Journal of Statistical Planning andInference 136 (2006) 555–569

www.elsevier.com/locate/jspi

An aging notion derived from the increasing convexordering: the NBUCA class

I.A. Ahmada,∗, A. Ahmedb, I. Elbatalb, M. KayidcaDepartment of Statistics and Act. Science, University of Central Florida, Orlando, FL 32816-2370, USA

bDepartment of Mathematical Statistics, Institute of Statistical Studies and Research, Cairo University, EGYPTcDepartment of Mathematics, Faculty of Education (Suez), Suez Canal University, EGYPT

Received 28 August 2003; accepted 7 July 2004Available online 12 October 2004

Abstract

In this paper, comparison between the life distribution of a new unit with that of the remaining lifeor a used unit in the increasing convex order leads us to introduce a new class of life distributionswhich we call new better than used in increasing convex average order and denote by NBUCA. Thisclass includes as subclasses the new better than used, NBU and the new better than used in increasingconvexordering,NBUC.Several propertiesof this classarepresented, including thepreservationunderconvolution, random maxima, mixing and formation of coherent structures. Stochastic comparisonsof the excess lifetime at different times of a renewal process when the interarrival times belong to theNBUCA class are established. We provide also some applications of Poisson shock models. Finallytesting exponentially against NBUCA is presented.© 2004 Elsevier B.V. All rights reserved.

MSC:primary: 62G15; 60E15; secondary: 6205

Keywords:Classes of life distributions; NBU, NBUC and NBUCA; Increasing convex order; Convolution;Mixing; Shock models; Life testing

1. Motivation and definitions

In the context of lifetime distributions, some orderings of distributions have been usedto give characterizations and new definitions of aging classes. By aging we mean the

∗ Corresponding author. Tel.: +1 4078232289.E-mail address:iahmad@mail.ucf.edu(I.A. Ahmad).

0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jspi.2004.07.015

556 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

phenomenon whereby an older system has a shorter remaining lifetime, in some statis-tical sense than a younger one (Bryson and Siddiqui, 1969). One of the most importantapproaches to the study of aging is based on the concept of additional residual life. LetX

be a lifetime random variable such that distribution functionF with F(0)= 0. Given a unitwhich has survived up to timet , itsadditional residual life(Barlow and Proschan, 1981) isgiven by

Xt = [X − t |X> t], t ∈ (�, �),

where� = sup{x : FX(x) = 0} and� = inf {x : FX(x) = 1}.The comparison of the additional residual life at different times has been used to produce

several notionsof aging.Below,wepresent someof the stochastic orderingandaging classesconsidered in this paper (seeShaked and Shanthikumar, 1994for a general reference).

Definition 1.1. LetX andY two non-negative random variables with distribution functionsF(x) andG(x), and survival functionsF(x) andG(x), respectively.X is said to be smallerthanY in the

(i) usual stochastic order, denoted byX�stY , if

F (x)�G(x) for all x;(ii) increasing convex order, denoted byX� icxY , if∫ ∞

x

F (u)du�∫ ∞

x

G(u)du for all x;

(iii) increasing convex average order, denoted byX� icxaY , if∫ ∞

0

∫ ∞

x

F (u)dudx�∫ ∞

0

∫ ∞

x

G(u)dudx for all x. (1.1)

In the economics theory, the above orders are respectively known asfirst-order stochasticdominancedenoted byX FSDY , second-order stochastic dominancedenoted byX SSDY , andweak third-order stochastic dominancedenoted byX WTSD Y (for more details,seeDeshpande et al., 1986; Kaur et al., 1994).

Definition 1.2. A random variableX or F is said to be:

(i) new better than used, denoted byX ∈ NBU, ifXt �stX for all t�0;

(ii) new better than used in the convex order, denoted byX ∈ NBUC, ifXt � icxX for all t�0;

(iii) new better than used in expectation, denoted byX ∈ NBUE, ifE(Xt)�E(X) for all t�0.

