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Unbiased Estimation of the DistributionFunction of an Exponential PopulationUsing Order Statistics with Application inRanked Set SamplingBikas K. Sinha a , Samindranath Sengupta b & Sujay Mukhuti ba Stat-Math Division , Indian Statistical Institute , Kolkata, Indiab Department of Statistics , Calcutta University , Kolkata, IndiaPublished online: 15 Feb 2007.

To cite this article: Bikas K. Sinha , Samindranath Sengupta & Sujay Mukhuti (2006) UnbiasedEstimation of the Distribution Function of an Exponential Population Using Order Statistics withApplication in Ranked Set Sampling, Communications in Statistics - Theory and Methods, 35:9,1655-1670, DOI: 10.1080/03610920600683663

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Communications in Statistics—Theory and Methods, 35: 1655–1670, 2006Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920600683663

Ordered Data and Inference

Unbiased Estimation of the Distribution Functionof an Exponential Population Using Order Statistics

with Application in Ranked Set Sampling

BIKAS K. SINHA1, SAMINDRANATH SENGUPTA2,AND SUJAY MUKHUTI2

1Stat-Math Division, Indian Statistical Institute, Kolkata, India2Department of Statistics, Calcutta University, Kolkata, India

In this paper we consider the problem of unbiased estimation of the distributionfunction of an exponential population using order statistics based on a randomsample. We present a (unique) unbiased estimator based on a single, say ith,order statistic and study some properties of the estimator for i = 2. We alsoindicate how this estimator can be utilized to obtain unbiased estimators when afew selected order statistics are available as well as when the sample is selectedfollowing an alternative sampling procedure known as ranked set sampling. It isfurther proved that for a ranked set sample of size two, the proposed estimator isuniformly better than the conventional nonparametric unbiased estimator, further,for a general sample size, a modified ranked set sampling procedure provides anunbiased estimator uniformly better than the conventional nonparametric unbiasedestimator based on the usual ranked set sampling procedure.

Keywords Distribution function; Exponential population; Order statistics;Ranked set sampling; Simple random sampling; Unbiased estimator.

AMS (2000) Classification: 62D05; 62F10.

1. Introduction

Consider a random variable X following an exponential distribution with finiteunknown mean ��>0�, to be written henceforth as an exp��� distribution. Theproblem is to estimate unbiasedly the distribution function of X at an arbitrarybut fixed finite time point t�>0�, viz., F�t� = P�X ≤ t� = 1− e−a, or equivalentlythe survival or the reliability function �F�t� = P�X > t� = e−a, where a = t/�.

Received April 8, 2005; Accepted January 20, 2006Address 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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1656 Sinha et al.

Let X1� X2� � � � � Xn be n independent observations on X. A simple unbiased estimatorof �F�t� based on this (simple) random sample (SRS) is

TSRS =1n

n∑i=1

I�Xi > t�� (1.1)

where I�A� is the indicator function of the set A. The variance of the estimator is

V�TSRS� =�F�t��1−�F�t��

n� (1.2)

The uniformly minimum variance unbiased estimator (UMVUE) is

T ∗SRS =

(1− t

W

)n−1

+(1.3)

where W = ∑ni=1 Xi and �d�+ = max�d� 0�. The variance of the UMVUE is

V�T ∗SRS� = �F�t��I�a � n�−�F�t�� (1.4)

where I�a � n� is defined as �n�I�a � n� = ∫ �0 e−uu2n−2/�u+ a�n−1du�

In many practical situations, however, instead of a complete randomsample of size n, only k �1 ≤ k < n� selected order statistics from it, viz.,X�i1�n�

� X�i2�n�� � � � � X�ik�n�

� 1 ≤ i1 < i2 < · · · < ik ≤ n are available. Some particularcases of interest are

1. Initial k order statistics, i.e., �X�1�n�� X�2�n�� � � � � X�k�n� are available.2. k consecutive order statistics starting from the ith one �2 ≤ i ≤ n− k+ 1�, i.e.,

�X�i�n�� X�i+1�n�� � � � � X�i+k−1�n� are observed.3. k sporadic nonconsecutive order statistics, i.e., �X�i1�n�

� X�i2�n�� � � � � X�ik�n�

�ir+1 �=ir + 1� r = 1� � � � � k− 1� are observed.

4. Only a single, say the ith, order statistic, i.e., X�i�n�, i = 2� 3� � � � � n is available.

Samples of type 1 are known as Type II censored samples where censoringresults in omission of the largest �n− k� observations. Similarly, a sample of type4 arises in the context of reliability estimation of a component on the basis of asystem life observation in an �n− i+ 1�-out-of-n system in which lives of all then components of the system are independently and identically distributed and thesystem life is the failure time of the ith failed component.

Although the problem of unbiased estimation of � based on such selected orderstatistics has been widely discussed in the literature, not much attention has beenpaid to the problem of unbiased estimation of �F�t� except for samples of type 1 (seeJohnson et al., 1994, Chap. 19). In this paper we consider this problem and presentin Sec. 2 the unique unbiased estimator of �F�t� based on a single, say the ith, orderstatistic along with some properties of the estimator for i = 2. We also indicate inSec. 3 how this unbiased estimator based on a single order statistic can be utilized toobtain unbiased estimators when a few selected order statistics are available, e.g., forsamples of other types mentioned above.

