Applied Mathematical Sciences, Vol. 6, 2012, no. 93, 4627 - 4640

Confidence Interval Estimation of the Shape

Parameter of Pareto Distribution

Using Extreme Order Statistics

Aissa Omar, Kamarulzaman Ibrahim and Ahmad Mahir Razali

School of Mathematical Sciences Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Abstract

The asymptotic distributions of several estimators of the shape parameter α of the Pareto distribution in the case where the scale parameter is known are derived. The study is carried out using simple random sampling (SRS), extreme ranked set sampling (ERSS) and on random samples that are based on minimum order statistics only or maximum order statistics only. These distributions are used to construct asymptotic confidence intervals (ACI) for α . A simulation study then compares these confidence intervals via their expected lengths and coverage probabilities. Keywords: Pareto distribution, shape parameter, simple random sampling, extreme ranked set sampling, minimum order statistics , maximum order statistics, asymptotic confidence interval, expected length, coverage probability

1 Introduction

For situations where the variable of interest is either costly or time consuming to be measured, but can easily be ranked, ranked set sampling (RSS) devised by McIntyre (1952) is generally useful in a sampling scheme utilizing the information of the ranks in addition to measurements. Since the performance of RSS is susceptible to error in ranking, various modifications of RSS have been

4628 A. Omar, K. Ibrahim and A. M. Razali proposed which are more practicable. Such modifications have shown to improve the precision of estimation in some cases as opposed to simple random sampling (SRS) and RSS, cf. Helu et al. (2010), Al-Saleh and Al-Kadiri (2000), Al-Saleh and Al-Omari (2002) and Muttlak (1997).

The Pareto distributions are quite commonly applied in many areas of research, such as actuarial science (cf. Hogg and Klugman (1984)), economic studies (cf. Arnold (1983)), hydrology (cf. Nadarajah and Ali (2008)) and reliability studies (cf. Harris (1968)).

In this paper, we construct the asymptotic confidence intervals for the shape parameter of Pareto distribution based on asymptotic distributions of several estimators of α with the scale parameter known and the data gathered under SRS, ERSS and on random samples that are based on minimum order statistics only or maximum order statistics only. A simulation study is used to compare these confidence intervals via their expected lengths and coverage probabilities.

Assume that the random variable ( ),X p γ α∼ , where 0α > and 0γ > are the shape and scale parameters respectively. Then the probability density function (pdf) of X is given by

1 ,( )0,

xf x xotherwise

α

α

αγ γ+

⎧≥⎪= ⎨

⎪⎩

,

and the cumulative distribution function (cdf) is given by

( ) 1 , .F x xx

αγ γ⎛ ⎞= − ≥⎜ ⎟⎝ ⎠

We are interested in making an inference for α when γ is known. This is applicable, for example in the area of actuarial sciences, when the reinsurer is interested in the information about all losses exceeding a certain limit which could for instance be the priority of the excess of loss treaty (see, Rytgaard (1990)). Thus it can be assumed without loss of generality that γ is equal to one. Then our pdf and cdf reduce respectively to

( )1 , 1, 0( ) ,

0,x xf x

otherwise

αα α− +⎧ ≥ >⎪= ⎨⎪⎩

(0.1)

and ( ) 1 , 1.F x x xα−= − ≥ (0.2)

The mean and variance of X are given by ( ) 1( ) 1 , 1E X α α α−= − > , (0.3)

2var( ) , 2( 1) ( 2)

X α αα α

= >− −

. (0.4)

respectively.

Confidence interval estimation 4629

2 Confidence Intervals based on SRS Let 1, , nX XK be a simple random sample of size n from ( )1,p α . Next, we consider asymptotic confidence intervals of α based on moment method (MOME), uniformly minimum variance unbiased estimator (UMVUE) and maximum likelihood estimator (MLE) as follows: 2.1 ACI based on the MOME When ( )~ 1,X p α , the mean of X is given by (1.3). Put ( ) srsE X X= and solve for α to get the MOME of α as

( ) 1,ˆ 1mom s srs srsX Xα

−= − (1.1)

where 1

1 n

srs ii

X Xn =

= ∑ . Rytgaard (1990) has shown that ,ˆmom sα is asymptotically

normal distributed by ( )

2

,2

Nn nα αα

α⎛ ⎞

+⎜ ⎟⎜ ⎟−⎝ ⎠ when α >2. This implies that

( ) ( )0 0

2 2

, ( /2) , ( /2)ˆ ˆ, .2 2mom s mom sZ Z

n n n nα αα α α αα α

α α

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(1.2) is an approximate 0100(1 )%α− confidence interval for α , where 0α is the significance level, and

