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Ratio estimation method is used to obtain a new estimator with higher precision for estimating the population mean or total. There are two methods for estimating ratios when the sampling design is simple stratified random sampling (SSRS), namely the combined and the separate ratio estimates. We study the performance of the combined and the separate ratio estimates using stratified ranked set sample (SRSS) introduced by Samawi (1996). Theoretical and simulation study as well as real data example are presented. It appears that using SRSS is more efficient than using SSRS for ratio estimation both in the case of combined and separate methods.
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METRON - International Journal of Statistics
2003, vol. LXI, n. 1, pp. 75-90
HANI M. SAMAWI MAHMOUD I. SIAM
Ratio estimation using stratified
ranked set sample
Summary - Ratio estimation method is used to obtain a new estimator with higher
precision for estimating the population mean or total. There aretwo methods for esti-
mating ratios when thesamplingdesign is simplestratified random sampling (SSRS),
namely the combined and the separate ratio estimates. We study the performance
of the combined and the separate ratio estimates using stratified ranked set sample
(SRSS) introduced by Samawi (1996). Theoretical and simulation study as well as
real data example are presented. It appears that using SRSS is more efficient than
using SSRS for ratio estimation both in the case of combined and separate methods.
Key Words - Concomitant variables; Order statistics; Ranked set sample; Ratio esti-
mators, Stratified sample.
1. Introduction
In many applications the quantity that is to be estimated from a random
sample is the ratio of two variables both of which vary from unit to unit. The
ratio of bilirubin level in jaundiced babies who stay in neonatal intensive care;
to their weight at birth or the ratio of acres of wheat to total acres on a farm
are two examples. Also, this method is used to obtain increased precision for
estimating the population mean or total by taking advantage of the correlation
between an auxiliary variable X and the variable of interest Y.
In the literature, ratio estimators are used in case of simple random sam-
pling (SRS) as well as in case of SSRS. (See for example Cochran, 1977).
There are two methods for estimating ratios that are generally used when the
sampling design is SSRS, namely the combined ratio estimate and the separate
ratio estimate.
McIntyre (1952) used the mean of n units based on a ranked set sample
(RSS) to estimate the population mean. Samawi and Muttlak (1996) used RSS
Received September 2001 and revised November 2002.
76 H. M. SAMAWI – M. I. SIAM
to estimate the population ratio and showed that it provided a more efficient
estimator compared to the one obtained by using SRS. Also, Samawi (1996)
introduced the concept of stratified ranked set sampling (SRSS), to improve the
precision of estimating the population mean.
In this paper we use the idea of SRSS instead of SSRS to improve the
precision of the two methods for estimating ratios. Also, we study the properties
of these estimators and compare them with other estimators.
In Section 2 and 3, we obtain the separate and combined ratio estimators
using SRSS sample respectively. We also derive the asymptotic mean and vari-
ance of these estimators. The comparison between the two estimators (separate
and combined) is discussed in terms of bias and efficiency in Section 4. Also,
the results of our simulation study and the use of the two methods using real
data about bilirubin level of baby’s, who stay in neonatal intensive care, is
discussed in Section 5 and 6 respectively.
1.1. Stratified ranked set sample
For the h-th stratum of the population, first choose r
h
independent samples
each of size r
h
h = 1, 2,... ,L. Rank each sample, and use RSS scheme to
obtain L independent RSS samples of size r
h
, one from each stratum. Let
r
1
+ r
2
+ ...+ r
L
= r. This complete one cycle of stratified ranked set sample.
The cycle may be repeated m times until n = mr elements have been obtained.
A modification of the above procedure is suggested here to be used for the
estimation of the ratio using stratified ranked set sample. For the h-th stratum,
first choose r
h
independent samples each of size r
h
of independent bivariate
elements from the h-th subpopulation, h = 1, 2,... ,L. Rank each sample with
respect to one of the variables say Y or X. Then use the RSS sampling scheme
to obtain L independent RSS samples of size r
h
one from each stratum. This
complete one cycle of stratified ranked set sample. The cycle may be repeated
m times until n = mr bivariate elements have been obtained. We will use the
following notation for the stratified ranked set sample when the ranking is on
the variable Y.Forthe k-th cycle and the h-th stratum, the SRSS is denoted by
{(Y
h(1)k
,X
h[1]k
)(Y
h(2)k
,X
h[2]k
), . . .,(Y
h(r
h
)k
,X
h[r
h
]k
) : k =1, 2,...,m; h=1, 2,...,L},
where Y
h(i)k
is the i-th order statistic from the i-th set in the h-th stratum and
X
h[i]k
is the corresponding concomitant variable (see Stokes, 1977).
