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Unbiased Estimation of P(X > Y) forExponential Populations Using OrderStatistics with Application in Ranked SetSamplingSamindranath Sengupta a & Sujay Mukhuti ba Department of Statistics , Calcutta University , Kolkata, Indiab Department of Statistics , St. Xavier's College , Kolkata, IndiaPublished online: 18 Mar 2008.

To cite this article: Samindranath Sengupta & Sujay Mukhuti (2008) Unbiased Estimation ofP(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling,Communications in Statistics - Theory and Methods, 37:6, 898-916, DOI: 10.1080/03610920701693892

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Communications in Statistics—Theory and Methods, 37: 898–916, 2008Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920701693892

Ordered Data Analysis

Unbiased Estimation of P�X > Y� for ExponentialPopulations Using Order Statistics with Application

in Ranked Set Sampling

SAMINDRANATH SENGUPTA1 ANDSUJAY MUKHUTI2

1Department of Statistics, Calcutta University, Kolkata, India2Department of Statistics, St. Xavier’s College, Kolkata, India

This paper addresses the problem of unbiased estimation of P�X > Y� = � fortwo independent exponentially distributed random variables X and Y . We present(unique) unbiased estimator of � based on a single pair of order statistics obtainedfrom two independent random samples from the two populations. We also indicatehow this estimator can be utilized to obtain unbiased estimators of � when onlya few selected order statistics are available from the two random samples as wellas when the samples are selected by an alternative procedure known as ranked setsampling. It is proved that for ranked set samples of size two, the proposed estimatoris uniformly better than the conventional non-parametric unbiased estimator andfurther, a modified ranked set sampling procedure provides an unbiased estimatoreven better than the proposed estimator.

Keywords Exponential population; Order statistic; Ranked set sampling; Simplerandom sampling; Unbiased estimation.

Mathematics Subject Classification Primary 62D05; Secondary 62F10.

1. Introduction

Consider two independent random variables X and Y following exponentialdistributions with finite unknown means �1�> 0� and �2�> 0�, to be writtenhenceforth as exp��1� and exp��2� distributions, respectively. The problem isto estimate unbiasedly P�X > Y� = E�F�Y�� = E�G�X�� = 1

1+�= �, where F�x� and

G�y� are respectively cumulative distribution functions (c.d.f’s) of X and Y , F�x� =1− F�x�, G�y� = 1−G�y� and � = �2

�1. The problem is of importance and has

Received July 6, 2006; Accepted June 22, 2007Address correspondence to Samindranath Sengupta, Department of Statistics,

Calcutta University, 35 Ballygunge Circular Road, Kolkata 700019, India; E-mail:samindras@yahoo.co.in

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Estimation of P�X > Y� for Exponential Populations 899

applications in different fields like reliability, biometry etc (see Gross and Clark,1975; Kotz et al., 2003).

Let X1� X2� � � � � Xn and Y1� Y2� � � � � Ym be independent observations on X andY respectively. A simple unbiased estimator of � based on these two independent(simple) random samples (SRS) from the two populations is

TSRS =1nm

n∑i=1

m∑j=1

I�Xi > Yj� (1)

with variance (see Kotz et al., 2003, Chapter 5)

V�TSRS� =1nm

��+ �n− 1�E�F2�Y��+ �m− 1�E�G2�X��− �n+m− 1��2�

= 1nm

[�+ �n− 1�

�1+ 2��+ 2�m− 1�

�1+ ���2+ ��− �n+m− 1��2

](2)

where I�A� is the indicator function of the set A. The uniformly minimum varianceunbiased estimator (UMVUE) of � is given by (Tong, 1974, 1975)

T ∗SRS =

m−2∑i=0

�−1�i�n�

�n+ i+ 1��m�

�m− i− 1�

(SxSy

)i+1

� if Sx ≤ Sy�

n−1∑i=0

�−1�i�n�

�n− i�

�m�

�m+ i�

(Sy

Sx

)i

� if Sx > Sy�

(3)

where Sx =∑n

i=1 Xi and Sy =∑m

j=1 Yj . An expression for the variance of theUMVUE is given in Sinha and Zielinski (1997).

In many practical situations, however, instead of complete random samplesonly k and l �1 ≤ k ≤ n� 1 ≤ l ≤ m� selected order statistics from them viz.Xi1n

� Xi2n� � � � � Xikn

and Yj1m� Yj2m� � � � � Yjlm, �1 ≤ i1 < i2 < · · · < ik ≤ n� 1 ≤ j1 <j2 < · · · < jl ≤ m� are, respectively available for the two populations, where Xrn

and Ysm are, respectively, the rth and the sth order statistics based on independentrandom samples of sizes n and m from exp��1� and exp��2� populations. Someparticular cases of interest are as follows:

1. Only a single pair of order statistics, say �Xrn� Ysm�1 ≤ r ≤ n� 1 ≤ s ≤ m isavailable.

2. k consecutive order statistics starting from the rth one (1 ≤ r ≤ n− k+ 1) i.e.,Xrn� Xr+1n� � � � � Xr+k−1n are available for X and l consecutive order statisticsstarting from the sth one (1 ≤ s ≤ m− l+ 1), i.e., Ysm� Ys+1m� � � � � Ys+l−1m areavailable for Y .

3. k sporadic non-consecutive order statistics for X, i.e., Xi1n� Xi2n

� � � � � Xikn(ir+1 �=

ir + 1� r = 1� 2� � � � � k− 1) and l sporadic non-consecutive order statistics for Yi.e., Yj1m� Yj2m� � � � � Yjlm (js+1 �= js + 1� s = 1� 2� � � � � l− 1) have been observed.

Samples of type 2 with r = s = 1 are known as Type II censored samples wherecensoring results in omission of the largest �n− k� and �m− l� observations fromthe two samples. Similarly, samples of type 1 arise in the context of stress-strengthreliability estimation of components on the basis of system life observations in an�n− r + 1�-out-of-n system and �m− s + 1�-out-of-m system in which lives of thecomponents of the systems are independently and identically distributed.