0( /2)Z α is the upper 0100(1 2)thα− percentile of

( )0,1N . 2.2 ACI based on the MLE and UMVUE It is straightforward to show that the MLE of α is given by

( )1

,1

ˆ lnn

mle s ii

n xα−

=

⎛ ⎞= ⎜ ⎟⎝ ⎠∑ (1.3)

Note that ( ),ˆ 1mle s

nn

α αΕ =−

when 1n > and ( )( ) ( )

22

, 2ˆV1 2mle s

nn n

α α=− −

when 2n > . According to Lehmann (1983) the asymptotic distribution of ,ˆmle sα

is [ ]( )1, ( )srsN Iα α − where ( )srsI α is the Fisher information of α and is given

by

, which implies that ,ˆmle sα is

asymptotically 2

,Nnαα

⎛ ⎞⎜ ⎟⎝ ⎠

. Also, it is a simple exercise to show that the UMVUE

of α exists and is given by

21

2 2

ln ( , , , )( ) nsrs

f x x nI E ααα α

⎛ ⎞∂= − =⎜ ⎟∂⎝ ⎠

K

4630 A. Omar, K. Ibrahim and A. M. Razali

( ) ( )1

, i1

ˆ 1 ln .n

umvue si

n xα−

=

⎛ ⎞= − ⎜ ⎟⎝ ⎠∑ (1.4)

and ,ˆumvue sα is asymptotically2

,Nnαα

⎛ ⎞⎜ ⎟⎝ ⎠

.Thus, the ACIs for MLE and UMVUE

are:

0 0, ( /2) , ( /2)ˆ ˆ, ,mle s mle sZ Z

n nα αα αα α⎡ ⎤− +⎢ ⎥⎣ ⎦

(1.5)

0 0, ( /2) , ( /2)ˆ ˆ, .umvue s umvue sZ Z

n nα αα αα α⎡ ⎤− +⎢ ⎥⎣ ⎦

(1.6)

respectively.

3 Confidence Intervals based on minimum order statistics Let (1: )1 (1: ), ,m m nX XK be a version of ranked set sample that consists of minimum order statistics of size n where m is the set size. Next, we consider the following asymptotic confidence intervals for α based on this random sample. 3.1 ACI based on MOME Note that (1: )1 (1: ), ,m m nX XK are independent identically distributed (iid) with

mean , 11

m mmα α

α>

− and variance 2

(1: ) 2 , 2( 1) ( 2)m

m mm m

ασ αα α

= >− −

then

( )(1: ) (1: )m mE X X= . Solving with respect to α we obtain the moment estimator given by

( )( ) 1

(1) (1) (1)ˆ 1mome X m Xα−

= −

where (1) (1: )1

1 .n

m ii

X Xn =

⎛ ⎞= ⎜ ⎟⎝ ⎠∑ By central limit theorem (CLT) we get

( )2(1) (1: )0,

1d

mmn X N

mα σ

α⎛ ⎞− ⎯⎯→⎜ ⎟−⎝ ⎠

,

Then applying the delta method with ( )( 1)

xg xm x

=−

we have

( ) ( )( )

2

(1)

1ˆ 0,2

dmome

mn N

m mα α

α αα

⎛ ⎞−− ⎯⎯→ ⎜ ⎟

⎜ ⎟−⎝ ⎠.

The ACI based on moment estimator is therefore given by

Confidence interval estimation 4631

( )( )

( )( )0 0

2 2

(1) ( /2) (1) ( /2)

1 1ˆ ˆ, .2 2mome mome

m mZ Z

nm m nm mα α

α α α αα α

α α

⎡ ⎤− −⎢ ⎥− +− −⎢ ⎥

⎣ ⎦(2.1)

3.2 ACI based on MLE The pdf of the minimum order statistic (1: )mX is given by

( ) ( ) ( )11: ; ,mg x m x x ααα α − +−⎡ ⎤= ⎣ ⎦

and log likelihood of (1: )1 (1: ), ,m m nX XK is

( ) ( ) ( ) ( )( )1: 1:1

ln ( ; ) ln 1 lnn

m m jj

L x n m m xα α α=

= − + ∑ ,

( )( )( )1:1:

1

ln ( ; )ln

nm

m jj

L x n m xα

α α =

∂= −

∂ ∑ . (2.2)