1.2. Ratio estimate using SSRS
The parameter of interest to be estimated in this paper is R =
µ
Y
µ
X
. Using
SSRS, we have two types of ratio estimators:
Ratio estimation using stratified ranked set sample 77
In separate case. Following Levy and Lemeshow (1991), we have the
following definition of separate ratio estimator. For the h-th stratum (h =
1, 2,... ,L), the ratio is defined by
ˆ
R
hsrs
=
Y
h
X
h
. Therefore, assuming that the
subpopulations totals of the variable X are known, the separate ratio estimator
using SSRS is given by
ˆ
R
SSRS(s)
=
L
h=1
T
h
T
ˆ
R
hsrs
=
L
h=1
Y
h
X
h
T
h
T
=
L
h=1
W
h
µ
Xh
µ
X
Y
h
X
h
(1.1)
where W
h
=
N
h
N
, N
h
is the h stratum size, N is the total population size,
T
h
= N
h
µ
Xh
, T = Nµ
X
, Y
h
=
r
h
i=1
Y
hi
r
h
, X
h
=
r
h
i=1
X
hi
r
h
Xh
, h = 1, 2,... ,L
are the known subpopulation specific means for the random variable X and
µ
X
=
L
h=1
W
h
µ
Xh
. Note that the subpopulations totals of the variable X need
to be known in order to compute
ˆ
R
SSRS(s)
for estimating the population mean
or total of the variable Y.Itcan be shown (see Hansen, 1953) that
E(
ˆ
R
SSRS
(s)) = R + O
1
Min
h
(mr
h
)
and the variance of ratio estimator can be approximated by
Var(
ˆ
R
SSRS(s)
)
L
h=1
W
2
h
µ
2
Xh
R
2
h
µ
2
X
n
h
×
C
2
Xh
+ C
2
Yh
2ρ
XhYh
C
Xh
C
Yh
(1.2)
where n
h
= mr
h
, C
Xh
=
σ
Xh
µ
Xh
, C
Yh
=
σ
Yh
µ
Yh
XhYh
=
Nh
i=1
(X
hi
µ
Xh
)(Y
hi
µ
Yh
)
Nhσ
Xh
σ
Yh
, R
h
=
µ
Yh
µ
Xh
, and σ
Xh
, and σ
Yh
are the standard deviations of the variable X and
Y, respectively in the h-th stratum. Note that, equation (1.2) also can be
derived easily from equation (6.45) in Cochran (1977) by dividing (6.45) by
the population total of the variable X and simple algebra.
In combined case. Combined ratio estimator using SSRS is defined by
ˆ
R
SSRS(c)
=
Y
SSRS
X
SSRS
=
L
h=1
W
h
Y
h
L
h=1
W
h
X
h
. (1.3)
It can be shown (see Hansen (1953) that
E(
ˆ
R
SSRS(c)
) = R + O(Max(n
1
h
))
and that the variance of the estimator is given by
Var (
ˆ
R
SSRS(c)
) R
2
L
h=1
W
2
h
n
h
σ
2
Yh
µ
2
Y
+
σ
2
Xh
µ
2
X
2
σ
Yh
µ
Y
σ
Xh
µ
X
ρ
XhYh
. (1.4)
78 H. M. SAMAWI – M. I. SIAM
2. Separate ratio estimation using SRSS
2.1. Ratio estimation when ranking is on variable Y
The separate ratio estimate, requires knowledge of the stratum totals T
h
(see Levy and Lemeshow, 1991) to estimate the population mean or the total.
Using the notation introduced in Section (1.1) ratio is estimated as follows. Let
Y
h(r
h
)
=
1
n
h
m
k=1
r
h
i=1
Y
h(i)k
,
X
h[r
h
]
=
1
n
h
m
k=1
r
h
i=1
X
h[i]k
,
and n
h
= mr
h
. Then the separate ratio estimator using SRSS when the ranking
on variable Y,isgivenby
ˆ
R
SSRS1(s)
=
L
h=1
W
h
µ
Xh
µ
X
Y
h(r
h
)
X
h[r
h
]
. (2.1)
It can be shown using Taylor expansion that
E(
ˆ
R
SSRS1
(s)) = R + O
1
Min
h
(mr
h
)
.