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 557

The NBU class, introduced byBryson and Siddiqui (1969)and independently byMarshall and Proschan (1972), has grown to become one of the most studied classes oflife distributions. Several extensions of the NBU class have been proposed in the literature.Cao and Wang (1991)suggested the NBUC class whileMarshall and Proschan (1972)in-troduced the NBUE class. The classes NBU, NBUC and NBUE have proved to be veryuseful in performing analyses of life lengths as well as usable in many replacement policieshence a lot of related results have been obtained in the literature (see, e.g.,Klar and Muller,2003; Utkin and Gurov, 2002; Barlow, 1999).The NBUE class, however, does not correspond to any of the orderings in Definition 1.1.

The NBU class corresponds toPart (i) of Definition 1.1 while the NBUC class correspondsto Part (ii). Hence another class corresponding toPart (iii) of the definition is needed andthat is what we do here. Thus the new class we develop and call “new better than used inincreasing convex average denoted by NBUCA” is an extension of the NBU and NBUCclasses and is different from the NBUE class. Precisely, we have:

Definition 1.3. A randomvariableX orF is said to benewbetter thanused in the increasingconvex average order denoted by NBUCA if

Xt � icxaX,

equivalently,X ∈ NBUCA iff∫ ∞

0

∫ ∞

x

F (u + t)dudx� F (t)

∫ ∞

0

∫ ∞

x

F (u)dudx for all t�0.

The above class expands the NBUC class to much bigger one but in a direction differentthan that of the NBUE class. There are no implications between our new class and that ofthe NBUE. Its dual class is new worse than used in increasing convex average ordering,denoted by NWUCA, which is defined by reversing the above inequality.It is obvious that,X ∈ NBUCA iff

X WTSDXt for all t�0

or

2∫ ∞

0�(x + t)dx��2F (t), (1.2)

where�2= E(X2), assumed finite, and�(x) = ∫ ∞x

F (u)du.To introduce a discrete version of this class defineN ={0,1, . . .} and let{Pk, k ∈ N} be

discrete distribution and definePk =1−Pk andpk = Pk − Pk+1, wherek�0. Suppose alsothat P0 = 1. Observe that our definition ofpk is different from that ofEsary et al. (1973),and corresponds to that ofAbouammoh and Ahmed (1988).

Definition 1.4. A discrete distribution{Pk, k ∈ N}, is said to be discrete NBUCA if,∞∑i=0

∞∑j=k+i

Pj � Pk

∞∑i=0

∞∑j=i

Pj for all k�0. (1.3)

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The paper is organized as follows: In Section 2, several properties of this class are pre-sented, including the preservation under convolution, random maxima, mixing and forma-tion of coherent structures. In that section, we give some reversed preservation propertiesof this class. In Section 3, we add a new result when the stochastic comparisons are given interms of increasing convex average. Some applications of Poisson shock models are givenin Section 4. Finally in Section 5, we address the question of testingH0 : F is exponentialagainstH1 : F ∈ NBUCA and not exponential. To allow for an easy flow of the material,all proofs are deferred to an appendix.

2. Preservation properties

Useful properties of aging classes of life distributions are the closure with respect totypical reliability operations, see, e.g.,Barlow and Proschan (1981). In this section wepresent some preservation results for the NBUCA class.

2.1. Convolution

As an important reliability operation, convolutions of life distributions of a certain classis often paid much attention. The closure properties of NBUC class were pointed out inCao andWang (1991)andHu and Xie (2002). In the next theorem we establish the closureproperty of the NBUCA class under the convolution operation.

Theorem 2.1. Suppose thatF1 and F2 are two independentNBUCA life distributions.Then their convolution is alsoNBUCA.

2.2. Random minima

Preservation properties of some stochastic orders and aging classes as well as their dualsunder random minima and maxima have been studied by several authors. For more details,onemay refer toShaked (1975), Bartoszewicz (2001)andLi and Zuo (2004). In the follow-ing, we give a preservation result for the NBUCA class under formation of randommaxima.First, we give the following preliminary result.