Equating (3.2) to zero then solving with respect to α gives

( )1

(1) (1: )1

ˆ ln .n

mle m ii

n m xα−

=

⎛ ⎞= ⎜ ⎟⎝ ⎠∑

Let (1: )1

1 n

m jj

V Vn =

= ∑ where (1: ) (1: )ln( ), 1, ,m j m jV X j n= = K iid with an exponential

distribution with mean are 1mα

.Then ( )2

(1)ˆ 0,dmlen N

nαα α

⎛ ⎞− ⎯⎯→ ⎜ ⎟

⎝ ⎠ thus

0 0(1) ( /2) (1) ( /2)ˆ ˆ , mle mleZ Z

n nα αα αα α⎡ ⎤− +⎢ ⎥⎣ ⎦

. (2.3)

as an approximate 0100(1 )%α− confidence interval for α .

4 Confidence Intervals based on maximum order statistics Let ( )1 ( ), ,m m nX XK be a version of ranked set sample that consists of maximum order statistics of size n where m is the set size. Next, we consider the following asymptotic confidence intervals forα based on this sampling technique. 4.1 ACI based on the MLE The pdf of the maximum order statistic ( : )m m jX is given by

( ) ( )1 1( : ) ; 1 .

m

m mg x m x x ααα α− − +−⎡ ⎤= −⎣ ⎦

Then the log-likelihood function is

( ) ( ) ( ) ( ) ( )( : ) ( : )1 1

ln ln ( 1) ln 1 1 ln .n n

m m j m m jj j

L n m m x m xαα α α= =

= + − − − +∑ ∑

4632 A. Omar, K. Ibrahim and A. M. Razali This implies that

( ) ( ) ( )( : ) ( : )( : )

1 ( : )

ln ; ( 1) lnln .

(1 )

nm m j m m j

m m jj m m j

L x m xn m xx α

α

α α −=

⎛ ⎞∂ −⎜ ⎟= + −⎜ ⎟∂ −⎝ ⎠

∑ (3.1)

Setting (4.1) equal to zero we get the solution denoted by ,ˆmle mα which is the MLE of α (Abu-Dayyeh et al. (2011)). Then, according to Lehmann (1983),

,ˆmle mα is asymptotically [ ]( )1, ( )moN Iα α − where

( ) ( )( )2 3

( : )2 2 3

0

31ln ;

( ) 1 1( 2)

k

mm m j

mok

mL x knI E m m

kα

αα α

−

=

⎡ − ⎤⎛ ⎞−⎢ ⎥⎜ ⎟⎛ ⎞∂ ⎝ ⎠⎢ ⎥⎜ ⎟= − = + −

⎜ ⎟ ⎢ ⎥∂ +⎝ ⎠ ⎢ ⎥⎣ ⎦

∑ . Therefore

we have

( )( )0

1

, ( /2)ˆ ( )mle m moZ Iαα α−

± . (3.2)

is an approximate 0100(1 )%α− confidence interval for α . 4.2 ACI based on the ad hoc estimator

It is easy to show that ( : ) ( : )ln( ), 1,...,m m j m m jY X j n= = are iid with mean mdα

and

variance 2mcα

where 1

20

1( 1)

( 1)

km

mk

mk

d mk

−

=

−⎛ ⎞−⎜ ⎟

⎝ ⎠=+∑ and

2

1 1

3 20 0

1 1( 1) ( 1)

2

( 1) ( 1)

k km m

mk k

m mk k

c m mk k

− −

= =

⎡ ⎤− −⎛ ⎞ ⎛ ⎞− −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥= −⎢ ⎥+ +⎢ ⎥⎣ ⎦

∑ ∑ .

,ˆAdhoc mα is asymptotically22,, mNn

σα⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

where

( )

1

, , ,1,

ˆ ; lnn

mAdhoc m rss m m m j

jrss m

d Y n XY

α−

=

⎛ ⎞= = ⎜ ⎟

⎝ ⎠∑ and

2

21 122, 3 2 2

0 0

1 1( 1) ( 1)

2

( 1) ( 1)

k km m

mk k m

m mk k

m mk k d

ασ− −

= =

⎡ ⎤− ⎛ − ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥− −⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎜ ⎟= −⎢ ⎥⎜ ⎟+ +⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑ .Then

0 0

2, 2,, ( /2) , ( /2)ˆ ˆ , m m

Adhoc m Adhoc mZ Zn nα α

σ σα α⎡ ⎤

− +⎢ ⎥⎣ ⎦

. (3.3)

Confidence interval estimation 4633 as an approximate 0100(1 )%α− confidence interval for α .