The unbiased estimator based on a single order statistic has yet another possibleapplication. There is (see Chiuv and Sinha, 1998; Sengupta and Mukhuti, 2006;

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Unbiased Estimation of the Distribution Function 1657

Sinha et al., 1996) for estimating the mean or the variance of an exponentialpopulation, an alternative sampling scheme that provides unbiased estimators withvariances uniformly smaller than the UMVUE based on an SRS of size n. Theprocedure is due to McIntyre (1952) and is known as ranked set sampling. Onefirst selects n independent SRS, each of size n, but observes only the ith smallestobservation from the ith set, i = 1� 2� � � � � n. Thus a ranked set sample (RSS) isgiven by �X1�1� X2�2� � � � � Xn�n, where Xi�j is the jth order statistic from the ith set,1 ≤ i, j ≤ n. It may be noted that although RSS requires identification of as manyas n2 sampling units, only n among them are actually measured, thus making acomparison of this procedure with SRS of the same size n meaningful. We refer toChiuv and Sinha (1998) and Chen et al. (2003) for a comprehensive review of thisprocedure.

Now suppose that a sample of size n is selected from an exp��� populationfollowing this alternative procedure. It is easy to verify that

TRSS =1n

n∑i=1

I�Xi�i > t� (1.5)

is an unbiased estimator of �F�t� and has variance

V�TRSS� =1n2

n∑i=1

[�F�i��t�{1−�F�i��t�

}](1.6)

where

�F�i��t� = P�Xi�i > t� =(

ni− 1

) i−1∑j=0

�−1�j

�ij

(i− 1j

)�F��n− i+ 1��ijt� (1.7)

with �ij = n−i+j+1n−i+1 , j = 0� 1� � � � � i− 1. Further, for n ≥ 2, V�TRSS� is smaller than

V�TSRS� uniformly in a (see El-Newihi and Sinha, 2000; Stokes and Sager,1988). However, numerical calculations reveal that V�TRSS� is greater than V�T ∗

SRS�uniformly in a for n = 2 and is smaller than V�T ∗

SRS� only for a small interval ofvalues of a for n ≥ 3. The lower and upper end points of the interval of values of afor which V�TRSS� < V�T ∗

SRS� are presented in Table 1 for different values of n.It is therefore natural to search for an improved unbiased estimator of �F�t�

based on the RSS whose variance is also smaller than V�T ∗SRS�, at least for an wider

interval of values of a. In Sec. 4 we suggest an alternative unbiased estimator of�F�t� based on RSS data using the results of Sec. 1 and prove that the estimatorTRSS can in fact be improved upon for an RSS of size two. We also prove that for

Table 1Table showing interval of values of a for which V�TRSS� < V�T ∗

SRS�

Sample size 2 3 4 5 6 7 8 9 10

LEP — 0.7405 0.6035 0.5395 0.4975 0.4660 0.4415 0.421 0.4040UEP — 1.3605 1.7020 1.9090 2.0590 2.1765 2.2730 2.3540 2.4250

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1658 Sinha et al.

a general sample size n�≥2�, it is possible to obtain an unbiased estimator of �F�t�based on a modified RSS whose variance is uniformly smaller than that of TRSS. Itmay be remarked that the problem of estimation of the distribution function basedon RSS has been studied in the literature by Stokes and Sager (1988) when X has acompletely unknown distribution and by El-Newihi and Sinha (2000) and Pal andSinha (2002) when the parent distribution is exponential or normal.

2. Unbiased Estimation Based on a Single Order Statistic

Let X�i�n� be the ith order statistic based on a random sample of size n drawn froman exp��� population. We first prove the following theorem on unbiased estimationof �F�t� based on X�i�n�.

Theorem 2.1. The unbiased estimator of �F�t� based on X�i�n�, denoted by hi�X�i�n��, isuniquely defined and satisfies the identity

I�x > t� = 1B�i� n− i+ 1�

i−1∑k=0

(i−1k

)�−1�khi

(x

n−i+k+1

)�n− i+ k+ 1�

(2.1)

where B(�� ) is the beta function.

Proof. Since hi�X�i�n�� is unbiased, we have by definition

1�

∫ �

0I�x > t�e−x/�dx = �F�t� = E�hi�X�i�n���

= 1�

∫ �

0hi�x�

(e

−�n−i+1�x�

)B�i� n− i+ 1�

(1− e−

x�

)i−1dx

= 1�

∫ �

0

i−1∑k=0

hi�x�

(i− 1k

)�−1�k

(e

−�n−i+k+1�x�

)B�i� n− i+ 1�

dx

= 1�

∫ �

0

i−1∑k=0

(i−1k

)�−1�khi

(x

n− i+ k+ 1

)B�i� n− i+ 1��n− i+ k+ 1�

e−x/�dx� ∀0 < � < ��

which gives (2.1). Uniqueness of the estimator follows from the completenessproperty of the exponential distribution. �

Solving the above identity one can find hi�X�i�n��. Alternatively, the estimatorcan be obtained as follows.