Theorem 2.2. LetX1, X2, . . .andY1, Y2, . . .eachbea sequenceofi.i.d. randomvariables,andN is independent ofXi ’s andYi ’s. If Xi ’s andYi ’s are both non-negative and withcommon left end point0, thenXi � icxaYi for i = 1,2, . . . , implies

max{X1, . . . , XN }� icxamax{Y1, . . . , YN }.

Theorem 2.3. LetX1, X2, . . . be a sequence ofi.i.d. random lives, andN is independentofXi ’s. IfX1 isNBUCA, thenmax{X1, . . . , XN } is also ofNBUCA property.

On the other hand, it is a well known fact that some aging notions are preserved underformation of parallel and/or series system (seeBarlow and Proschan, 1981; Abouammoh

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 559

and El-Neweihi, 1986; Hendi and Mashhour, 1993; Li and Kochar, 2001; Pellerey andPetakos, 2002). As a consequence to Theorem 2.3, Corollary 2.1 below is immediate.

Corollary 2.1. LetX1, X2, . . . , Xn bea set ofNBUCA independent identically distributedcomponents and considerTn =max{X1, X2, . . . , Xn}. ThenTn ∈ NBUCA.2.3. Reversed preservation properties

Recently,Li and Yam (2004)developed some reversed preservation of some stochasticorders and aging conceptions for the parallel (series) systems which are composed of in-dependent and identical elements (i.i.d). Next, we will make a discussion on the reversedpreservation property of the NWUCA aging class of life distributions.

Theorem 2.4. Assume thatX1, . . . , Xn are i.i.d copies ofX. For any integern>1, ifmin{X1, . . . , Xn} isNWUCA, thenX is alsoNWUCA.

2.4. Mixture

The NBUCA class is not closed under mixtures, since mixtures of some exponential lifedistributions often belong to the DFR class (Barlow and Proschan, 1981).

3. Stochastic comparisons of excess lifetimes of renewal processes

Let us consider a renewal process with independent and identically distributed non-negative interarrival timesXi with common distributionF andF(0) = 0. Let S0 = 0andSk = ∑k

i=1Xi and consider the renewal counting processN(t) = Sup{n : Sn� t}.Several papers have investigated some characteristics of the renewal process related toaging properties ofF (see, for instance,Barlow and Proschan, 1981; Shaked and Zhu,1992, Brown, 1980,1981).Chen (1994)investigated the relationship between the behavior of the renewal function

M(t) = E(N(t)) and aging property ofF . Some other results are given for the excesslifetime at timet�0, that is,�(t) = Sn(t)+1 − t , which is the time of the next event a timet . Some examples of such results are the following:

(a) Chen (1994)showed that

(i) if �(t) is stochastically decreasing int�0, thenF ∈ NBU and(ii) if E[�(t)] is stochastically decreasing int�0, thenF ∈ NBUE;

(b) Li et al. (2000)showed that, if�(t) is decreasing int�0 in the increasing convex orderthenF ∈ NBUC.Whereas these results give sufficient conditions for the aging property ofF , in practical

situations it would be more interesting to derive some properties for�(t) from the ageingproperty ofF . In fact, given a renewal process it is more feasible to study it ifF has some

560 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

ageing property than if�(t) has some of the previous properties. A result in such a directionis the following one:

(a) Barlow and Proschan (1981, p. 169)showed that, ifF ∈ NBU, then�(t)�st�(0), forall t�0.

(b) Belzunce et al. (2001)showed that:

(i) if F ∈ NBUE, thenE[�(t)]�E[�(0)], for all t�0;(ii) if F ∈ NBUC, then�(t)� icx�(0), for all t�0.

The following two theorems present the parallel results for the NBUCA class. Since thetwo results can be proved in a similar way to Theorem 7 ofLi et al. (2000)and Theorem2.2 ofBelzunce et al. (2001), respectively, the proofs are omitted.

Theorem 3.1. If �(t) is decreasing int for all t�0 in the increasing convex average orderthenF ∈ NBUCA.