5 Confidence Intervals based on ERSS Samawi et al. (1996) introduced a modification of RSS called extreme ranked set sampling (ERSS). Under ERSS complete ranking is not required, but only extremes are determined. The idea of ERSS is as follows. If the sample size m is even, select from the first m/2 samples the smallest units and select from last m/2 samples the largest units. If the sample size m is odd, select from the first (m-1)/2 samples the smallest units and from the next (m-1)/2 samples select the largest units. From the last sample, we select the median. Next, we will consider the following asymptotic confidence intervals for α : 5.1 ACI based on the MLE Let (1)1 ( )(1) ( ) 1

2 2

, , , , ,m m m mmX X X X

+K K be an extreme ranked set sample of size

n=2m where m is the set size and m is even and (1) ( ) ( ( 1)/2), , , 1, 2,..., ( 1) / 2,ij ij m j mX X X i m+ = − 1,...,j r= be an extreme ranked set

sampling of size n=rm where m is the set size and m is odd. The MLE were derived by Omar et al. (2012). (a) The case when m is even The joint pdf of (1) ( ) and ij ij mX X is given by

( ) ( ) ( )1 11 1(1) ( ) (1) (1) ( ) ( ), ; 1 .

m m

ij ij m ij ij ij m ij mg x x m x x m x xα αα αα α α− −− + − +− −⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦

Then the log-likelihood function is given by

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

(1) ( ) (1)1 1

( ) ( )1 1

ln ; , ln 1 ln

1 ln 1 1 ln .

r l

ij ij m ijj i

r m

ij m ij mj i l

L x x mr m m x

m x xα

α α α

α

= =

−

= = +

= − +

⎡ ⎤+ − − − +⎣ ⎦

∑∑

∑∑

This implies that

( )

( ) ( ) ( )

2

(1)1 1

( ) ( )( )

1 1 ( )

ln ln

1 ln ln 0.

1

l mr

ijj i

r mij m ij m

ij mj i l ij m

L mr m x

m x xx

x

α

α

α α

=

= =

−

−= = +

∂= −

∂

⎡ ⎤−+ − =⎢ ⎥

−⎢ ⎥⎣ ⎦

∑ ∑

∑∑ (4.1)

Then (5.1) has exactly one root, i.e. ,ˆMLE eα is the MLE of α since

( ) ( )( )( )

22

( ) ( )

22 21 ( )

1 lnln 01

r m ij m ij m

j i l ij m

m x xL mr

x

α

αα α

−

−= +

⎛ ⎞−∂ ⎜ ⎟= − − <⎜ ⎟∂ −⎜ ⎟⎝ ⎠

∑∑ .

The estimator ,ˆmle eα cannot be obtained in a closed form. According to

4634 A. Omar, K. Ibrahim and A. M. Razali Lehmann (1983) ,ˆmle eα is asymptotically [ ]( )1, ( )ERSSN Iα α − where ( )ERSSI α

is the Fisher information of α under ERSS which can be found as follows:

( ) ( ) ( )( )( )

22

( ) ( )(1) ( )22 2

( )

2 30

lnln , ;1 ,

1

3( 1)

1 ( 1)( 2)

r m ij m ij mij ij mERSS

j i ij m

sm

s

x xL x x mrI E m Ex

msmr m ms

α

α

αα

α α

α

−

−

=

⎛ ⎞⎡ ⎤⎛ ⎞∂ ⎣ ⎦⎜ ⎟⎜ ⎟= − = + − ⎜ ⎟⎜ ⎟∂ ⎜ ⎟−⎝ ⎠ ⎝ ⎠−⎛ ⎞

−⎜ ⎟⎝ ⎠= + −

+

∑∑

3

.−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑

This implies that

( ) ( )0 0

1 1

, ( /2) , ( /2)ˆ ˆ( ) , ( ) .mle e ERSS mle e ERSSZ I Z Iα αα α α α− −⎡ ⎤− +⎢ ⎥⎣ ⎦

(4.2)

(b) The case when m is odd The joint pdf of (1) ( ) ( 1)( )

2

and,ij ij m mjX X X + is given by

( ) ( )1 11 1(1) ( ) (1) (1) ( ) ( )1

2

12

1 122 2

, , ; 1

! 11 !