Also, the approximate variance of separate ratio is given by
Var (
ˆ
R
SSRS1(s)
)
L
h=1
W
2
h
µ
2
Xh
µ
2
X
R
2
h
n
h
C
2
Xh
+ C
2
Yh
2ρ
XhYh
C
Xh
C
Yh
m
n
h
r
h
i=1
(M
Xh[i]
M
Yh(i)
)
2
(2.2)
where, M
Xh[i]
=
µ
Xh[i]
µ
Xh
µ
Xh
, M
Yh(i)
=
µ
Yh(i)
µ
Yh
µ
Yh
, E(Y
h(i)
) =µ
Yh(i)
and E(X
h[i]
) =
µ
Xh[i]
.For evaluating the terms in the variances based on RSS see Stokes
(1980). For evaluating the expectation and the variance of the concomitant
variable of the order statistics see Stokes (1977).
Ratio estimation using stratified ranked set sample 79
2.2. Ratio estimation when ranking is on variable X
By changing the notation ( ) of perfect ranking by [ ] of imperfect ranking
for X and Y (i.e. changing the role of the order statistics with the role of the
concomitant variable of the order statistics) ratio is given by
ˆ
R
SSRS2(s)
=
L
h=1
W
h
µ
Xh
µ
X
Y
h[r
h
]
X
h(r
h
)
(2.3)
where
Y
h[r
h
]
=
1
n
h
m
k=1
r
h
i=1
Y
h[i]k
and X
h(r
h
)
=
1
n
h
m
k=1
r
h
i=1
X
h(i)k
.
Again we get the following results:
E(
ˆ
R
SSRS2(s)
) =
µ
Y
µ
X
+ O
1
Min
h
(mr
h
)
and
Var (
ˆ
R
SSRS2(s)
)
L
h=1
W
2
h
µ
2
Xh
µ
2
X
R
2
h
n
h
C
2
Xh
+ C
2
Yh
2ρ
XhYh
C
Xh
C
Yh
m
n
h
r
h
i=1
(M
Xh(i)
M
Yh[i]
)
2
(2.4)
where, M
Xh(i)
=
µ
Xh(i)
µ
Xh
µ
Xh
, M
Yh[i]
=
µ
Yh[i]
µ
Yh
µ
Yh
, E(Y
h[i]
) =µ
Yh[i]
and E(X
h(i)
) =
µ
Xh(i)
.
Theorem2.1. Assumethattheapproximationtothevarianceoftheratioestimators
in (1.2), (2.2) and (2.4) are valid and the bias of the estimators can be ignored for
large m. Then
Var (
ˆ
R
SRSS1(s)
) Va r (
ˆ
R
SSRS(s)
)
and
Var (
ˆ
R
SRSS2(s)
) Va r (
ˆ
R
SSRS(s)
).
Proof. Take
Var (
ˆ
R
SSRS(s)
)Va r (
ˆ
R
SRSS1(s)
) =m
L
h=1
W
2
h
µ
2
Xh
µ
2
X
R
2
h
n
2
h
r
h
i=1
M
Xh[i]
M
Yh(i)
2

> 0.
Therefore,Var(
ˆ
R
SRSS1(s)
) Var(
ˆ
R
SSRS(s)
).Similarly,Var(
ˆ
R
SRSS2(s)
) Var(
ˆ
R
SSRS(s)
).
80 H. M. SAMAWI – M. I. SIAM
2.3. Which variable to rank?
Since we can not rank on both variables at the same time and some time
it is easier to rank on one variable than the other, we need to decide to rank
on variable X or Y.
Theorem 2.2. Let us assume that there are L linear relationships between Y
h
and
X
h
, i.e., |ρ
h
| > 0 and it is easy to rank on the variable X. Also, assume that
the approximation to the variance of the ratio estimators
ˆ
R
SRSS1(s)
and
ˆ
R
SRSS2(s)
as given in equations (2.2) and (2.4) respectively are valid and the bias of the
estimators can be ignored. Then
Var (
ˆ
R
SRSS2(s)
) Va r (
ˆ
R
SSRS1(s)
).
Proof. By looking at the two variances in equations (2.2) and (2.4) we need
oly to compare
r
h
i=1
(M
Xh[i]
M
Yh(i)
)
2
with
r
h
i=1
(M
Xh(i)
M
Yh[i]
)
2
.Atthis
end, we note firstly that
µ
Xh[i]
=
µ
Xh(i)
for perfect ranking of X in the i-th set
µ
Xh
for imperfect ranking of X in the i-th set.
(2.5)
Consider the simple linear regression model of Y
h
on X
h
Y
hi
= α
h
+ β
h
X
hi
+ ε
hi
, (2.6)
µ
Yh
= α
h
+ β
h
µ
Xh
(2.7)
where α
h
and β
h
are parameters and ε
hi
is a random error with E
hi
) =
0, Var
hi
) = σ
2
h
, and Cov
hi
hj
) = 0 for i = j, i = 1, 2,... ,r
h
. Also, ε
hi
and X
hi
are independent.