Theorem 3.2. Given a renewal process as above, if F ∈ NBUCA, then

E(�2(t))�E(�2(0)), f or all t�0.

4. Application to shock models

Shockmodels areof great interest in the context of reliability theory.Suppose that adeviceis subjected to shocks occurring randomly as events in a Poisson process with constant�.Suppose further that the device has probabilityPk of surviving the firstk shocks. Then thesurvival function of the device is given by

H (t) =∞∑k=0

Pk

e−�t (�t)k

k! , t�0. (4.1)

For the discrete distribution{Pk, k ∈ N}, it is well known that the properties ofPk

are reflected in corresponding properties of the continues life distributionH(t). For moredetails see for instance,Esary et al. (1973), Klefsjo (1981, 1983)andFagiuoli and Pellerey(1993). Next, we give a similar result for the NBUCA class.

Theorem 4.1. The survival functionH (t) in (4.1) is NBUCA if and only if {Pk, k ∈ N}is discreteNBUCA.

Let N(t) be the number of shocks interval(0, t]. The kth shock arrives at timeTk,Uk+1 = Tk+1 − Tk, k = 0,1, . . . . We assume thatU1, U2, . . . are mutually independentand Limn→∞ Tn = +∞ with probability 1. Let

ak(t) = P [N(t) = k, k = 0,1, . . . , ]

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 561

and

Ak(x) =∫ ∞

x

ak(t)dt.

Proposition 4.1. H ∈ NBUCA if for anyk andx the following conditions hold,

(a) Pk

∑∞j=0 Pj

∫ ∞0 Aj(x)dx�

∑∞j=k Pj

∫ ∞0 Aj(x)dx, and

(b) Akj (x)∑k

j=0 aj (t)�Ak(x + t).

Theorem 4.2. Let Uk ∈ NBUCA, ∫ ∞0 Ak(x)dx decreasing and bounded ink. If Pk is

discreteNBUCA, then the Condition(a)holds.

5. Testing in the NBUCA class

In the context of reliability and life testing, the hazard rate of a life distribution plays animportant role for stochasticmodeling and classification. Being a ratio of probability densityfunction and the corresponding survival function, it uniquely determines the underlyingdistribution and exhibits different monotonic behaviors. The concept of theagelessnotionis equivalent to the phenomenon that age has no effect on the hazard rate. Thus theagelessproperty is equal to constant hazard rate, that is, the distribution is exponential. Hencetesting non-parametric classes is done by testing exponentially versus some kind of classes.This applies to many non-parametric classes such as the NBU, NBUC and NBUE, amongothers.For a recent literature on testing the above classes as well as others we refer the readers

toAhmad (2001),Ahmad and Mugdadi (2004)andKayid andAhmad (2004). Much of theearlier literature is cited in those papers where definitions, inter-relations and discussion ofabove classes and others are presented.The problem we propose in this section is that we want to testH0 : F in exponential

againstH1 : F is NBUCA and not exponential. The following result gives a measure ofdeparture fromH0 in favor ofH1.

Theorem 5.1. Suppose thatF isNBUCA life distribution such that its�r+3 the(r + 3)rdmoment is finite for some integerr�0, then

2�r+3�(r + 2)(r + 3)�2�r+1.

According to the above theorem, we set the following measure of departure fromH0:

�(r) = (r + 2)(r + 3)�2�r+1− 2�r+3.

Note that�(r) = 0 underH0 and is positive forH1. To make the test scale invariant, weadjust it by�r+3 and define the measure

�(r) = �(r)

�r+3 ,

where� = �1.

562 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

To estimate�(r), letX1, . . . , Xn be a random sample from the life distributionF . LetX = 1/n ∑n

i=1Xi be the sample mean. We estimate�(r) by

�(r) = 1

Xr+3n(n − 1)∑i �=j

∑{(r + 2)(r + 3)X2i Xr+1

j − 2Xr+3i }.

Theorem 5.2. Asn → ∞,√n(�(r)−�(r)) is asymptotically normal with mean zero andvariance2(r) given in(A.5).UnderH0, the variance is given by(A.6).