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900 Sengupta and Mukhuti

Although the problem of unbiased estimation of the mean of an exponentialpopulation based on such selected order statistics have been widely discussed inthe literature (see Johnson et al., 1994, Chapter 19), not much attention has beenpaid on the problem of unbiased estimation of functions of means of one or moreexponential populations except for samples of type 2 with r = s = 1. Only recentlythe problem of unbiased estimation of the c.d.f of a single exponential populationhas been studied in Sinha et al. (2006). In this article we consider the problemof unbiased estimation of � based on such selected order statistics and present inSec. 2 the (unique) unbiased estimator based on a single pair of order statistics, i.e.,samples of type 1 along with a few properties of the estimator for some special cases.We also indicate in Sec. 3 how this unbiased estimator can be utilized to obtainunbiased estimators of � for samples of other types mentioned above.

The unbiased estimator based on a single pair of order statistics has yet anotherpossible application. In situations where the experimental or sampling units can beeasily ranked by judgment without actual measurements, an alternative samplingscheme often used to achieve more efficiency as compared to simple randomsampling is as follows. The procedure is originally due to McIntyre (1952) and isknown as ranked set sampling (RSS). In this procedure, to select a sample of size nfrom an exp��1� population, one first selects n independent sets of SRS each of size nfrom the population but observes only the ith smallest observation from the ith set,1 ≤ i ≤ n. An RSS of size n from the population is thus given by �X11� X22� � � � � Xnn�where Xir is the rth order statistic from the ith set, 1 ≤ i� r ≤ n. Similarly, a RSS ofsize m from an exp��2� population is given by �Y11� Y22� � � � � Ymm�� where Yjs is thesth order statistic from the jth set in m-independent sets of SRS from the populationeach of size m, 1 ≤ j� s ≤ m. It may be noted that although ranked set samplingrequires identification of n2 and m2 sampling units for the two populations, only nand m of them are actually measured, thus making a comparison of this procedurewith simple random sampling of same number of units meaningful. We refer toChiuv and Sinha (1998) and Chen et al. (2004) for a comprehensive review of thisprocedure.

Now suppose that two independent RSS of sizes n and m are selectedfrom exp��1� and exp��2� populations, respectively. Clearly Xii’s and Yjj’s areindependently distributed and marginally Xir is distributed as Xrn and Yjs isdistributed as Ysm. Hence it is easy to verify that

TRSS =1nm

n∑i=1

m∑j=1

I�Xii > Yjj� (4)

is an unbiased estimator of � and has variance given by

n2m2V�TRSS� = nm�+ E

[n2

( m∑j=1

F�Yjj�

)2

−n∑

i=1

( m∑j=1

Fi�Yjj�

)2]

+ nE

[m2G2�X�−

m∑j=1

G2j �X�

]− n2m2�2 (5)

where Fi�x� and Gj�y� are respectively the c.d.f’s of Xin and Yjm, Fi�x� = 1− Fi�x�

and Gj�y� = 1−Gj�y�. Further, it is known that V�TRSS� is smaller than V�TSRS�uniformly in � unless n = m = 1 (see Sengupta and Mukhuti, 2006). Numerical

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Estimation of P�X > Y� for Exponential Populations 901

comparison of the variances of TRSS and T ∗SRS , presented in Table 1 of Appendix 2,

however, reveal that V�TRSS� is smaller than V�T ∗SRS� only for a range of values of �.

For exponential populations it is quite natural to search for an improvedunbiased estimator of � whose variance is uniformly smaller than that of thenonparametric unbiased estimator TRSS and, hence, is also smaller than the varianceof T ∗

SRS at least for an wider range of values of �. In Sec. 4 we suggest an alternativeunbiased estimator of � based on RSS data using the results of Sec. 2 and provethat the estimator TRSS can, in fact, be improved upon for n = m = 2. We mayremark that in RSS the sample sizes are usually kept small to avoid errors in rankingby judgment. For samples of size two, we further suggest a modification of theRSS procedure, which provides an unbiased estimator of �, whose variance is evensmaller than the improved unbiased estimator based on RSS. Similar results for theproblem of unbiased estimation of the c.d.f of a single exponential population aregiven in Sinha et al. (2006).

2. Unbiased Estimation Based on a Single Pair of Order Statistics

Let Xrn and Ysm be the rth and sth order statistics obtained from independentrandom samples of sizes n and m from exp��1� and exp��2� populations respectively.We first prove the following theorem on unbiased estimation of � based on the pair�Xrn� Ysm�.

Theorem 2.1. The unbiased estimator of � based on the pair �Xrn� Ysm�, denoted byhrs�Xrn� Ysm�, is uniquely defined and satisfies the identity

I�x > y� = 1B�r� n− r + 1�B�s�m− s + 1�

×r−1∑i=0

s−1∑j=0

�−1�i+j� r−1i �� s−1

j �hrs

(x

�n−r+i+1� �y

�m−s+j+1�

)�n− r + i+ 1��m− s + j + 1�

(6)

where B�a� b� is the beta function.

Proof. Since hrs�Xrn� Ysm� is unbiased for �, we have

1�1�2

∫ �

0

∫ �

0I�x > y�e

−(

x�1

+ y�2

)dx dy = � = E�hrs�Xrn� Ysm��

= 1�1�2B�r� n− r + 1�B�s�m− s + 1�

×∫ �

0

∫ �

0hrs�x� y�e

−(�n−r+1�x

�1+ �m−s+1�y

�2

)�1− e

− x�1 �r−1�1− e

− y�2 �s−1dx dy

= 1�1�2B�r� n− r + 1�B�s�m− s + 1�

×∫ �

0

∫ �

0hrs�x� y�

r−1∑i=0

s−1∑j=0

�−1�i+j

(r − 1i

)(s − 1j

)e−(�n−r+i+1�x

�1+ �m−s+j+1�y

�2

)dx dy

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902 Sengupta and Mukhuti

= 1�1�2B�r� n− r + 1�B�s�m− s + 1�

×∫ �

0

∫ �

0

r−1∑i=0

s−1∑j=0

�−1�i+j � r−1i �

�n− r + i+ 1�

� s−1j �

�m− s + j + 1�e−(

x�1

+ y�2

)

×hrs

(x

n− r + i+ 1�

y

m− s + j + 1

)dx dy ∀ 0 < �1� �2 < �

which gives (6). Uniqueness of the estimator follows from the completeness of theexponential family of distributions (see Lehmann, 1983, Theorem 5.6, Chapter 1).