Let us first consider the smallest order statistic, viz., X�1�n�. Then h1�Z�1�n��is simply the indicator function I�Z�1�n� > t�, where Z�1�n� = nX�1�n�. Further, thevariance of h1�Z�1�n�� is �F�t��1−�F�t��. Next, we consider the second order statisticX�2�n�. In what follows we present the (unique) unbiased estimator of �F (t) based onZ�2�n� = �n− 1�X�2�n� and study a few properties of the estimator.

Theorem 2.2. The (unique) unbiased estimator of �F�t� based on Z�2�n� is

h2

(Z�2�n�

) = �∑k=0

1n�k

I(Z�2�n� > �kt

)� (2.2)

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Unbiased Estimation of the Distribution Function 1659

where � = n/�n− 1�. Further, the variance of h2�Z�2�n�� is given by

Var�h2�Z�2�n��� =1n

{�F�t�+

(1�+ 1

�2

)�F��t�+

(1�3

+ 1�4

)�F��2t�+ · · ·

}−�F 2�t��

(2.3)

Proof. Note that, by (1.7), P�Z�2�n� > t� = �F�2�

(t

n−1

) = n[�F�t�− 1

��F��t�]. Hence

�F�t� = E1n�I�Z�2�n� > t��+ 1

��F��t�

= E1n�I�Z�2�n� > t��+ E

1n�

�I�Z�2�n� > �t��+ 1�2�F��2t�

= · · · = E�∑k=0

1n�k

[I(Z�2�n� > �kt

)]�

Also,

E(h22�Z�2�n��

) = 1n2

{ �∑k=0

1�2k

P(Z�2�n� > �kt

)+ 2�∑

k>

∑l=0

1�k�l

P(Z�2�n� > �kt

)}

= 1n2

{ �∑k=0

1�2k

P(Z�2�n� > �kt

)+ 2�∑k=1

�k − 1�2k−1��− 1�

P(Z�2�n� > �kt

)}

= 1n

{ �∑k=0

1�2k

[�F��kt�− 1

��F��k+1t�

]

+ 2�∑k=1

�k − 1�2k−1��− 1�

[�F��kt�− 1

��F��k+1t�

]}

= 1n

{�F�t�+

�∑k=1

(1

�2k−1+ 1

�2k

)�F��kt�

}upon simplification. Hence follows the result. �

Corollary 2.3. For n ≥ 2, V�h2�Z�2�n��� < V�h1�Z�1�n��� uniformly in a.

Proof. Since �F��kt� < �F�t�� k = 1� 2� � � � � it follows that

V�h2�Z�2�n��� <1n

(1− 1

�

)−1

�F�t�−�F 2�t� = �F�t��1−�F�t� = V�h1�Z�1�n���� �

Corollary 2.4. For n = 2, V�h2�Z�2�n��� <12V�h1�Z�1�n��� uniformly in a.

Proof. For n = 2, �F��kt� < �F��t� = �F�2t�, k = 2� 3� � � � , and hence

V�h2�Z�2�n��� <12��F�t�+�F�2t�−�F 2�t� = 1

2�F�t��1−�F�t� = 1

2V�h1�Z�1�n����

�

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1660 Sinha et al.

Corollary 2.5. An unbiased estimator of the variance of h2�Z�2�n�� is V̂ �h2�Z�2�n��� =h22�Z�2�n��− h2�Z�2�n�/2�, which is uniformly nonnegative for n = 2.

Proof. The unbiasedness of V̂ follows trivially, since V�h2�Z�2�n��� = E�h22�Z�2�n���−�F�2t�.

Now for n = 2,

h2�Z�2�n�� = 0� if Z�2�n� < t

=r∑

k=0

12k+1

= 1− 12r+1

� if 2r t < Z�2�n� < 2r+1t� r = 0� 1� 2� � � �

(2.4)

Similarly,

h2�Z�2�n�/2� = 0� if Z�2�n� < 2t

= 1− 12r� if 2r t < Z�2�n� < 2r+1t� r = 1� 2� � � �

(2.5)

Therefore, for n = 2,

V̂ �h2�Z�2�n��� = 0 if Z�2�n� < t

= 122r+2

if 2r t < Z�2�n� < 2r+1t� r = 0� 1� 2� � � �

(2.6)

Hence the result. �

It may be remarked that for n ≥ 3, the unbiased variance estimator may assumenegative values. For example, the variance estimate is negative in the interval�2t� 9t/4� for n = 3.

In order to obtain an unbiased estimator of �F�t� using X�i+1�n�, we defineZ�i+1�n� = �n− i�X�i+1�n� and present the (unique) unbiased estimator based on Z�i+1�n�

in Theorem 2.6 �0 ≤ i ≤ n− 1�.