To carry out this test, we calculaten1/2�(r)/0(r) and reject if larger thanZ�, the�thnormal variate. The choice ofr is a question that needs to be addressed. There are threepossible routs. Either to choose a small value liker =0 orr =1 to make calculations simpleor to try to find the value ofr that gives the maximum power or efficiency if we have somebelief about an alternative. To chooser which maximizes the power one can use empiricalcalculations by simulating sampling from an alternative we believe in and calculating theempirical powers for various sample sizes atr = 0,1, . . . etc. and choose ther that givesthe best power.However, choosing ther that maximizes the efficiency is easy to do theoretically if we

use the Pitman notion of efficiency. Let us consider two alternatives that are commonly usedand are in the NBUCA class:

(i) The linear failure rate:

F(1) (x) = e−x−�

2x2, x�0, ��0;

(ii) The Makehaun:

F(2) (x) = e−x−�(x+e−x−1), x�0, ��0.

Note that the null is at�=0 for these alternatives. The Pitman asymptotic efficacy of thetest based on�(r) is given by

PAE(��(r)) = | 1d���(r)|�→�0

0(r)

= |(r + 2)(r + 3)(2�′r+1,�0 + (r + 1)!�′

2,�0) − 2�′

r+3,�0|0(r)

,

where�′r,�0

= | dd��r,�|�→�0.Working with the above two alternatives we get:

PAE (Linear failure rate) = 4(r + 1)(r + 3)!0(r)

(5.1)

and

PAE (Makehaun) = [7− 3(12)r+1](r + 3)!20(r)

. (5.2)

Both (5.1) and (5.2) are decreasing inr. Thusr = 0 is what we choose here.

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 563

In this case the PAEs are respectively equal to 0.894 and 0.615. The corresponding valuesof the NBUE test ofHollander and Proschan (1975)are 0.866 and 0.289. Clearly our test isbetter and also much simpler to do.Ahmad and Mugdadi (2004)gave a test for the NBUCclass which is smaller than one and reported PAEs values of 0.894 and 0.167. Thus our testis better and works for a much larger class, the NBUCA.

Acknowledgements

The authors are indebted to the Editor and two referees for their careful and detailedremarks which helped improve both content and presentation of this work.

Appendix A.

Proof of Theorem 2.1. The survival function of the convolution of two life distributionF1andF2 is

F (w) =∫ ∞

0F1(w − z)dF2(z), for all w�0,

for any fixedt, x�0,∫ ∞

0

∫ ∞

x

F (t + y)dy dx

=∫ ∞

0

∫ ∞

0

∫ ∞

0F1(x + t + y − u)dF2(u)dy dx

=∫ ∞

0

∫ ∞

0

∫ x

0F1(x + t + y − u)dF2(u)dy dx

+∫ ∞

0

∫ ∞

0

∫ ∞

x

F1(x + t + y − u)dF2(u)dy dx

=∫ ∞

0

∫ x

0

[∫ ∞

0F1(x + t + y − u)dy

]dF2(u)dx

+∫ ∞

0

∫ ∞

0

[∫ ∞

0F1(t + y − v)dv F2(v + x)

]dy dx

:= A + B.

Observe that

A� F1(t)

∫ ∞

0

∫ x

0

[∫ ∞

0F1(x + y − u)dy

]dF2(u)dx

� F (t)

∫ ∞

0

∫ ∞

0

∫ x

0F1(x + y − u)dF2(u)dy dx,

564 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

where the first inequality follows from the NBUCA property and the second inequalitytrivially follows from the fact thatF (t)� Fi(t) for all t andi = 1,2. The rest of the proofnow is similar to that of Theorem 2.1 inHu and Xie (2002). �

For the proof of Theorem 2.2 and 2.4 we will need the following lemma. The proof issimilar to Lemma 7.2(a) inBarlow and Proschan (1981).

