Question

Argue that if the ranking (in Ranked Set Sampling) is based on the auxiliary variables x, and the value of y itself is not used in the ranking, then y-bar-hat sub RSS is an unbiased estimator of the population mean.

Answer 1

2. Estimation of Population Mean Based
on Two New RSS
In this study two new estimators of population mean based
on modified ranked set sampling methods (i) choosing both
extremes of each sample and (ii) choosing extreme form two
independent samples have been developed.
2.1. Ranked Set Sampling by Choosing Extremes of
Samples (RSS(E))
Balci et al. [14] introduced ranked set sample by choosing
extremes of the samples and they have called this sampling
scheme as RSS(E). The procedure of RSS(E) is described as
follows:
1. Select m random samples each of size m.
2. Each sample is ranked in itself as in ranked set sampling
design.
3. Then smallest and largest order statistics from each
sample are observed.
4. Repeat above steps r times until the desired sample size
n= 2rm is obtained.
We assume that the lowest and largest units of this set can
be detected visually, or by any other means easily.
Let 12 2 XX X , ,..., m be a random sample of size 2m
with probability density function f(x) with mean µ and
variance 2 σ . Let { 1 2 , ,..., } XX X i i im , i=1,2,…m be m
sets of independent random samples each of size m from a
population with distribution function F(x) and pdf f(x) with
mean µ and variance 2 σ . Denote
X XX X i(1) = min{ , ,..., } i i im 1 2 and
() 1 2 max{ , ,..., } X XX X i m = i i im i=1,2,…,m. Then
1(1) 1( ) 2(1) 2( ) (1) ( ) { , , ,..., , } X X XX X X m m m mm is a
RSS(E) of size 2m. Note that the order statistics within the
sample are dependent and between the samples are
independent. For all i= 1,2,…,m let µ = ( ) E Xi ,
2 ( ) σ =Var Xi , (1) (1) ( ) µ = E Xi , () () µ m im = E X( ),
2
(1) (1) ( ) σ =Var Xi , 2
() () σ m im =Var X( ) an
(1, ) (1) ( ) σ m i im = Cov X X (, ) .
Let X be the mean of the SRS of size 2m. The mean and
variance of X are E X( ) = µ and
2 Var X m ( ) /2 =σ , respectively. The estimator of the
population mean based on RSS(E) can be defined as
(1) ( )
1
1 ( ) 2
m
E i im
i
X XX
m =
= ∑ + (1)
XE can be written as
XE = (1) ( )
1 ( )
2
X X + m ,
where (1) (1)
1
1 m
i
i
X X
m =
= ∑ and () ()
1
1 m
m i m
i
X X
m =
= ∑ .
The mean and variance of XE can be shown to be
(1) ( )
1 () ( ) 2
E XE m = + µ µ (2)
2 2
(1) ( ) (1, )
1 () ( 2 ) 4
Var XE mm
m
= ++ σσ σ (3)
If the underlying distribution is symmetric about zero,
then (1) (m) X d -X . Arnold et al. [15] have shown that
(1) (m) µ µ - = and 2 2
(1) ( ) σ σ = m . Using these
results
( )0 E XE = ,
and
2
(1) (1, )
1 () ( ) 2
Var XE m
m
= + σ σ . (4)
Thus, if the underlying distribution is symmetric about its
mean then XE is an unbiased estimator of the population
mean.
2.2. Ranked Set Sampling by Choosing Extremes of Tow
Independent Samples (IERSS)
Here we introduce another modified RSS called
independent Extremes ranked set sampling (IERSS) based
on two independent samples of size 2m: first we select 2m
random samples of size m each and then identify the
maximum within each set of first m samples by visual
inspection or by some other cheap method, without actual
measurement of the variable of interest. Repeat this for other
m simple random samples but for the minima. Repeat above
steps for r times until the desired sample size n=2rm is
obtained. Let 1 2 { , ,..., } XX X i i im and
1 2 { , ,..., } YY Y i i im i=1,2,…m be 2m sets of random samples
each of size m from a population with distribution function
F(x) and pdf f(x) with mean µ and variance 2 σ . Denote
X XX X i(1) = min{ , ,..., } i i im 1 2 and
() 1 2 max{ , ,..., } Y YY Y i m = i i im , i=1,2,…,m. Then
1(1) 2(1) (1) 1( ) 2( ) ( ) { , ,..., , , ,..., } XX X YY Y m m m mm beIERSS of size 2m. Note that elements of sample are
independent of each other. The estimator of the population
mean based on IERSS with one cycle can be defined as:
(1) ( )
1
1 ( ) 2
m
IE i i m
i
X XY
m =
= ∑ + (5)
The mean and variance of XIE can be shown to be
(1) ( )
1 ()( ) 2
E XIE = + µ µ m (6)
2 2
(1) ( )
1 () ( ) 4
Var XIE m m
= + σ σ (7)
Where 2 σ (1) is as defined above and
2
() () σ m im =Var Y( ) . If the underlying distribution is
symmetric about zero then using the above results of Arnold
et al.(1992) , we have ( )0 E XE = and
2
(1)
1 ( ) 2
Var XIE
m = σ . (8)
We can easily see that if the underlying distribution is
symmetric about its mean then XIE is an unbiased