2

m m

ij ij m ij ij ij m ij mmj

m

m mj j

g x x x m x x m x x

m x xm

α α

α αα αα α α

− −

− −− + − +− −+⎛ ⎞

⎜ ⎟⎝ ⎠

−

+ +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎡ ⎤ ⎡ ⎤= −⎜ ⎟ ⎣ ⎦ ⎣ ⎦⎜ ⎟

⎝ ⎠

⎡ ⎤ ⎡• −⎢ ⎥⎡ − ⎤ ⎢ ⎥⎛ ⎞ ⎣ ⎦ ⎣⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

12

( 1)1

2

.

m

mjx αα

−

− ++⎛ ⎞

⎜ ⎟⎝ ⎠

⎤⎢ ⎥⎢ ⎥⎦

Then the log-likelihood function is

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

(1)1 1

1

( ) ( )1 1

11 (22

ln 1 ln 1 ln ln

( 1) ln 1 ( 1) ln ln

( 1) ln 1 ( 1) ln2

r d

ijj i

r m

ij m ij mj i d

mm jj

L r m m m x k

m x x r

m x xα

α

α α α

α α

α−

= =

−−

= = +

++⎛ ⎞⎜ ⎟⎝ ⎠

= − − + +

⎡ ⎤+ − − − + +⎣ ⎦

⎛ ⎞−+ − − +⎜ ⎟⎜ ⎟

⎝ ⎠

∑∑

∑ ∑

)1.

r

j=

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑

Then, we have

( ) ( ) ( ) ( )1

( ) ( )(1) ( )

1 1 1 ( )

11 ( )22

1( )21

2

1 lnln ln ln1

1 ln2 1 ln

21

r d r mij m ij m

ij ij mj i j i d ij m

mm jj

mjmj

m x xL rm m x xx

m x xm x

x

α

α

α

αα α

−

−

−−

−= = = +

++⎛ ⎞⎜ ⎟⎝ ⎠

+

+⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤−∂= − + −⎢ ⎥

∂ −⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞−⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟

⎛ ⎞⎝ ⎠ +⎛ ⎞⎝ ⎠⎢ ⎥+ − ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠− ⎝ ⎠⎢ ⎥⎢⎣ ⎦

∑∑ ∑ ∑

1

0.r

j=

=

⎥

∑

(4.3)

Confidence interval estimation 4635 The solution of (5.3) with respect to α is the MLE of α , i.e. ,ˆMLE eα which cannot be obtained in a closed form. The Fisher information of α under ERSS in this case which can be found as follows:

( ) ( ) ( )( )( )

( )

22 1 ( ) ( )

22 21 ( )

2

11 ( )22

12

lnln1

1

ln1

21

r m ij m ij mERSS

j i d ij m

mm jj

mj

x xL rmI E m Ex

x xm E

x

α

α

α

α

αα

α α

−

−

−−

−= +

++⎛ ⎞⎜ ⎟⎝ ⎠

+⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞∂ ⎜ ⎟= − = + −⎜ ⎟ ⎜ ⎟∂ −⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

− ⎢ ⎥⎛ ⎞ ⎝ ⎠⎣ ⎦+ ⎜ ⎟⎝ ⎠ ⎛

−⎝

∑ ∑

21

,r

j=

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎞

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎠⎝ ⎠

∑

( ) ( )( )5

3 2222 3 3

0 0

3 5 / 2( 1)( 1)

1 !1 1 ,

( 2) ( 2)( 1) !2

sv mm

v s

m mm mv srm m

v m smα

−−

= =

⎡ ⎤− ⎛ − ⎞⎛ ⎞−−⎢ ⎥⎜ ⎟⎜ ⎟ −⎝ ⎠ ⎝ ⎠⎢ ⎥= + − +⎢ ⎥+ + +⎛ − ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

∑ ∑

Therefore, the ACI for α is as in (5.2). 5.2 ACI based on an ad hoc estimator We suggest the following ad hoc estimator: (a) In the case when m is even,

( ) ( )( )/ 2

(1: ) ( : ) / 21 1

ˆ ,ln ln

evenAdhoc r m

m i m m i nj i

rQ

x xα

+= =

=⎛ ⎞+⎜ ⎟⎝ ⎠

∑ ∑

where21

1

0

1 1 ( 1) .2 ( 1)

mk

k

mmQ m mk k

−−

=

⎡ ⎤−⎛ ⎞ ⎡ ⎤= + −⎢ ⎥⎜ ⎟ ⎢ ⎥+⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

∑ Note that the distribution of

( )(1: )lni m iY X= is the same as the distribution of the minimum order statistics based on simple random sample of size m from an exponential distribution with

mean 1α

. Therefore, it is easy to show 1( )iE Ymα

= . Also, note that the

distribution of ( )( : )lni m m iZ X= is the same as the distribution of the maximum order statistics of a simple random sample of size m from an exponential

distribution with mean 1α

.