Lemma A.1. Assume thatW(x) is aLebesgue–Stieltjesmeasure, not necessarily positive.If h(x) is non-negative and increasing, and∫ ∞

0

∫ ∞

t

dW(x)dt�0, f or all t�0,

then∫ ∞0

∫ ∞t

h(x)dW(x)dt�0.

Proof of Theorem 2.2. By (1.1), the orderX1� icxaY1 states that∫ ∞

0

∫ ∞

t

[G(u) − F (u)]dx dt =∫ ∞

0

∫ ∞

t

[F(u) − G(u)]dx dt�0, t�0.

Since the function

Fn−1(x) + Fn−2(x)G(x) + · · · + F(x)Gn−2(x) + Gn−1(x)

is increasing and positive for allx�0, it follows from LemmaA.1 that, for allt�0,∫ ∞

0

∫ ∞

t

[GN :N(u) − FN :N(u)]dx dt

=∫ ∞

0

∫ ∞

t

[FN :N(u) − GN :N(u)]dx dt

=∫ ∞

0

∫ ∞

t

∞∑n=1

pN(n)[Fn(x) − Gn(x)]dx dt

=∫ ∞

0

∫ ∞

t

[ ∞∑n=1

pN(n)

n−1∑k=0

Fn−k−1(x)Gk(x)

][F(x) − G(x)]dx dt�0,

wherepN is the discrete density function ofXi . That is to say,

max(X1, . . . , XN)� icxamax(Y1, . . . , YN). �

Proof of Theorem 2.3.X1 is NBUCA; thus, for allt�0,

(Xi)t � icxaXi, i = 1,2, . . . .

By Theorem 2.2, we have,

max{(X1)t , . . . , (XN)t }� icxamax{X1, . . . , XN }, t�0.

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 565

According toLi and Zuo (2004), it holds that, for any random numberN ,

(max{X1, . . . , XN })t �stmax{(X1)t , . . . , (XN)t }, t�0.

From transitivity, it follows that

(max{X1, . . . , XN })t � icxamax{X1, . . . , XN }, t�0.

That is, max{X1, . . . , XN } is NBUCA. �

Proof of Theorem 2.4.min{X1, . . . , Xn} is NWUCA, it holds that, for anyt�0,

(min{X1, . . . , Xn})t � icxamin{X1, . . . , Xn}.

By the fact that

(min{X1, . . . , Xn})t st=min{(X1)t , . . . , (Xn)t },

we have, for anyt�0,

min{(X1)t , . . . , (Xn)t }� icxamin{X1, . . . , Xn}.

That is, for anyt�0 andx�0,∫ ∞

0

∫ ∞

x

F nt (y)dy dx�

∫ ∞

0

∫ ∞

x

F n(y)dy dx,

equivalently,

∫ ∞

0

∫ ∞

x

[F nt (y) − F n(y)]dy dx =

∫ ∞

0

∫ ∞

x

[Ft (y) − F (y)]h(y)dy dx,

where

h(y) = F n−1(y) + F n−2(y)Ft (y) + · · · + F (y)F n−2t (y) + F n−1

t (y).

Notice that the function[h(y)]−1 is non-negativeand increasing, it is follows fromLemmaA.1 that, for allx�0 andt�0,

∫ ∞

0

∫ ∞

x

[Ft (y) − F (y)]dy dx�0.

Thus,Xt � icxaX, and henceX is also of NWUCA property. �

566 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

In the proof of the following theorem we use the variation diminishing property of thetotally positive kernelk(i, t) = e−�t (�t)k/k! (seeKarlin (1968) for a discussion of thisproperty).

Proof of Theorem 4.1.Observes the following relations, which can be easily verified:

∫ ∞

0

∫ ∞

t

H (s)ds dt = 1

�2

∞∑k=0

∞∑j=k

Pj (A.1)

and

∫ ∞

0

∫ ∞

t

H (x + s)ds dt = 1

�2

∞∑i=0

∞∑k=0

∞∑

j=k+i

Pj

e−�t (�t)i

i! . (A.2)

By assumption, from (1.3) we have

Pk

∞∑i=0

∞∑j=i

Pj −∞∑i=0

∞∑j=k+i

Pj �0.