estimator of the population mean.
3. Efficiency
The efficiency of XE with respect to X for estimating
the population mean is defined as:
Eff( XE , X ) = ( )
( ) E
Var X
MSE X . (9)
If the distribution is symmetric then ( ) MSE XE =
( ) Var XE .
Similarly,
Eff( XIE , X ) = ( )
( ) IE
Var X
MSE X . (10)
If the distribution is symmetric then ( ) MSE XIE =
( ) Var XIE .
And finally
Eff( XIE , XE ) = ( )
( )
E
IE
MSE X
MSE X . (11)
For uniform distribution over (0,1) the efficiency of XE
with respect to X is given by
Eff( XE , X ) = ( 1)( 2) 1
12
m m + +
> , for m>2. (12)
Similarly the efficiency of XIE with respect to X is
given by
Table 1. The Relative Efficiency of estimators of population mean using RSS(E) and IERSS
Distribution N ( ) Biase XE ( ) Biase XIE ( ,) Eff X X E ( ,) Eff X X IE (,) Eff X X IE E
Uniform
2
4
6
8
10
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
2.500
4.667
7.500
11.000
1.500
3.120
5.440
8.437
12.100
1.500
1.250
1.167
1.125
1.110
Exponential
2
4
6
8
10
0.00
0.167
0.308
0.421
0.514
0.00
0.167
0.308
0.421
0.514
1.000
0.977
0.518
0.276
0.159
1.330
1.039
0.526
0.277
0.165
1.333
1.024
1.014
1.004
1.036
Normal
2
4
6
8
10
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
1.677
2.117
2.441
2.695
1.467
2.030
2.404
2.710
2.904
1.4671
1.2142
1.3548
1.0987
1.0774
Logistic
2
4
6
8
10
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
1.435
1.613
1.706
1.763
1.437
1.706
1.801
1.850
1.880
1.4368
1.1872
1.1170
1.0831
1.0662Eff( XIE , XE ) =
1 1 m
m
+
> , for m>1 and tends to 1 as
m goes to infinity. This indicates that as the set size m
increases (1, ) (1) ( ) σ m i im = Cov X X (, ) tends to zero and
both estimators XIE and XE are equally efficient. The
results of efficiency and bias of the estimators are presented
in Table 1 for uniform, exponential, normal and logistic
distributions using SRS, RSSE and IERSS sampling
schemes. From Table 1 we observed that in the case of
uniform, normal and logistic distributions the estimators
based on XIE and XE are both more efficient than X
and XIE is more efficient than XE . For exponential
distribution ( ,) Eff X X E and ( ,) Eff X X IE decrease
as m increases. This is because XIE and XE are bias
estimators and bias diverges as m goes to∞ .
We can easily see that the estimator of population mean
based on XIE is equivalent to the one of the estimators
proposed by Samawi et al. [10] based on extreme ranked set
sample with number of cycles equal two and when set size m
is even.
4. Extreme RSS with Errors in Ranking
Dell and Clutter [4] considered the case in which there are
errors in ranking; that is, the quantified observation from the
i-th sample in the j-th cycle may no be the i-th order statistic
but rather the i-th judgement order statistic. They showed
that the sample mean of RSS with errors in ranking is
unbiased estimator of the population mean regardless of the
errors in ranking and has smaller variance than the usual
estimator based on SRS with the same sample size.
Let Xi[1] and Xi m[ ] denote the smallest and largest
judgment order statistic of the sample respectively
(i=1,2,…,m) . Then
1[1] 1[ ] 2[1] 2[ ] [1] [ ] { , , ,..., , } X X XX X X m m m mm and
1[1] 2[1] [1] 1[ ] 2[ ] [ ] { , ,..., , , ,..., } XX X YY Y m m m mm denote
RSS(E) and IERSS samples with errors in ranking. The
estimators of the population mean using RSS(E) and IERSS
with errors in ranking can be defined as
[1] [ ]
1
1 ( ) 2
m
E i im
i
X XX
m =
 = ∑ + , (13)
and
[1] [ ]
1
1 ( ) 2
m
IE i i m
i
X XY
m =
 = ∑ + (14)
The variance of XE
 and XIE
 can be defined as
Var ( XE
 )= 2 2
[1] [ ] [1, ]
1 ( 2) 4 m m m
σσ σ + + (15)
and
2 2
[1] [ ]
1 () ( ) 4
Var XIE m m
 = + σ σ (16)
where 2
[1] [1] ( ) σ =Var Xi , 2
[] [] σ m im =Var X( ) and
[1, ] [1] [ ] σ m i im = Cov X X (, ) for i=1,2,…m.
It can easily seen that XE
 and XIE
 are unbiased
estimators of the population mean if the underlying
distribution is symmetric about its mean.
5. Conclusions
We proposed two new estimators of the population mean
using two modified ranked set sampling methods. The
proposed estimators are unbiased of the population mean
when the underlying distribution is symmetric about its mean.
We showed that both estimators have smaller variances than
the estimator using SRS and provide more efficient
estimators. The estimators using extremes RSS (RSS (E) and
IERSS) will reduce errors in ranking compared to RSS,
MRSS, QRSS and BGRSS, since we have to identify and
measure the smallest and largest of the ith sample.