The identity (6) can be solved to obtain hrs�Xrn� Ysm�. Following Sinha et al.(2006), the estimator can, however, be obtained in a simpler alternative way asfollows.

Let us first consider the smallest order statistic for each of the two randomsamples i.e., X1n and Y1m. Then h11�Z1n�W1m� = h11, is simply the indicatorfunction I�Z1n > W1m�, where Z1n = nX1n and W1m = mY1m. Further, the varianceof the estimator is ��1− ��, since Z1n and W1m are independent random variablesfollowing exp��1� and exp��2� distributions respectively.

We next consider the rth �r ≥ 2� order statistic for one population, say exp��1�,and the smallest order statistic for the other, i.e., the pair �Xrn� Y1m�� r ≥ 2 andpresent in the following theorem (unique) unbiased estimator of � based on Zrn =�n− r + 1�Xrn and W1m defined above.

Theorem 2.2. The (unique) unbiased estimator of � based on the pair �Zrn�W1m� is

hr1 = hr1�Zrn�W1m� =�∑

i1=0

�∑i2=0

· · ·�∑

ir−1=0

di1i2���ir−1I�Zrn >

i11

i22 � � �

ir−1r−1W1m� (7)

where i = n−r+i+1n−r+1 � 1 ≤ i ≤ r − 1� di1i2���ir−1

= �−1��1ik �i1+i2+···+ir−1�!(n

r−1

)i1!···ir−1!

(( r−11

) 1

)i1(( r−1

2

) 2

)i2 · · ·(( r−1r−1

) r−1

)ir−1

and �1 stands over all even suffixes of i.

Proof. From Theorem 2.6 of Sinha et al. (2006) we get E�hr1 �W1m� = F�W1m�whence the theorem follows, since W1m has the same distribution as that of Y . �

In particular, for r = 2, the estimator in (7) reduces to

h21 = h21�Z2n�W1m� =1n

�∑i=0

1 iI�Z2n > iW1m�

=

0� if Z2n ≤ W1m

1n

k∑i=0

1 i

= 1− 1 k+1

� if kW1m < Z2n ≤ k+1W1m� k = 0� 1� 2� � � �(8)

where = nn−1 . An expression for the variance of h21 is presented in the next

theorem. It appears difficult to obtain a tractable expression for the variance ofhr1� r > 2.

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Estimation of P�X > Y� for Exponential Populations 903

Theorem 2.3. The variance expression of h21, defined in (8), is

V�h21� =1nE

[F�Y�+ + 1

2

�∑i=1

1 2�i−1�

F� iY�

]− �2

= 1n

[�+ + 1

2

�∑i=1

1 2�i−1��1+ i��

]− �2� (9)

Proof. From Theorem 2.2 of Sinha et al. (2006) we have

E�h221 �W1m� =

1nE

[F�W1m�+

+ 1 2

�∑i=1

F� iW1m�

2�i−1�

]�

which gives (9).

Corollary 2.4. For n ≥ 2, V�h21� < V�h11� uniformly in �.

Proof. Since F� ix� < F�x� ∀x > 0 ∀i ≥ 1,

V�h21� <1n

(1+ + 1

2

�∑i=1

1 2�i−1�

)E�F�Y��− �2 = E�F�Y��− �2 = ��1− �� = V�h11��

Hence the proof follows.

We now consider second order statistic from each of the two samples, i.e., thepair �X2n� Y2m�. In what follows we present the unique unbiased estimator of �based on Z2n = �n− 1�X2n and W2m = �m− 1�Y2m and study a few properties ofthe estimator for some special cases.

Theorem 2.5. The (unique) unbiased estimator of � based on the pair �Z2n�W2m� is

h22 = h22�Z2n�W2m� =1nm

�∑i=0

�∑j=0

1 i�j

I

(Z2n >

i

�jW2m

)� (10)

where = nn−1 and � = m

m−1 .

Proof. Let h22 = 1nm

∑�j=0

gj�j, where

gj = gj�Z2n�W2m� =�∑i=0

1 iI

(Z2n >

i

�jW2m

)� (11)

Then, as in the proof of Theorem 2.2,

E�gj� = nE

[F

(W2m

�j

)]= nmE

[F

(Y

�j

)− 1

�F

(Y

�j+1

)]� (12)

since P�W2m ≤ w� = G2

(w

m− 1

)= m

[G�w�− 1

�G��w�

]� (13)

Hence� E�h22� =�∑j=0

1�j

E

[F

(Y

�j

)− 1

�F

(Y

�j+1

)]= E�F�Y�� = ��

which proves the theorem.

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904 Sengupta and Mukhuti

For m = n, we have = � so that h22 simplifies to

h22 =1

2n− 1

[I�Z2n > W2n�+

�∑i=1

1 i

{I�Z2n > iW2n�+ I

(Z2n >

W2n

i

)}]

=

12n− 1

�∑i=k+1

1 i

= n

�2n− 1� k+1�

if1

k+1W2n < Z2n ≤

1 k

W2n� k = 0� 1� 2� � � �

12n− 1

{ �∑i=1

1 i

+k∑

i=0

1 i

}= 1− n

�2n− 1� k+1�

if kW2n < Z2n ≤ k+1W2n� k = 0� 1� 2� � � �

(14)

The variance of h22 for this special case is derived below.