Theorem 2.6. The (unique) unbiased estimator of �F�t� based on Z�i+1�n�, denoted byhi+1�Z�i+1�n��, is given by

hi+1�Z�i+1�n�� =�∑

y1=0

�∑y2=0

· · ·�∑

yi=0

dy1y2���yiI(Z�i+1�n� > �

y11 �

y22 � � � �

yii t)� (2.7)

where �k = n−i+kn−i

� k = 1� 2� � � � � i, dy1y2���yi= �−1�

∑1 yi(

ni

) �y1+y2+···+yi�!y1!y2!���yi!

�i1�

y1

�y11

�i2�

y2

�y22

� � ��ii�

yi

�yii

, and∑1 extends over all even suffixes of y.

Proof. The unbiased estimator of �F�t� can be obtained as in the proof of Theorem2.2 using (1.7). We however directly verify the unbiasedness of hi+1�Z�i+1�n�� by

E�hi+1�Z�i+1�n��� =�∑

y1=0

�∑y2=0

· · ·�∑

yi=0

dy1y2���yiP[Z�i+1�n� > �

y11 �

y22 � � � �

yii t]

=�∑

y1=0

�∑y2=0

· · ·�∑

yi=0

ey1y2���yi�F(�y11 �y22 � � � �

yii t)say. (2.8)

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Unbiased Estimation of the Distribution Function 1661

Note that, by (1.7),

P(Z�i+1�n� > �

y11 �

y22 � � � �

yii t) = �F�i+1�

(�y11 �

y22 � � � �

yii t

n− i

)=

(n

i

){�F(�y11 �y22 � � � �

yii t)− (

i

1

)�1

�F(�y1+11 �

y22 � � � �

yii t)

+(i

2

)�2

�F(�y11 �y2+12 � � � �

yii t)− · · ·

+ �−1�i(i

i

)�i�F(�y11 �y22 � � � �

yi+1i t

)}� (2.9)

so that e00···0 = 1 and

ey1y2���yi =(n

i

){dy1y2���yi

−(i

1

)�1

dy1−1y2���yn+

(i

2

)�2

dy1y2−1���yi· · · + �−1�i

1�idy1y2���yi−1

}=

(n

i

)dy1y2���yi

{1− y1 + y2 + · · · + yi

y1 + y2 + · · · + yi

}= 0� for �y1� y2� � � � � yi� �= �0� 0� � � � � 0��

Thus E�hi+1�Z�i+1�n��� = �F�t�. �

For example, the (unique) unbiased estimator of �F�t� based on Z�3�n� =�n− 2�X�3�n� is given by

h3�Z�3�n�� =�∑

y1=0

�∑y2=0

dy1y2I(Z�3�n� > �

y11 �

y22 t

)(2.10)

where

�1 =n− 1n− 2

� �2 =n

n− 2� and dy1y2

= 2n�n− 1�

�−1�y2(y1 + y2

y2

)2y1

�y11 �

y22

�

It may be noted that for i ≥ 2, the estimator hi+1�Z�i+1�n�� is improper in the sensethat it may produce estimates outside the interval �0� 1�. For example, the value ofh3�Z�3�3�� exceeds 1 in the interval �9t� 12t�. Also, it appears difficult to obtain atractable expression for the variance of the estimator. Further, an unbiased varianceestimator is given by V̂ �hi+1�Z�i+1�n��� = h2

i+1�Z�i+1�n��− hi+1�Z�i+1�n�/2�, which alsomay assume negative values.

It is of interest to compare the unbiased estimator hi�Z�i�n�� with an alternativebiased estimator of �F�t�, viz.,

h∗i �X�i�n�� = e

(− c1n�i�t

X�i�n�

)� (2.11)

where c1n�i� is the mean of X�i�n� based on a random sample of size n from an exp�1�distribution. Following Zacks and Even (1966), the mean square error (MSE) ofh∗i �X�i�n��, for i ≥ 2, can be obtained as (see Appendix 1)

MSE�h∗i �X�i�n��� = M∗

i1�2a � n�− 2e−aM∗i1�a � n�+ e−2a� (2.12)

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1662 Sinha et al.

Table 2Table showing the interval of values of a outside which

V�h2�Z�2�n��� < MSE�h∗2�X�2�n���

Sample size 2 3 4 5 6 7 8 9 10

LEP 0.300 0.320 0.335 0.335 0.335 0.335 0.335 0.340 0.340UEP 5.510 4.850 4.540 4.405 4.335 4.295 4.270 4.250 4.240

where

M∗i1�a � n� = E�h∗

i �X�i�n���

= i

(n

i

) i−1∑r=0

�−1�r(i− 1r

)2

√ac1n�i�

n− i+ r + 1K1

(2√ac1n�i��n− i+ r + 1�

)(2.13)

and K1�y� is the modified Bessel function of the second kind of order 1 (seeGradshteyn and Ryzhik, 1994, pp. 968–974). Table 2 shows the interval of values ofa outside which V�h2�Z�2�n�� is smaller than MSE�h∗

2�X�2�n��. The results are similarto those in Zacks and Even (1966), wherein the efficiency of the UMVUE of �F�t�was compared with that of the maximum likelihod estimator for a complete SRS.

