ >0.

References

[1] Acz´el, J. (1966). Lectures on Functional Equations and Their Applications. Academic Press, New York.

[2] Ahsanullah, M. (1978a ). Record values and the exponential distribution. Ann. Inst. Statist.

Math. 30, A, 429-433.

[3] Ahsanullah, M. (1978b ).On a characterization of the exponential distribution by spacings.

Ann. Inst. Statist. Math. 30, A, 163-166.

[4] Ahsanullah,M. (2000). Generalized order statistics from exponential distribution. JSPI 85,85- 91.

[5] Ahsanullah,M. (2004).Record Values - Theory and Applications. University Press of America, Lanham, Maryland, USA.

[6] Cramer, E, Kamps,U. and Raqab,M.Z. (2003).Characterizations of exponential by spacings of generalized order statistics. Applications Mathematicae, 30,3, 257-265.

[7] David, H.A. (1981). Order Statistics (second edition). Wiley, New York.

[8] Galambox, J. and Kotz, S. (1978). Characterizations of Probability Distributions. Lecture Notes in Mathematics, No. 675, Springer Verlag, 1978, New York.

[9] Kamps, U. (1995). A concept of generalized order statistics. B. G. Teubner Stuttgart, Germany.

[10] Rossberg, H.J. (1977). Characterization of distribution functions by the independence of order statistics, , A, 34, 111- 120.