Case 1. If we are ranking on the Y
h
variable we get the following model
from equation (2.6)
Y
h(i)
= α
h
+ β
h
X
h[i]
+ ε
h[i]
, (2.8)
where ε
h[i]
is a random error with, Var
h[i]
) = σ
2
[r
h
]i
and Cov
h[i]
h[j]
) = 0
for i = j, i = 1, 2,... ,r
h
also ε
h[i]
and X
h[i]
are independent. Then,
µ
Yh(i)
= α
h
+ β
h
µ
Xh[i]
. (2.9)
From (2.7) and (2.9) we get
M
Yh(i)
=
β
h
M
Xh[i]
R
h
(2.10)
Ratio estimation using stratified ranked set sample 81
and now (2.2) can be written as
Var (
ˆ
R
SRSS1(s)
)
L
h=1
W
2
h
µ
2
Xh
µ
2
X
R
2
h
n
h
C
2
Xh
+ C
2
Yh
2ρ
XhYh
C
Xh
C
Yh
m
n
h
r
h
i=1
µ
2
Xh
M
2
Xh[i]
1
µ
Xh
β
h
µ
Yh
2
.
Case 2. If we are ranking on the variable X we get the following model:
Y
h[i]
= α
h
+ β
h
X
h(i)
= ε
h[i]
. (2.11)
The expected value of Y
h[i]
is
µ
Yh[i]
= α
h
+ β
h
µ
Xh(i)
+ E
h[i]
). (2.12)
Similarly, we can show that
M
Yh[i]
=
β
h
M
Xh(i)
R
h
(2.13)
and now (2.4) can be written as
Var (
ˆ
R
SRSS2(s)
)
L
h=1
W
2
h
µ
2
Xh
µ
2
X
R
2
h
n
h
C
2
Xh
+ C
2
Yh
2ρ
XhYh
C
Xh
C
Yh
m
n
h
r
h
i=1
µ
2
Xh
M
2
Xh(i)
1
µ
Xh
β
h
µ
Yh
2
.
Therefore, from (2.5) it is clear that Var(
ˆ
R
SRSS2(s)
) Va r (
ˆ
R
SRSS1(s)
).
3. Combined ratio estimation using SRSS
3.1. Ratio estimation when ranking on variable Y
The combined ratio estimate using SRSS is defined by
ˆ
R
SRSS1(c)
=
Y
(SRSS)
X
[SRSS]
(3.1)
82 H. M. SAMAWI – M. I. SIAM
where
Y
(SRSS)
=
L
h=1
W
h
Y
h(r
h
)
and
X
[SRSS]
=
L
h=1
W
h
X
h[r
h
]
Therefore,
ˆ
R
SRSS1(c)
=
Y
(SRSS)
X
[SRSS]
=
L
h=1
W
h
Y
h(r
h
)
L
h=1
W
h
X
h[r
h
]
. (3.2)
For fixed r, assume that we have finite second moments for X and Y.
Since the ratio is a function of the means of X and Y, i.e., R =
µ
Y
µ
X
, and
hence R has at least bounded second order derivatives of all types in some
neighborhood of
Y
X
) provided that µ
X
= 0. Then, assuming large m,
and by using the multivariate Taylor series expansion, we can approximate the
variance and get the order of the bias of the ratio estimator as follows:
E(
ˆ
R
SRSS(c)
) = R + O(Max
h
(n
1
h
))
and
Var (
ˆ
R
SRSS1(c)
) R
2
L
h=1
W
2
h
n
h
σ
2
Yh
µ
2
Y
+
σ
2
Xh
µ
2
X
2
σ
Xh
σ
Yh
µ
X
µ
Y
ρ
XhYh
m
n
h
r
h
i=1
D
Yh(i)
D
Xh[i]
2
(3.3)
where, D
Xh[i]
=
µ
Xh[i]
µ
Xh
µ
X
and D
Yh(i)
=
µ
Yh(i)
µ
Yh
µ
Y
.
3.2. Ratio estimation when ranking on variable X
In this case the estimate is given by:
ˆ
R
SRSS2(c)
=
Y
[SRSS]
X
(SRSS)
where
Y
[SRSS]
=
L
h=1
W
h
Y
h(r
h
)
Ratio estimation using stratified ranked set sample 83
and
X
(SRSS)
=
L
h=1
W
h
X
h[r
h
]
.