3. Unbiased Estimation Based on a Few Selected Order Statistics

We now consider the problem of unbiased estimation of �F�t� using a few, say k �2 ≤k < n�, selected order statistics based on a random sample of size n drawn from anexp��� population, viz., X�i1�n�

� X�i2�n�� � � � � X�ik�n�

, 1 ≤ i1 < · · · < ik ≤ n. Earlier, in theintroduction, we have described different situations in which such a problem mayarise.

We first consider a sample of Type 1, mentioned in the introduction, whichconsists of initial k order statistics, i.e., �X�1�n�� X�2�n�� � � � � X�k�n�. Noting thatY1 = nX�1�n�, Yi = �n− i+ 1��X�i�n� − X�i−1�n��� i = 2� 3� � � � � k, are iid exp��� randomvariables, the UMVUE of �F�t� is (see Johnson et al., 1994, Chapter 19)

T ∗k =

(1− t

Wk

)�k−1�

+where Wk =

k∑r=1

Yr� (3.1)

The expression for the variance of T ∗k is as in (1.4), except that n is replaced by k.

Consider next the problem of unbiased estimation of �F�t� when kconsecutive order statistics, starting from the ith one, are available, i� k ≥ 2. Let�X�i�n�� X�i+1�n�� � � � � X�i+k−1�n� be the sample of consideration. To obtain an unbiasedestimator of �F�t�, let Yr = �n− r − i+ 1��X�r+i�n� − X�r+i−1�n��� r = 1� 2� � � � � k− 1,and note that the Yr’s are iid exp��� random variables that are also distributedindependently of X�i�n�. The UMVUE of �F�t� based on the Yr’s is Tik, which isas in (3.1) except that k is replaced by k− 1. The variance of Tik is obtained byreplacing n by k− 1 in (1.4). The fact that Tik and hi�Z�i�n�� are two independentunbiased estimators of �F�t� makes it tempting to combine the two estimators in a

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Unbiased Estimation of the Distribution Function 1663

suitable way. However, the ratio of their variances takes a very difficult form, andit involves the unknown parameter in a complicated way. So it is not obvious howbest to combine the two. An unbiased estimator of �F�t� based on this whole sampleis therefore simply suggested as

T ′ik =

12�hi�Z�i�n��+ Tik�� (3.2)

whose variance is 14 �Var�hi�Z�i�n��+ Var�Tik��, where hi�Z�i�n�� is defined as in (2.7).

It may be noted that a biased estimator of �F�t� is given by

T ′′ik = e

− t

�̂ik � (3.3)

where �̂ik is the best linear unbiased estimator (BLUE) of � based on a sample ofType 2, viz.,

�̂ik =1�

[c1n�i�

c2n�i�X�i�n� +

k−1∑r=1

Yr

]� (3.4)

where c1n�i� and c2n�i� are the mean and the variance of X�i�n� obtained from anexp(1) distribution and

� = c21n�i�

c2n�i�+ k− 1� (3.5)

The MSE of T ′′ik is given by (see Appendix 1)

MSE�T ′′ik� = Mik�2a � n�− 2e−aMik�a � n�+ e−2a� (3.6)

where Mik�a � n� is defined as

Mik�a � n� = E�T ′′ik�

= i(n

i

)c2n�i�

c1n�i�

i−1∑r=0

�−1�r(i−1r

)�1− �ir�

k−1

×[2√a�

�irK1�2

√�n− i+ r + 1�ac1n�i��

−k−2∑j=0

2�a��j+12 �1− �ir�

j

j! Kj+1�2√a��

]� (3.7)

where �ir = n−i+r+1�

and K��z� is the modified Bessel function of the second kind oforder � (see Gradshteyn and Ryzhik, 1994, pp. 968–974). Table 3 shows the intervalof values of a outside which V�T ′

2k� < MSE�T ′′2k� for different values of n and k.

Finally, consider a sample composed of k sporadic nonconsecutive orderstatistics, viz., X�i1�n�

, X�i2�n�� � � � � X�ik�n�

, where i1 < i2 < · · · < ik� ir+1 �= ir + 1, andr = 1� 2� � � � � k− 1. To construct an unbiased estimator of �F�t� based on sucha sample, note that Yr = X�ir+1�n�

− X�ir �n�, r = 1� 2� � � � � k− 1, are independently

distributed as the �ir+1 − ir�th order statistic based on a random sample of size

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1664 Sinha et al.