Theorem 2.6. For n = m �≥2� the variance of h22 is

V�h22� = �− 3 − 1 3� + 1�

E

[ �∑i=1

1 2�i−1�

�G� iX�− F� iY��

]− �2

= �− 3 − 1 3� + 1�

[ �∑i=1

1 2�i−1�

{ i

�+ i− 1

1+ � i

}]− �2� (15)

Proof. Let n = m, i.e., = � and gj be defined as in (11). Then, as in the proof ofTheorem 2.3,

E�g2j � = nE

[F

(W2n

j

)+ + 1

2

�∑i=1

1 2�i−1�

F

(W2n

j−i

)]� (16)

Also it may be easily seen that, for j < j′,

E�gjgj′ � =j′−j−1∑i=0

1 iE�gj�+

1 j′−j

E�g2j �

and hence

n4E�h222� =

�∑j=0

1 2j

E�g2j �+ 2∑j′>

�∑j=0

1 j+j′ E�gjgj′ �

= 2 + 1� + 1�� − 1�

�∑j=0

1 2j

E�g2j �+2 2

� + 1�� − 1�2

�∑j=0

1 2j

E�gj�

which gives upon simplification using (12), (13), and (16).

n2E�h222� =

2

� − 1�2E�h2

22�

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Estimation of P�X > Y� for Exponential Populations 905

= 2 + 1� + 1�� − 1�

E

[ �∑j=0

1 2j

{F

(Y

j

)− 1

F

(Y

j+1

)

+ + 1 2

�∑i=1

1 2�i−1�

{F

(Y

j−i

)− 1

F

(Y

j−i+1

)}}]

+ 2 2

� + 1�� − 1�2E

[ �∑j=0

1 2j

{F

(Y

j

)− 1

F

(Y

j+1

)}]

= 2 + 1� + 1�� − 1�

E

[ �∑j=0

1 2j

{− 1

F

(Y

j+1

)+ 3 − − 1

3F

(Y

j

)

+ � + 1�� 3 − 1� 3

�∑i=1

1 2i

F

(Y

j−i

)}]

+ 2 2

� + 1�� − 1�2E

[F�Y�− � − 1�

�∑j=1

1 2j

F

(Y

j

)]

= 2 + 1� + 1�� − 1�

E

[ � + 1�� 3 − 1�

4 − 1

�∑j=1

1 2j

F� jY�+ 2

2 + 1F�Y�

− � 2 − + 1� 2 + 1

�∑j=1

1 2j

F

(Y

j

)]

+ 2 2

� + 1�� − 1�2E

[F�Y�− � − 1�

�∑j=1

1 2j

F

(Y

j

)]

= 2

� − 1�2E

[F�Y�− 3 − 1

� + 1�

�∑j=1

1 2j

{G� jX�− F� jY�

}]�

since E�F� Y j�� = E�G� jX��� Hence the theorem follows.

Corollary 2.7. For n = m�≥ 2�, V�h22� < V�h21� < V�h11� uniformly in �.

Proof. Let � = V�h21�− V�h22�. Then since E�F� iY�� = E�G�X i�� ∀i ≥ 0, from (9)

and (15), we have

� + 1�� = E

[ 3 − 1 2

�∑i=1

1 2�i−1�

G� iX�+ � − 1�

�∑i=1

1 2�i−1�

G

(X

i

)− � + 1�G�X�

]

= E

[{ 3 − 1 2

�∑i=1

1 2�i−1�

G� iX�−G�X�

}

−

{G�X�− � − 1�

2

�∑i=1

1 2�i−1�

G

(X

i

)}]

= E

[ �∑i=0

1 2i

{G� i+1X�− 1

G� iX�

}−

�∑i=0

1 2i

{G

(X

i

)− 1

G

(X

i+1

)}]

=

nE

[ �∑i=0

1 2i

{G� i+1Z2n�−G

(Z2n

i

)}](17)

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using the fact that P�Z2n ≤ z� = F2�z

n−1 � = n�F�z�− 1 F� z��� Clearly (17) is positive

and this along with Corollary 2.4 proves the result.

For n �= m, it again appears difficult to find an explicit expression for V�h22�.However, in the following theorem we prove that the result in Corollary 2.7 is trueeven when the sample sizes are unequal.

Theorem 2.8. For every n�m ≥ 2, V�h22� < V�h21� < V�h11� uniformly in �.

Proof. Let 1− h22 = 1nm

∑�i=0

g∗i i, where g∗i = g∗i �Z2n�W2m� =

∑�j=0

1�jI�W2m ≥ �j

i

Z2n�. Note that E�g∗i � = mE�G�Z2n i�� and E�g∗i

2� < m2E�G�Z2n i�� as in the proofs

of Theorem 2.2 and Corollary 2.4. Also since 0 ≤ g∗i ≤ �1− 1��−1 = m�∀iE�g∗i g∗j � ≤

mE�g∗i � for i < j. Hence

E�1− h22�2 <

1n2

E

[ �∑i=0

1 2i

G

(Z2n

i

)+ 2

∑j>

�∑i=0

1 i+j

G

(Z2n

i

)]

= E

[1n

�∑i=0

1 iI

(W1m ≥ Z2n

i

)]2

= E�1− h21�2

or equivalently, E�h222� < E�h2

21�. Thus we have V�h22� < V�h21�� The theorem nowfollows from Corollary 2.4.

We finally present below the (unique) unbiased estimator of � based on�Xrn� Ysm� or equivalently �Zrn�Wsm� where Zrn = �n− r + 1�Xrn and Wsm = �m−s + 1�Ysm.

Theorem 2.9. The (unique) unbiased estimator of � based on the pair �Zrn�Wsm� is

hrs = hrs�Zrn�Wsm�

=�∑

i1=0

�∑i2=0

· · ·�∑

ir−1=0

�∑j1=0

�∑j2=0

· · ·�∑

js−1=0

di1i2���ir−1d′j1j2���js−1

I

(Zrn >

i11

i22 � � �

ir−1r−1

�j11 �

j22 � � � �

js−1s−1

Wsm

)

(18)

where i and di1i2���ir−1are defined in Theorem 2.2, �j = m−s+j+1

m−s+1 , 1 ≤ j ≤ s − 1,

d′j1j2���js−1

= �−1��′1jl(

ms−1

) �j1+j2+···+js−1�!j1!j2!���js−1!

(( s−11

)�1

)j1(( s−1

2

)�2

)j2� � �

(( s−1s−1

)�s−1

)js−1

, �′1 stands over all

even suffixes of j.