Let ( ) ( )(1: ) ( : ) / 2[ln lni m i m m i nV x x += + and 1 2 /2, , , nV V VK are iid with mean

4636 A. Omar, K. Ibrahim and A. M. Razali

21

1

0

11 1 ( 1) ( 1)

2

mk

k

m Qm mmk kα α

−−

=

⎡ ⎤−⎛ ⎞ ⎡ ⎤⎛ ⎞ + − =⎢ ⎥⎜ ⎟⎜ ⎟ ⎢ ⎥+ ⎛ ⎞⎝ ⎠ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎜ ⎟⎝ ⎠

∑ and variance 2 2(1: ) ( : )m m mσ σ+

where 2(1: )mσ and 2

( : )m mσ are the variances of the minimum and maximum order statistics of a simple random sample of size m from an exponential distribution

with mean1 ,α

respectively.

Then by CLT

( )/2

2 2(1: ) ( : ) /2

1

N 0, ,m

dm i m m i m

i

Qr V σ σα +

=

⎛ ⎞⎛ ⎞− ⎯⎯→ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑

and by using the delta method we have

( )

2 / 22 2(1: ) ( : ) / 2

1

4 / 22 2(1: ) ( : ) / 22

1

0, ' , V V

N 0, .

md

m i m m i mi

m

m i m m i mi

Q Qr N g

rQ

α σ σ

α σ σ

+=

+=

⎛ ⎞⎛ ⎞ ⎛ ⎞− ⎯⎯→ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎡ ⎤= +⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∑

∑

Thus a 0100(1 )%α− confidence interval for α is

( ) ( )0 0, ( /2) , , ( /2) ,ˆ ˆ ˆ ˆ , Adhoc e Adhoc e Adhoc e Adhoc eZ SE Z SEα αα α α α⎡ ⎤− +⎣ ⎦ (4.4)

where ( )4 / 2

2 2, (1: ) ( : ) / 22

1

ˆ .m

Adhoc e m i m m i mi

SErQαα σ σ +

=

⎛ ⎞= +⎜ ⎟⎝ ⎠∑

(b) In the case when m is odd

( ) ( ) ( )( 1)/2

(1: ) ( : )( /2) (( 1)/2: )1 1

ˆ ,ln ln ln

evenAdhoc mr

m ij m m i n j m m jj i

rU

x x xα

−

+ += =

=⎛ ⎞⎡ ⎤+ +⎜ ⎟⎣ ⎦⎝ ⎠

∑ ∑

where 21

1

0

2( 1)/2

20

11 1 1 ( 1) 2 2 ( 1)

14 ! 1 ( 1) .2

( 2 1)1 !2

mk

k

ms

s

mm mU m mk k

mm

m sm s

−−

=

−

=

−⎛ ⎞ ⎡ ⎤− −⎛ ⎞ ⎛ ⎞= + −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢ ⎥+⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦−⎛ ⎞ ⎡ ⎤⎜ ⎟+ −⎢ ⎥⎜ ⎟ + +⎛ − ⎞ ⎣ ⎦⎛ ⎞ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑

∑

Following the same procedures in the case where m is even we get a 0100(1 )%α− confidence interval for α as (5.5) where

( )4 1

2 2, (1: ) ( : ) (( 1) / 2: )2

1 1

ˆd m

Adhoc e m i m m i m mi i d

SErUαα σ σ σ

−

+= = +

⎛ ⎞= + +⎜ ⎟⎝ ⎠∑ ∑ and 2

(( 1) / 2: )m mσ + is the

Confidence interval estimation 4637 variance of the median of order statistics of a simple random sample of size m

from an exponential distribution with mean1 .α

6 Comparison of the Confidence Intervals We now compare the asymptotic confidence intervals via their lengths and their coverage probabilities. Since all the confidence intervals above are of the form

0 0ˆ ˆ( / 2) ( / 2)ˆ ˆ( , )Z Zα αα αα σ α σ− + , it follows that the lengths are given by

0ˆ ( / 2)2 .L Z αασ= The coverage probability of the confidence interval

0 0ˆ ˆ( / 2) ( / 2)ˆ ˆ( , )Z Zα αα αα σ α σ− + is defined as

0 0ˆ ˆ( / 2) ( / 2)ˆ ˆ( ).P Z Zα α αα αα σ α α σ− < < + The smaller the length and the larger the coverage probability is the better the confidence interval. A simulation conducted to compute the lengths and coverage probabilities of the confidence intervals for α =3, 4, 5 and 6, 2,3,4m = ,

=3,4,6,8,12r , where n rm= and 0 0.05α = . The results are summarized in Table1 and 2 below. Note that all confidence intervals obtained are dependent on α which is unknown, therefore, we replaced α by their corresponding estimators. From tables 1 and 2 we may conclude the following:

• The estimator ,ˆmle eα has the closer coverage probability to the nominal value

01 α− .