Table

3Tab

leshow

inginterval

ofvalues

ofaou

tsidewhich

V�T

′ 2k�<

MSE

�T′′ 2k�fordifferentnan

dk

k=

2k=

3k=

4k=

5k=

6k=

7k=

8k=

9

nLEP

UEP

LEP

UEP

LEP

UEP

LEP

UEP

LEP

UEP

LEP

UEP

LEP

UEP

LEP

UEP

30�000

6�870

40�000

6�810

0�000

9�325

50�000

6�775

0�000

9�270

0�000

6�725

60�000

6�755

0�000

9�240

0�000

6�580

0�000

6�920

70�000

6�745

0�000

9�15

0�000

6�500

0�000

6�765

0�000

8�090

80�000

6�735

0�000

9�200

0�000

6�445

0�000

6�670

0�000

7�910

0�000

10�015

90�000

6�73

0�000

9�190

0�000

6�410

0�000

6�610

0�000

7�790

0�000

9�810

0�000

12�405

100�000

6�725

0�000

9�185

0�000

6�385

0�000

6�565

0�000

7�710

0�000

9�655

0�000

12�260

0�000

13�816

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Unbiased Estimation of the Distribution Function 1665

�n− ir�. Also the Yr’s are independent of X�i1�n�. Hence an unbiased estimator of

�F�t� is given by

Ti1�i2�����ik= 1

k

[hi1

�Z�i1�n��+

k−1∑r=1

hir+1−ir��n− ir+1 + 1�Yr�

](3.8)

with variance 1k2

[V(hi1

�Z�i1�n��)+∑k−1

r=1 V(hir+1−ir

(Z�ir+1−ir �n−ir �

))].

4. Application in Ranked Set Sampling

In this section, we consider the problem of unbiased estimation of �F (t) based on anRSS of size n drawn from an exp��� population. Recall that an RSS of size n isgiven by �X1�1� X2�2� � � � � Xn�n, where Xi�j is the jth order statistic based on the ithrandom sample of size n in n independent sets of random samples drawn from thepopulation, 1 ≤ i� j ≤ n. Clearly, the observations are independent, and marginallyXi�j is distributed as X�j�n�, the jth order statistic in a random sample of size n.An unbiased estimator of �F (t) based on an RSS of size n is therefore given by

T ′RSS =

1n

n∑i=1

hi�Zi�i�� (4.1)

where Zi�i = �n− i+ 1�Xi�i and hi is defined as in (2.7). The variance of T ′RSS is

V�T ′RSS� =

1n2

n∑i=1

V�hi�Zi�i��� (4.2)

and marginally Zi�i is distributed as Z�i�n� defined in Sec. 1. In the following theoremwe prove that for n = 2, the estimator T ′

RSS is uniformly better than TRSS defined in(1.5).

Theorem 4.1. For an RSS of size 2, V�T ′RSS� < V�TRSS� uniformly in a.

Proof. Consider an RSS of size 2. Using (1.5) and (1.6), we get on simplification

4V�TRSS� = 2�F�t�− 4�F�2t�+ 4�F�3t�− 2�F�4t��

Also from (4.2) we have, by Corollary 2.4,

4V�T ′RSS� <

32V�h1�Z1�1�� =

32�F�t��1−�F�t��

Hence

4�V�TRSS�− V�T ′RSS�� >

12�F�t�− 5

2�F�2t�+ 4�F�3t�− 2�F�4t�

= 12�F�t��1−�F�t�− 2�F 2�t��1−�F�t��2

= 12�F�t��1−�F�t��1− 4�F�t��1−�F�t�� ≥ 0�

This proves the theorem. �

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1666 Sinha et al.

In fact, for an RSS of size 2, the unbiased estimator T ′RSS can be further

improved as follows. Consider an estimator of the form

T�RSS = �h1�Z1�1�+ �1− ��h2�Z2�2�� 0 ≤ � ≤ 1� (4.3)

Then V�T�RSS� = �2V�h1�Z1�1��+ �1− ��2V�h2�Z2�2��, and it is easy to verifythat V�T�RSS� is increasing in � for � ≥ V�h2�Z2�2��

V�h1�Z1�1��+V�h2�Z2�2��= �0 (say). Since, by

Corollary 2.4, �0 <13 , it follows that V�T�RSS� is increasing in � ∈ � 13 � 1� implying

that the estimator

T ′′RSS =

13h1�Z1�1�+

23h2�Z2�2� (4.4)

is best in the class of estimators �T�RSS � 13 ≤ � ≤ 1 uniformly in a. In particular, we

have

Theorem 4.2. For an RSS of size 2, V�T ′′RSS� < V�T ′

RSS� < V�TRSS� uniformly in a.

On comparing T ′RSS and T ′′

RSS based on an RSS of size 2 with the UMVUE T ∗SRS

based on an SRS of same size, it is found by direct computations that T ′RSS performs

better than T ∗SRS for a ∈ �0�875� 1�550� and T ′′

RSS performs better than T ∗SRS for a ∈

�0�1885� 2�865�. It has been noted earlier that for samples of size 2, TRSS is worsethan T ∗

SRS uniformly in a.It may again be noted that, for an RSS of size 2, a biased estimator of �F�t� is

T ′′′RSS = e

− t

�̂RSS � (4.5)

where

�̂RSS =(c12�2�c22�2�

X2�2 + Z1�1

)( c212�2�c22�2�

+ 1) � (4.6)

where c1n�i� and c2n�i� are defined as in Sec. 3. Since Z1�1 is an exp(�) randomvariable that is distributed independently of X2�2, which is marginally distributed asX�2�2�, the MSE of T ′′′