Proof. Let hrs =∑�

j1=0

∑�j2=0 · · ·

∑�js−1=0 d

′j1j2���js−1

gj1j2���js−1where

gj1j2���js−1= gj1j2���js−1

�Zrn�Wsm�

=�∑

i1=0

�∑i2=0

· · ·�∑

ir−1=0

di1i2���ir−1I

(Zrn >

i11

i22 � � �

ir−1r−1

�j11 �

j22 � � � �

js−1s−1

Wsm

)�

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Estimation of P�X > Y� for Exponential Populations 907

Then as in the proof of Theorem 2.2,

E�gj1j2���js−1� = E

[F

(Wsm

�j11 �

j22 � � � �

js−1s−1

)]�

Now since P�Wsm ≤ w� = Gs

(w

m−s+1

) = (ms−1

)∑s−1j=0

�−1�j

�j

(s−1j

)G��jw� with �0 = 1, we

have

E�hrs� =�∑

j1=0

�∑j2=0

· · ·�∑

js−1=0

d′j1j2���js−1

E

[F

(Wsm

�j11 �

j22 � � � �

js−1s−1

)]

=�∑

j1=0

�∑j2=0

· · ·�∑

js−1=0

ej1j2���js−1E

[F

(Wsm

�j11 �

j22 � � � �

js−1s−1

)]

where e00���0 = 1 and

ej1j2���js−1=

(m

s − 1

){d′j1j2���js−1

−(s−11

)�1

d′j1−1j2���js−1

+(s−12

)�2

d′j1j2−1���js−1

− · · · + �−1�s−1

(s−1s−1

)�s−1

d′j1j2���js−1−1

}

=(

ms − 1

)d′j1j2���js−1

{1− j1 + j2 + · · · + js−1

j1 + j2 + · · · + js−1

}= 0

if �j1j2 � � � js−1� �= �0� 0� � � � � 0��

Thus E�hrs� = E�F�Y �� = � and this proves the theorem.

When a complete random sample is available from one population, say exp��1�,but only a single order statistic is available for the other, say, Ysm, an unbiasedestimator of �, as in Theorem 2.2, is

h�s = h�s�X1�Wsm� =�∑

j1=0

�∑j2=0

· · ·�∑

js−1=0

d′j1j2���js−1

I

(X1 >

Wsm

�j11 �

j22 � � � �

js−1s−1

)�

where Wsm = �m− s + 1�Ysm and �j and d′j1j2���js−1

are defined in Theorem 2.9. SinceSx =

∑ni=1 Xi and Wsm are complete and sufficient for the problem, the UMVUE

of �, obtained through Rao–Blackwellisation, is given by

h∗�s = h∗

�s�Wsm� Sx� = E�h�s � Sx�Wsm�

=�∑

j1=0

�∑j2=0

· · ·�∑

js−1=0

d′j1j2���js−1

(1− Wsm

�j11 �

j22 � � � �

js−1s−1Sx

)n−1

+(19)

where a+ = Max�a� 0�.

3. Unbiased Estimation Based on Selected Order Statistics

We consider in this section the problem of unbiased estimation of � using a few,say k and l �1 ≤ k ≤ n� 1 ≤ l ≤ m� selected order statistics based on independent

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908 Sengupta and Mukhuti

random samples of sizes n and m drawn respectively from exp��1� and exp��2�

populations. Earlier, in Sec. 1, we have described different situations where such aproblem may arise.

We first consider samples of type 2 mentioned in Sec. 1, which consistsof k and l consecutive order statistics obtained from two random samples,i.e., �Xrn� Xr+1n� � � � � Xr+k−1n� and �Ysm� Ys+1m� � � � � Ys+l−1m�� r� s ≥ 1. We note thatX∗

i = �n− i+ 1��Xin − Xi−1n�, i = 1� 2� � � � � n and Y ∗j = �m− j + 1��Yjm − Yj−1m�,

j = 1� 2� � � � � m are independently distributed following exp��1� and exp��2�

distributions respectively, where X0n = Y0m = 0. For r = s = 1, the UMVUE of �,as is well known, is, therefore given by (3) with n�m� Sx, and Sy replaced respectivelyby k� l� S∗

x , and S∗y , where S∗

x = ∑ki=1 X

∗i and S∗

y = ∑lj=1 Y

∗j .

For r = 1� s ≥ 2, an unbiased estimator of � that can similarly be obtained isT ∗1s which is given by (3) with n�m� Sx, and Sy replaced respectively by k� l− 1� S∗

x ,and S∗∗

y , where S∗∗y = ∑s+l−1

j=s+1 Y∗j . Also an unbiased estimator of � based on S∗

x andYsm is h∗

�s�S∗x�Wsm� defined by (19) with n and Sx replaced respectively by k and S∗

x .A combined unbiased estimator of � based on the whole data can, therefore, besuggested as

T ∗∗1s = 1

2�T ∗

1s + h∗�s�S

∗x�Wsm��� (20)

For r� s ≥ 2, a natural unbiased estimator similarly obtained is T ∗rs defined by (3)

with n�m� Sx, and Sy replaced respectively by k− 1� l− 1� S∗∗x , and S∗∗

y , where S∗∗x =∑r+k−1

i=r+1 X∗i . Also an unbiased estimator based on S∗∗

x and Ysm is h∗�s�S

∗∗x �Wsm� defined

by (19) with n and Sx replaced respectively by k− 1 and S∗∗x and an unbiased

estimator based on S∗∗y and Xrn is h∗

r��Zrn� S∗∗y � defined as

h∗r��Zrn� S

∗∗y � = 1−

�∑i1=0

�∑i2=0

· · ·�∑

ir−1=0

di1i2���ir−1

(1− Zrn

i11

i22 � � �

ir−1r−1S

∗∗y

)l−2

+�

where i and di1i2���ir−1are defined in Theorem 2.2. Further, an unbiased estimator

based on Xrn and Ysm is hrs defined in (18). Hence, a pooled unbiased estimator of �can be suggested as

T ∗∗rs = 1

4�T ∗

rs + h∗r��Zrn� S

∗∗y �+ h∗

�s�S∗∗x �Wsm�+ hrs�� (21)