Since this holds for everyi, k ∈ N , it follows that for everyx�0,

∞∑k=0

∞∑

i=0

∞∑j=i

PkPj −∞∑i=0

∞∑j=k+i

Pj

e−�t (�t)i

i! �0.

By (A.1) and (A.2) this means

∫ ∞

0

∫ ∞

t

H (x + s)ds dt�H (t)

∫ ∞

0

∫ ∞

t

H (s)ds dt for all t�0.

i.e.,H (t) is NBUCA. �

Proof of Theorem 4.2. The theorem is obviously true fork = 0. Fork�1, let

∫ ∞

0A(x)dx = lim

k→∞

∫ ∞

0Ak(x)dx.

I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569 567

we have

Pk

∞∑j=0

Pj

∫ ∞

0Aj(x)dx −

∞∑j=k

Pj

∫ ∞

0Aj(x)dx

= Pk

∞∑j=0

Pj

∫ ∞

0[Aj(x) − A(x)]dx −

∞∑j=k

Pj

∫ ∞

0[Aj(x) − A(x)]dx

= Pk

∞∑j=i

Pj

∞∑j=i

∫ ∞

0[Aj(x) − Ai+1(x)]dx

−∞∑j=k

Pj

∞∑l=j

∫ ∞

0[Al(x) − Al+1(x)]dx

= Pk

∞∑l=0

∫ ∞

0[Al(x) − Al+1(x)]dx −

l∑j=0

∫ ∞

0[Al(x) − Al+1(x)]dx

l∑j=k

Pj

=[Pk

k−1∑l=0

∫ ∞

0[Al(x) − Al+1(x)]dx

]l∑

j=k

Pj

−∞∑l=k

∫ ∞

0[Al(x) − Al+1(x)]dx

Pk

l∑j=0

Pj −l∑

j=k

Pj

�0. �

Proof of Theorem 5.1. From (1.2) we have, for any integerr�0

2∫ ∞

0xr

∫ ∞

x

�(u)du��2

∫ ∞

0xr F (x)dx. (A.3)

But ∫ ∞

0xr F (x)dx = E

∫ x

0xr dx = 1

r + 1 E(Xr+1) = �r+1r + 1. (A.4)

Next, ∫ ∞

0xr

∫ ∞

x

�(u)du =∫ ∞

0�(x)

∫ x

0ur dv dx

=∫ ∞

x

�(x)xr+1

r + 1 dx

= 1

r + 1∫ ∞

0xr+1

∫ ∞

x

F (u)dudx

= 1

r + 1∫ ∞

0F (x)

∫ x

0ur+1 dudx

= 1

(r + 1)(r + 2)∫ ∞

0xr+2F (x)dx

= 1

(r + 1)(r + 2)(r + 3) �r+3.

568 I.A. Ahmad et al. / Journal of Statistical Planning and Inference 136 (2006) 555–569

The result follows from (A.3) and (A.4). �

Proof of Theorem 5.2. The result follows directly from the central limit theory ofU -statistics cf.Lee (1989).We only need to calculate the variance. Let us evaluate the varianceof �(r) where

�(r) = 1

n(n − 1)∑i �=j

∑�r (Xi,Xj ),

where

�r (X1, X2) = (r + 2)(r + 3)X21Xr+12 .

But

E[�r (X1, X2)|X1] = (r + 2)(r + 3)X21�r+1− 2Xr+31 = �(1)

r (X1)

and

E[�r (X2, X1)|X1] = (r + 2)(r + 3)�2Xr+11 − 2�r+3= �(2)

r (X1).

Hence2(r) is given by

2(r) = �−2r−6V {�(1)r (X1) + �(2)

r (X1)}. (A.5)

UnderH0, and by direct calculation, the null variance is equal to

20(r) = (r + 2)(r + 3)(9r2+ 41r + 42)× (2r + 2)! − 4(4r + 9)[(r + 3)!]2. � (A.6)

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