RSS is given by (3.6) with i = k = n = 2. On comparing the MSEof T ′′′

RSS with the variances of T ′RSS and T ′′

RSS, it is found that T ′RSS performs better

than T ′′′RSS for values of a outside the interval (0.000, 6.910) and T ′′

RSS performs betterthan T ′′′

RSS for values of a outside the interval (0.000, 6.480].We finally remark that a modification of the RSS provides an unbiased

estimator even better than T ′′RSS for samples of size 2. A modified RSS of size 2 is

given by �X1�2� X2�2 that differs from an RSS only in observing X1�2 instead of X1�1

from the first random sample of size 2. For this modified RSS, the estimator

TMRSS =12�h2�X1�2�+ h2�X2�2�

is unbiased for estimating �F�t� and, by Corollary 2.3, it follows that

V�T�RSS� > ��2 + �1− ��2V�h2�X2�2�� ≥12V�h2�X2�2�� = V�TMRSS�

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Unbiased Estimation of the Distribution Function 1667

uniformly in a for every 0 ≤ � ≤ 1. It has been observed empirically that thevariance of TMRSS is smaller than that of T ∗

SRS based on an SRS of size 2 for valuesof a in [0.0000, 5.1960].

For n ≥ 3, it appears difficult to compare the variances of TRSS and T ′RSS, as it is

difficult to obtain a tractable expression for V�hi�Zi�i�� for i > 2. However, in whatfollows we prove that it is possible to obtain an unbiased estimator of �F�t� based ona modified RSS whose variance is uniformly smaller than that of TRSS. For n ≥ 2, amodified RSS of size n is defined as �X1�2� X2�2� � � � � Xn�n, which again differs froman RSS only in observing X1�2 instead of X1�1 from the first random sample of size n.Note that

�F�t� = 1n

n∑i=1

�F�i��t� =1n

[�F�1��t�+�F�2��t�+

n∑i=3

�F�i��t�

]

= 1n

[�F�nt�+�F�2��t�+

n∑i=3

�F�i��t�

]�

An unbiased estimator of �F�t� based on this modified RSS is therefore

T ′MRSS =

1n

[T21 + T22

2+

n∑i=3

I�Xi�i > t�

]� (4.7)

where

T2j = h2

(Zj�2

n

)+ I�Xj�2 > t�

= h2

(Zj�2

n

)+ I�Zj�2 > �n− 1�t�� j = 1� 2� (4.8)

since for each j = 1� 2, h2�Zj�2/n� is an unbiased estimator of �F�1��t� and I�Zj�2 >

n− 1t� is an unbiased estimator of �F�2��t�. Now, by Theorem 2.2,

V

(h2

(Z1�2

n

))= V

(h2

(Z2�n

n

))= 1

n

[�F�1��t�+

�∑k=1

(1

�2k−1+ 1

�2k

)�F�1���

kt�

]−�F 2

�1��t�� (4.9)

where � = n/�n− 1�. Also

Cov(h2

(Z1�2

n

)� I(Z1�2 > n− 1t

)) = Cov(1n

�∑k=0

1�k

I(Z1�2 > �knt

)� I(Z1�2 > n− 1t

))

= E

[1n

�∑k=0

1�k

I(Z1�2 > �knt

)]−�F�1��t��F�2��t�

= �F�1��t�{1−�F�2��t�

}�

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1668 Sinha et al.

so that

V�T21� = V

(h2

(Z1�2

n

))+ V

(I(Z1�2 > n− 1t

))+ 2Cov(h2

(Z1�2

n

)� I(Z1�2 > n− 1t

))= V

(h2

(Z1�2

n

))+�F�2��t�

{1−�F�2��t�

}+ 2�F�1��t�{1−�F�2��t�

}� (4.10)

where V�h2�Z1�2/n�� is given by (4.9). Hence

V�T ′MRSS� =

1n2

[V�T21�

2+

n∑i=3

�F�i��t�{1−�F�i��t�

}]� (4.11)

where V�T21� is given by (4.10). In the following theorem we prove that T′MRSS is

uniformly better than TRSS.

Theorem 4.3. For n ≥ 2, V�T ′MRSS� < V�TRSS� uniformly in a.

Proof. We have, by (4.11),

V�TRSS�− V�T ′MRSS� =

1n2

[ n∑i=1

�F�i��t�{1−�F�i��t�

}]− V�T ′MRSS�

= 1n2

[ 2∑i=1

�F�i��t�{1−�F�i��t�

}− V�T21�

2

]� (4.12)

Now, by Corollary 2.3,

V

(h2

(Z1�2

n

))= V

(h2

(Z�2�n�

n

))< �F�nt��1−�F�nt� = �F�1��t��1−�F�1��t��

Hence, by (4.10) and (4.12),

V�TRSS�− V�T ′MRSS� >

1n2

[12

2∑i=1

�F�i��t�{1−�F�i��t�

}−�F�1��t�{1−�F�2��t�

}]= 1

2n2

(�F�2��t�−�F�1��t�)[1− {�F�2��t�−�F�1��t�

}]> 0

as 0 < �F�2��t�−�F�1��t� < 1−�F�1��t� < 1�

Hence the result. �

Table 4Table showing interval of values of a for which V�T ′

MRSS� < V�T ∗SRS�.