Finally consider a sample of type 3 mentioned in Sec. 1 which comprisesof k sporadic nonconsecutive order statistics Xi1n

� Xi2n� � � � � Xikn

�i1 < i2 < · · · <ik� ir+1 �= ir + 1� r = 1� 2� � � � � k− 1� from the random sample for X and l sporadicnonconsecutive order statistic Yj1m� Yj2m� � � � � Yjlm �j1 < j2 · · · < jl; js+1 �= js + 1� s =1� 2� � � � � l− 1) from the random sample for Y . To construct an unbiased estimatorof � for such a sample note that X∗∗

r = Xir+1n− Xir n

, r = 1� 2� � � � � k− 1 areindependently distributed as the �ir+1 − ir�th order statistic based on a randomsample of size �n− ir� from an exp��1� population independently of Xi1n

and Y ∗∗s =

Yjs+1m− Yjsm, s = 1� 2� � � � � l− 1 are independently distributed as the �js+1 − js�th

order statistics based on a random sample of size �m− js� from an independent

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Estimation of P�X > Y� for Exponential Populations 909

exp��2� population independently of Yj1m. Hence, an unbiased estimator of � isgiven by

T ∗∗∗kl = 1

kl

[hi1j1

�Zi1n�Wj1m

�+k−1∑r=1

h�ir+1−ir �j1�Z∗

�ir+1−ir ��n−ir �� Wj1m

�

+l−1∑s=1

hi1�js+1−js��Zi1n

�W ∗�js+1−js��m−js�

�

+k−1∑r=1

l−1∑s=1

h�ir+1−ir ��js+1−js��Z∗

�ir+1−ir ��n−ir �� W ∗

�js+1−js��m−js��

]� (22)

where Z∗�ir+1−ir ��n−ir �

= �n− ir+1 + 1�X∗∗r , W ∗

�js+1−js��m−js�= �m− js+1 + 1�Y ∗∗

s and thefunction hrs is defined in (18).

4. Application in Ranked Set Sampling

In this section we consider the problem of unbiased estimation of � based ontwo independent RSS of sizes n and m, drawn respectively from exp��1� andexp��2� populations. Recall that RSS of size n from the exp��1� population isgiven by �X11� X22� � � � � Xnn� where Xir is the rth order statistic based on the ithrandom sample of size n in n independent sets of random samples drawn fromthe population, 1 ≤ i� r ≤ n and RSS of size m from the exp��2� population isgiven by �Y11� Y22� � � � � Ymm� where Yjs is the sth order statistic based on the jthrandom sample of size m in m-independent sets of random samples drawn from thepopulation, 1 ≤ j� s ≤ m. Also the observations are independent and marginally Xir

is distributed as Xrn, the rth order statistic in a random sample of size n from anexp��1� population, and Yjs is distributed as Ysm, the sth order statistic in a randomsample of size m from an exp��2� population.

An unbiased estimator of � based on these RSS observations is, therefore,given by

T ′RSS =

1nm

n∑r=1

m∑s=1

hrs�Zrr�Wss� =1nm

n∑r=1

m∑s=1

hrs� (23)

where Zir = �n− r + 1�Xir , Wjs = �m− s + 1�Yjs and hrs is defined in (18). In thefollowing theorem we derive an expression for the variance of T ′

RSS for n = m = 2.

Theorem 4.1. For n = m = 2, the variance of the estimator T ′RSS is given by

16V�T ′RSS� = 4�− 2

3E

[ �∑i=1

122�i−1�

�G�2iX�− F�2iY��]+ 4E�F

2�Y�+G2�X��

−E

[{ �∑i=1

12iF

(Y

2i

)− F�Y�

}2

+{G�X�−

�∑i=1

12iG

(X

2i

)}2]− 12�2�

(24)

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910 Sengupta and Mukhuti

Proof. From the results of Sec. 2 we have E�h211� = � and E�h2

21� and E�h222� are

respectively given by (9) and (15) with n = = 2. Similarly we have,

E�h212� =

32�− 3

8

�∑i=1

122�i−1�

E�G�2iX���

Also simple calculations yield

E�h11h22� = E�h12h21� = �2�

E�h11h12� = E�F2�W11�� = E�F

2�Y���

E�h11h21� = E�G2�Z11�� = E�G2�X���

E�h12h22� =14E

[ �∑i=0

12iF

(W22

2i

)]2

= 14E

2

{ �∑i=0

12iF

(Y

2i

)}2

−{ �∑

i=0

12iF

(Y

2i+1

)}2 [using (13)]

= 14E

2F 2

�Y�− 2

{ �∑i=1

12iF

(Y

2i

)}2

+ 4F�Y��∑i=1

12iF

(Y

2i

)

= E�F2�Y��− 1

2E

[ �∑i=1

12iF

(Y

2i

)− F�Y�

]2

�

and similarly

E�h21h22� = E�G2�X��− 12E

[G�X�−

�∑i=1

12iG

(X

2i

)]2

�

Finally, using the above expressions, we get upon simplification

16E�T ′RSS

2� =

2∑r=1

2∑s=1

2∑r ′=1

2∑s′=1

E�hrshr ′s′ �

= 4�− 23E

[ �∑i=1

122�i−1�

{G�2iX�− F�2iY�

}]+ 4�2 + 4E�F2�Y�+G2�X��

−E

{ �∑

i=1

12iF

(Y

2i

)− F�Y�

}2

+{G�X�−

�∑i=1

12iG

(X

2i

)}2

which proves (24).

Comparing (24) with (2) it is clearly seen that for n = m = 2, V�T ′RSS� is also

uniformly smaller than V�TSRS�. In the following theorem we, in fact, prove that for

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Estimation of P�X > Y� for Exponential Populations 911

n = m = 2, V�T ′RSS� is even uniformly smaller than V�TRSS�. It may be noted that for

n = m = 2, (5) simplifies to

16V�TRSS� = 4�+ 4E�F2�Y�+G2�X��− 4E�F�Y�F�Y��2 − 4E�G�X�G�X��2

− 8E�F�Y�F�Y�G�X�G�X��− 12�2� (25)

Theorem 4.2. For n = m = 2, V�T ′RSS� < V�TRSS� uniformly in �.