• The estimator ,ˆmle mα has the shorter expected length.

• The expected length of ACIs for α based on RSS and ERSS is smaller than the

expected length of their counterparts estimators based on SRS.

• The coverage probability of ACIs based on RSS and ERSS is more close to the nominal value in the case of ACIs based on SRS.

4638 A. Omar, K. Ibrahim and A. M. Razali

Table1. The coverage probabilities of the confidence intervals

α n ,ˆmom sα ,ˆmle sα ,ˆumvue sα (1)ˆmomeα (1)ˆmleα ,ˆAdhoc mα ,ˆmle mα ,ˆAdhoc eα ,ˆmle eα

3

12

m=2 , r=6 0.9164 0.9126 0.9498

0.9678 0.9560 0.9694 0.9538 0.9597 0.9737m=3 , r=4 0.9570 0.9594 0.9558 0.9546 0.9560 0.9647m=4 , r=3 0.9592 0.9578 0.9476 0.9530 0.9500 0.9720

24

m=2,r=12 0.9414 0.9284 0.9468

0.9580 0.9538 0.9804 0.9526 0.9557 0.9787m=3 , r=8 0.9514 0.9496 0.9476 0.9572 0.9620 0.9683m=4 , r=6 0.9536 0.9542 0.9458 0.9556 0.9540 0.9480

4

12

m=2 , r=6 0.9112 0.9450 0.9450

0.9660 0.9566 0.9734 0.9534 0.9567 0.9767m=3 , r=4 0.9628 0.9568 0.9478 0.9500 0.9417 0.9713m=4 , r=3 0.9560 0.9500 0.9504 0.9516 0.9577 0.9593

24

m=2,r=12 0.9288 0.9334 0.9420

0.9588 0.9558 0.9812 0.9572 0.9580 0.9700m=3 , r=8 0.9576 0.9480 0.9404 0.9534 0.9407 0.9490m=4 , r=6 0.9554 0.9498 0.9460 0.9512 0.9540 0.9290

5

12

m=2 , r=6 0.9084 0.9124 0.9434

0.9572 0.9566 0.9716 0.9522 0.9527 0.9777m=3 , r=4 0.9586 0.9510 0.9526 0.9504 0.9380 0.9700m=4 , r=3 0.9542 0.9538 0.9424 0.9502 0.9507 0.9437

24

m=2,r=12 0.9276 0.9232 0.9470

0.9544 0.9470 0.9826 0.9518 0.9503 0.9810m=3 , r=8 0.9548 0.9540 0.9424 0.9542 0.9320 0.9707m=4 , r=6 0.9530 0.9536 0.9410 0.9550 0.9493 0.9540

6

12 m=2 , r=6

0.9142 0.9050 0.94500.9612 0.9538 0.9722 0.9524 0.9603 0.9790

m=3 , r=4 0.9560 0.9526 0.9534 0.9564 0.9337 0.9690m=4 , r=3 0.9568 0.9598 0.9436 0.9516 0.9560 0.9543

24 m=2,r=12

0.9278 0.9242 0.95340.9618 0.9512 0.9800 0.9560 0.9530 0.9823

m=3 , r=8 0.9520 0.9518 0.9500 0.9566 0.9333 0.9713m=4 , r=6 0.9554 0.9518 0.9510 0.9522 0.9550 0.9450

Confidence interval estimation 4639

Table2. The expected length of the confidence intervals

α n ,ˆmom sα ,ˆmle sα ,ˆumvue sα (1)ˆmomeα (1)ˆmleα ,ˆAdhoc mα ,ˆmle mα ,ˆAdhoc eα ,ˆmle eα

3

12

m=2 , r=6 3.9200 3.3948 3.3948

3.8227 3.3948 3.3948 2.6536 3.1218 3.5928m=3 , r=4 3.7698 3.3948 2.2119 2.2296 2.6349 2.8777m=4 , r=3 3.7869 3.3948 1.9637 1.9907 2.6219 2.8846

24

m=2,r=12 2.7718 2.4002 2.4002

2.5709 2.4002 2.4005 1.8258 2.1810 2.4829m=3 , r=8 2.5477 2.4002 1.5182 1.5571 1.9066 2.0020m=4 , r=6 2.5339 2.4002 1.3609 1.3962 1.8240 1.7026