Sample size 2 3 4 5 6 7 8 9 10

LEP 0.0000 0.4355 0.4275 0.4165 0.4115 0.4045 0.3970 0.3890 0.3805UEP 5.1960 1.7845 1.7655 1.9150 2.0595 2.1765 2.2730 2.3540 2.4250

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Unbiased Estimation of the Distribution Function 1669

Table 4 shows lower and upper end points of the interval of values of a forwhich V�T ′

MRSS� < V�T ∗SRS�.

Comparing the lower and upper end points in Table 4 with those in Table 1, itis observed that the intervals in Table 4 are wider. However, for larger values of nthe gain is not so significant.

Appendix 1

Proof of (2.13) and (3.7). For � = 1 and i ≥ 2, the joint p.d.f of �X�i�n� and Y =∑k−1r=1 Yr is given by

fX�i�n��Y�x� y� = i

(n

i

)��k− 1�

�1− e−x� �i−1e

−�n−i+1�x� e−yyk−2

= i(n

i

)��k− 1�

i−1∑r=0

�−1�r(i− 1r

)e−�ir �x+y�e−�1−�ir �yyk−2�

where � = c1n�i�/c2n�i� and �ir is defined as in Sec. 3. Hence the p.d.f of ��̂ik = Z isgiven by

fZ�z� =i(n

i

)��k− 1�

i−1∑r=0

�−1�r(i− 1r

)e−�ir z

∫ z

0e−�1−�ir �yyk−2dy

= i(n

i

)�

i−1∑r=0

�−1�r(i− 1r

)e−�ir z

�1− �ir�k−1

[1−

k−2∑j=0

�1− �ir�j

j! e−�1−�ir �zzj]

Thus finally for i ≥ 2,

Mik�a � n�

= E�T ′′ik� =

i(n

i

)�

i−1∑r=0

�−1�r(i−1r

)�1− �ir�

k−1

[ ∫ �

0e−��ir z+ a�

z �dz−k−2∑j=0

�1− �ir�j

j!∫ �

0e−�z+ a�

z �zjdz

]

= i(n

i

)c2n�i�

c1n�i�

i−1∑r=0

�−1�r(i−1r

)�1− �ir�

k−1

[2

√a�2

n− i+ r + 1K−1�2

√�n− i+ r + 1�ac1n�i��

−k−2∑j=0

2�a��j+12 �1− �ir�

j

j! K−j−1�2√a��

]� (4.13)

using (see Zacks and Even, 1966) that

∫ �

0x−�e−ax− b

x dx = 2(a

b

)1/2��−1�

K�−1�2√ab��

where K��y� is the modified Bessel function of the 2nd kind of order �. SinceK−��y� = K��y� for all � = 0� 1� 2� � � � , we get (3.7). For k = 1, we obtain (2.13) from(4.13).

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1670 Sinha et al.

Acknowledgment

Research of Sujay Mukhuti is supported by Council of Scientific and IndustrialResearch, India (Sanction no. 9/28(566)/2002/EMR-I).

References

Chen, Z., Bai, Z., Sinha, B. K. (2003). Ranked Set Sampling: Theory and Applications.New York: Springer-Verlag.

Chiuv, N. N., Sinha, B. K. (1998). On some aspects of ranked set sampling in parametricestimation. In: Balakrishnan, N., Rao, C. R., eds. Handbook of Statistics. Vol. 17.New York: North Holland, pp. 337–377.

El-Newihi, E., Sinha, B. K. (2000). Reliability estimation based on ranked set sampling.Commun. Statist. Theory Meth. 29(7):1583–1595.

Gradshteyn, I. S., Ryzhik, I. M. (1994). Tables of Integrals, Series and Products. Boston:Academic press.

Johnson, N. L., Kotz, S., Balakrishnan, N. (1994). Continuous Univariate Distributions. Vol. 1.2nd ed., New York: John Wiley.

McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets. Austral.J. Agricul. Res. 3:385–390.

Pal, M., Sinha, B. K. (2002). Estimation of survival function based on multiple cycle rankedset samples in normal population. J. Statist. Stud. Special Vol. in Honour of Prof. MirMasoom Ali’s 65th Birthday, pp. 239-252.

Sengupta, S., Mukhuti, S. (2006). Unbiased variance estimation in a simple exponentialpopulation using ranked set samples. J. Statist. Plan. Inf. 136:1526–1553.

Sinha, B. K., Sinha, B. K., Purakayastha, S. (1996). On some aspects of ranked set samplingfor estimation of normal and exponential parameters. Statist. Decisions 14:223–240.

Stokes, S. L., Sager, T. W. (1988). Characterization of a ranked set sample with applicationto estimating distribution functions. J. Amer. Statist. Asso. 83:374–381.

Zacks, S., Even, M. (1966). Minimum variance unbiased and maximum likelihood estimatorsof reliability functions for systems in series and in parallel. J. Amer. Statist. Asso.61(316):1052–1062.

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