The proof of the theorem is given in Appendix 1.

We finally remark that a modification of the rankled set sampling procedureprovides an unbiased estimator of � which is even better than T ′

RSS for samplesof size two. A modified RSS of size two from the exp��1� population is given by�X12� X22� which differs from RSS only in observing X12 instead of X11 from the firstrandom sample of size two from the population. A modified RSS of size two fromthe exp��2� population is similarly given by �Y12� Y22�. For this modified RSS, theestimator

TMRSS =14

2∑i=1

2∑j=1

h22�Zi2�Wj2�� (26)

is unbiased for � and, following the proof of Theorem 4.1, it can be seen that

16V�TMRSS� = 4�− 76E

[ �∑i=1

122�i−1�

{G�2iX�− F�2iY�

}]+ 4E�F2�Y�+G2�X��

− 2E

{ �∑

i=1

12iF

(Y

2i

)− F�Y�

}2

+{G�X�−

�∑i=1

12iG

(X

2i

)}2− 12�2�

(27)

Comparing (27) and (24), it obviously follows that V�TMRSS� < V�T ′RSS�, so that we

have the following theorem.

Theorem 4.3. For n = m = 2, V�TMRSS� < V�T ′RSS� < V�TRSS� < V�TSRS� uniformly

in �.

A direct proof of the fact that TMRSS has a uniformly smaller variance than TRSS

is, however, much simpler, which is indicated below.

Theorem 4.4. For n = m = 2, V�TMRSS� < V�TRSS� uniformly in �.

Proof. Let � = V�TRSS�− V�TMRSS�. Note that

4E��F�Y�F�Y��2 + �G�X�G�X��2 + F�Y�F�Y�G�X�G�X��

≤ 32E[F�Y�F�Y�+G�X�G�X�

](28)

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912 Sengupta and Mukhuti

since F�x�F�x� ≤ 14 ∀x and G�y�G�y� ≤ 1

4 ∀y. Also

E

[ �∑i=1

122�i−1�

{G�2iX�− F�2iY�

}]>

43E�G�2X�− F�2Y�� = 4�

�1+ 2���2+ ��(29)

and

E

{ �∑

i=1

12iF

(Y

2i

)− F�Y�

}2

+{G�X�−

�∑i=1

12iG

(X

2i

)}2

> E

[{F

(Y

2

)− F�Y�

}2

+{G�X�−G

(X

2

)}2]

≥ 2E[{

F

(Y

2

)− F�Y�

}{G�X�−G

(X

2

)}]

= 2E�F�Y�F�Y�G�X�G�X��� (30)

Hence, by (5), (27), (28), (29), and (30), we have 16� ≥ �6 �1+ 2��−1�2+ ��−1 > 0,

which proves the theorem.Tables 2 and 3 in Appendix 2 present numerical comparisons of the variances of

T ′RSS and TMRSS with variances of TRSS and T ∗

SRS . It is seen that the relative efficienciesof T ′

RSS and TMRSS with respect to TRSS in increasing in � for � ≥ 1 and is decreasingin � for � ≤ 1, while the relative efficiencies of TRSS , T

′RSS , and TMRSS with respect to

T ∗SRS is decreasing in � for � ≥ 1 and is increasing in � for � ≤ 1. Also the ranges of

values of � for which the variances of T ′RSS and TMRSS are smaller than the variances

of T ∗SRS is considerably wider than the range of values of � for which the variance

of TRSS is smaller than that of T ∗SRS .

Appendix 1

Proof of Theorem 4.2. Not to obscure the essential steps of the proof we first provesome necessary results in the following lemmas. In what follows we write A =E�F�Y�F�Y��2 = 2�2

�1+2���1+3���1+4�� and B = E�G�X�G�X��2 = 2��2+���3+���4+��

.

Lemma 1.

E�G�2X�− F�2Y�� ≥ 12E�F�Y�F�Y�G�X�G�X��� (A.1)

Proof. Simple calculations give

LHS of (A.1) = 3��2+ ���1+ 2��

≥ 12�2

�1+ ��2�2+ ���1+ 2��= RHS of (A.1)

which proves (A.1).

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Estimation of P�X > Y� for Exponential Populations 913

Lemma 2.

E

{ �∑

j=1

12jF

(Y

2j

)− F�Y�

}2

+{G�X�−

�∑j=1

12jG

(X

2j

)}2

>92E�F�Y�F�Y�G�X�G�X��� (A.2)

Proof. Since F�x� is strictly convex,∑�

j=112j

(F(x2j

))> F

(x3

)for x > 0 and similarly∑�

j=112j

(G(

y

2j

))< G

(y

3

)for y > 0. Hence

LHS of (A.2) > E

[{F

(Y

3

)− F�Y�

}2

+{G�X�−G

(X

3

)}2]

≥ 2E[{

F

(Y

3

)− F�Y�

}{G�X�−G

(X

3

)}]

= 8�2

�1+ ��2�3+ ���1+ 3��

so that

LHS–RHS >�2

�1+ ��2

[8

�3+ ���1+ 3��− 9

2�2+ ���1+ 2��

]

= 5�2�1− ��2

2�1+ ��2�2+ ���3+ ���1+ 2���1+ 3��≥ 0

which proves (A.2).

Lemma 3.

E�G�2X�− F�2Y�� =(92+ �1− ��2

2��1+ ��+ �

1+ �

)A+

(92+ �1− ��2

2�1+ ��+ 1

1+ �

)B�

(A.3)

Proof. We note that

�

�1+ ���1+ 2��− 9

2A = ���1− ��2 + 2�2�

�1+ ���1+ 2���1+ 3���1+ 4��

= 11+ �

{�1− ��2

2�+ �

}A

and similarly, by substituting 1�in place of �,

�

�1+ ���2+ ��− 9

2B = 1

1+ �

{�1− ��2

2+ 1

}B�

Hence LHS of (A.3) = 3��2+���1+2�� = �

1+�

(1

1+2� + 12+�

) = RHS of (A.3) whichcompletes the proof.