4

12

m=2 , r=6 4.8009 4.5263 4.5263

4.5734 4.5263 4.5263 3.5427 4.1810 4.8200m=3 , r=4 4.5452 4.5263 2.9472 2.9826 3.3938 3.8759m=4 , r=3 4.5364 4.5263 2.6281 2.6593 3.7864 3.2830

24

m=2,r=12 3.3948 3.2006 3.2006

3.2338 3.2006 3.2006 2.4420 2.8639 3.2846m=3 , r=8 3.2139 3.2006 2.0242 2.0675 2.3770 2.6738m=4 , r=6 3.2077 3.2006 1.8146 1.8578 2.4054 2.2646

5

12

m=2 , r=6 5.8435 5.6579 5.6579

5.6932 5.6579 5.6579 4.4047 5.2661 5.9654m=3 , r=4 5.6724 5.6579 3.6929 3.7231 4.2191 4.8205m=4 , r=3 5.6658 5.6579 3.2077 3.3266 4.3762 4.0753

24

m=2,r=12 4.1320 4.0008 4.0008

4.0257 4.0008 4.0008 3.0433 3.5816 4.1226m=3 , r=8 4.0110 4.0008 2.5303 2.5959 2.9079 3.3406m=4 , r=6 4.0063 4.0008 2.2682 2.3147 3.0436 2.8387

6

12 m=2 , r=6

6.9295 6.7895 6.78956.8178 6.7895 6.7895 5.3115 6.2779 7.1862

m=3 , r=4 6.8013 6.7895 4.4099 4.4833 5.0452 5.8161m=4 , r=3 6.7959 6.7895 3.9328 4.0001 5.2789 4.9157

24 m=2,r=12

4.9000 4.8009 4.80094.8209 4.8009 3.8009 3.6580 4.2919 4.9317

m=3 , r=8 4.8092 4.8009 3.0883 3.1180 3.4772 4.0019m=4 , r=6 4.8055 4.8009 2.7219 2.7924 3.6316 3.4011

4640 A. Omar, K. Ibrahim and A. M. Razali Acknowledgments. We thank Universiti Kebangsaan Malaysia for providing partial support for this research under the research grants UKM-Gup-2011-213. References [1] W. Abu-Dayyeh, A. Assrhani, and K. Ibrahim, Estimation of the shape and scale parameters of Pareto distribution using ranked set sampling, Statistical Papers, (2011), 1-19. [2] M. F. Al-Saleh and M. A. Al-Kadiri, Double-ranked set sampling, Statistics &amp; Probability Letters, 48 (2000), 205-212. [3] M. F. Al-Saleh and A. I. Al-Omari, Multistage ranked set sampling, Journal of Statistical Planning and Inference, 102 (2002), 273-286. [4] B. C. Arnold, Pareto Distributions, International Co-operative Publishing House, Fairland MD, 1983. [5] C. M. Harris, The Pareto Distribution as a Queue Service Discipline, Operations Research, 16 (1968), 307-313. [6] A. Helu, M. Abu-Salih, and O. Alkam, Bayes Estimation of Weibull Distribution Parameters Using Ranked Set Sampling, Communications in Statistics - Theory and Methods, 39 (2010), 2533-2551. [7] R. V. Hogg and S. A. Klugman, Loss Distributions, Wiley, New York, 1984. [8] E. L. Lehmann, Theory of point estimation, J. Wiley, New York, 1983. [9] G. A. McIntyre, A method for unbiased selective sampling, using, ranked sets, Australian Journal of Agricultural Research 3 (1952), 385-390. [10] H. A. Muttlak, Median ranked set sampling, Journal of Applied Statistical Sciences, 6 (1997), 245-255. [11] S. Nadarajah and M. Ali, Pareto Random Variables for Hydrological Modeling, Water Resources Management, 22 (2008), 1381-1393. [12] A. Omar, K. Ibrahim, and A. F. Shahabuddin, Estimation of the Shape and Scale Parameters of the Pareto Distribution using Extreme Ranked Set Sampling "Manuscript submitted for publication", (2012). [13] M. Rytgaard, Estimation in the Pareto Distribution, ASTIN BULLLETIN a Journal of International Actuarial Association, 20 (1990), 201-215. [14] H. M. Samawi, M. S. Ahmed, and W. Abu-Dayyeh, Estimating the Population Mean Using Extreme Ranked Set Sampling, Biometrical Journal, 38 (1996), 577-586. Received: April, 2012