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914 Sengupta and Mukhuti

Lemma 4.

E��F�2Y�− F�4Y��+ �G�4X�−G�2X��� =(3+ 1

�

)A+ �3+ ��B� (A.4)

Proof. We note that E�F�2Y�− F�4Y��− 3A = 2��1+2���1+4�� − 3A = A

�and similarly,

by substituting 1�in place of �, E�G�4X�−G�2X��− 3B = �B whence the lemma

follows.

Lemma 5.

E

[ �∑i=3

122�i−1�

{(F�4Y�− F�2iY�

)+ (G�2iX�−G�4X�

)}]

>

(17+ 5

48�

)A+

(17+ 5�

48

)B� (A.5)

Proof. We note that for i ≥ 3,

E�F�4Y�− F�2iY��− 3�2i − 4�2i

A = �2i − 4�{1+ (

5− 32i−1

)�}

2��1+ 2i��A

>�2i − 4�

�

(5

2i+1− 3

22i

)A

so that E

[ �∑i=3

122�i−1�

{(F�4Y�− F�2iY�

)}]>

�∑i=3

[�2i − 4�23i−2

{3+ 1

�

(52− 3

2i

)}]A

=(17+ 11

105�

)A >

(17+ 5

48�

)A�

(A.6)

Similarly we have

E

[ �∑i=3

12�i−1�

{G�2iX�−G�4X�

}]>

(17+ 5�

48

)B� (A.7)

The lemma now follows from (A.6) and (A.7).We now proceed to prove Theorem 4.2. We first note that we can write

�∑i=1

122�i−1�

{G�2iX�− F�2iY�

}

= 43�G�2X�− F�2Y��+ 1

3��G�4X�−G�2X��+ �F�2Y�− F�4Y���

+�∑i=3

12�i−1�

�(G�2iX�−G�4X�

)+ (F�4Y�− F�2iY�

)��

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Estimation of P�X > Y� for Exponential Populations 915

Hence from (24) and (25), using the above lemmas we get

16� = 16�V�TRSS�− V�T ′RSS�� >

4372

E�G�2X�− F�2Y��+ 29

{(3+ 1

�

)A+ �3+ ��B

}

+ 23

{(17+ 5

48�

)A+

(17+ 5�

48

)B

}− 4�A+ B�

= 29

[{4316

(�1− ��2

2��1+ ��+ �

1+ �

)+ 21

16�

}A

+{4316

(�1− ��2

2�1+ ��+ 1

1+ �

)+ 21�

16

}B

]

− 185336

�A+ B�� (A.8)

Now

4316

{�1− ��2

2��1+ ��+ �

1+ �

}+ 21

16�= 85− 44�+ 129�2

32��1+ ��>

52

(A.9)

since �85− 124�+ 49�2� > �9− 7��2 ≥ 0 and similarly

4316

{�1− ��2

2�1+ ��+ 1

1+ �

}+ 21�

16>

52� (A.10)

Finally, by (A.8), (A.9), and (A.10), we have 16� > 51008 �A+ B� > 0, which proves

Theorem 4.2.

Appendix 2

Table 1Table showing the values of the ratio �1��� = V�T∗

SRS�

V�TRSS�

n m � = 0�1 � = 0�4 � = 0�8 � = 0�9 � = 1�0 � = 1�1 � = 1�2 � = 3�5 � = 5�5

2 2 0.573 0.973 1.073 1.078 1.079 1.078 1.075 0.889 0.7563 0.585 1.0499 1.188 1.197 1.201 1.201 1.199 0.961 0.7815 0.589 1.113 1.300 1.316 1.327 1.332 1.334 1.115 0.91510 0.581 1.138 1.362 1.386 1.403 1.415 1.423 1.315 1.159

3 2 0.541 1.072 1.196 1.201 1.201 1.198 1.192 0.944 0.7873 0.542 1.193 1.387 1.397 1.400 1.398 1.391 1.040 0.8155 0.530 1.299 1.596 1.620 1.633 1.639 1.638 1.261 0.97710 0.504 1.340 1.732 1.773 1.802 1.821 1.833 1.600 1.333

5 2 0.623 1.230 1.333 1.332 1.326 1.318 1.306 0.985 0.8063 0.612 1.441 1.636 1.639 1.633 1.621 1.605 1.099 0.8285 0.592 1.675 2.056 2.077 2.083 2.078 2.065 1.394 1.00810 0.537 1.812 2.447 2.509 2.548 2.570 2.580 1.987 1.512

10 2 0.872 1.389 1.426 1.420 1.403 1.388 1.371 0.997 0.8053 0.915 1.742 1.837 1.823 1.802 1.776 1.747 1.107 0.8105 0.912 2.288 2.581 2.572 2.548 2.513 2.471 1.433 0.97410 0.818 2.920 3.774 3.823 3.840 3.826 3.795 2.327 1.566

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916 Sengupta and Mukhuti

Table 2Table showing the ratios �2��� = V�TRSS�

V�T ′RSS�

and �3��� = V�TRSS�

V�TMRSS�

� 0.1 0.4 0�8 0�9 1�0 1�1 1�2 3�5 5�5

�2��� 1.50 1.28 1.24 1.24 1.24 1.24 1.24 1.31 1.38�3��� 2.46 1.76 1.66 1.66 1.66 1.66 1.66 1.86 2.05

Table 3Table showing the ratios �4��� = V�T∗

SRS�

V�T ′RSS�

and �3��� = V�T∗SRS�

V�TMRSS�

� 0.1 0.4 0�8 0�9 1�0 1�1 1�2 3�5 5�5

�4��� 0.86 1.24 1.33 1.34 1.34 1.34 1.34 1.17 1.04�5��� 1.41 1.71 1.78 1.79 1.79 1.79 1.78 1.65 1.55

Acknowledgment

Research of the second author is supported by Council of Scientific and IndustrialResearch, India (Sanction no. 9/28(566)/2002/EMR-I). We also thank the refereefor his helpful comments.

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