Therefore, in combined case, we get
ˆ
R
SRSS2(c)
=
Y
[SRSS]
X
(SRSS)
=
L
h=1
W
h
Y
h[r
h
]
L
h=1
W
h
X
h(r
h
)
. (3.4)
Using the same argument as in Section (3.1), E(
ˆ
R
SRSS2(c)
) = R+ O(Max
h
(n
1
h
))
and
Var (
ˆ
R
SRSS2(c)
) R
2
L
h=1
W
2
h
n
h
σ
2
Yh
µ
2
Y
+
σ
2
Xh
µ
2
X
2
σ
Xh
σ
Yh
µ
X
µ
Y
ρ
XhYh
m
n
h
r
h
i=1
D
Yh[i]
D
Xh(i)
2
,
(3.5)
where, D
Xh(i)
=
µ
Xh(i)
µ
Xh
µ
X
and D
Yh[i]
=
µ
Yh[i]
µ
Yh
µ
Y
.
Theorem 3.1. Assume that the approximations to the variance of the ratio estima-
tors in (1.4), (3.3) and (3.5) are valid and the bias of the estimators can be ignored
for large m. Then
Var (
ˆ
R
SRSS1(c)
) Va r (
ˆ
R
SSRS(c)
)
and
Var (
ˆ
R
SRSS2(c)
) Va r (
ˆ
R
SSRS(c)
).
Proof. Similar to that for Theorem 2.1.
3.3. Ranking on which variable?
Again, since we cannot rank both variables at the same time we need to
decide which variable we should rank. Therefore, we need to compare the
variance of
ˆ
R
SRSS1(c)
in equation (3.3) and variance
ˆ
R
SRSS2(c)
in equation (3.5).
Theorem 3.2. Assume that there are L linear relationships between Y
h
and X
h
,
i.e., |ρ
h
| > 0 and it is easy to rank variable X. Also assume that the approximation
to the variance of the ratio estimators
ˆ
R
SRSS1(c)
and
ˆ
R
SRSS2(c)
given in equations
(3.3) and (3.5) respectively are valid and the bias of the estimators can be ignored
for large m. Then
Var (
ˆ
R
SRSS2(c)
) Va r (
ˆ
R
SRSS1(c)
).
Proof. Is similar to that of Theorem (2.2).
84 H. M. SAMAWI – M. I. SIAM
4. Comparison of the combined and separate estimates
Consider the case when ranking is on variable X. Equations (2.4) and
(3.5) can be written respectively as
Var (
ˆ
R
SRSS2(s)
) =
L
h=1
W
2
h
m
n
2
h
µ
2
X
R
2
h
r
h
i=1
σ
2
Xh(i)
+
r
h
i=1
σ
2
Yh[i]
2R
h
r
h
i=1
σ
Xh(i)Yh[i]
and
Var (
ˆ
R
SRSS2(c)
) =
L
h=1
W
2
h
m
n
2
h
µ
2
X
R
2
r
h
i=1
σ
2
Xh(i)
+
r
h
i=1
σ
2
Yh[i]
2R
r
h
i=1
σ
Xh(i)Yh[i]
where
σ
2
Xh(i)
=Var
X
h(i)
2
Yh[i]
=Var
Y
h[i]
and σ
Xh(i)Yh[i]
=Cov
X
Xh(i)
, Y
Yh[i]
.
Thus,
Var(
ˆ
R
SRSS2(c)
)Va r (
ˆ
R
SRSS2(s)
) =
m
µ
2
X
L
h=1
W
2
h
n
2
h
(R
2
R
2
h
)
r
h
i=1
σ
2
Xh(i)
2(R R
h
)
r
h
i=1
σ
Xh(i)Yh[i]
=
m
µ
2
X
L
h=1
W
2
h
n
2
h
(R R
h
)
2
r
h
i=1
σ
2
Xh(i)
+2(R
h
R)
×
r
h
i=1
σ
Xh(i)Yh[i]
R
h
r
h
i=1
σ
2
Xh(i)

.
As in the case of SSRS (see Cochran, 1977), if the ratio estimate is valid, the
last term on the right is usually small. (It vanishes if within each stratum the
relationship between Y
h[i]
and X
h(i)
is a straight line through the origin).
Also, as in Cochran (1977), unless R
h
is constant from stratum to stratum,
the use of a separate ratio estimate in each stratum is likely to be more precise
if the sample in each stratum is large enough so that the approximate formula
for Var(
ˆ
R
SRSS2(s)
) is valid and the cumulative bias that can effect the ratio
estimate is negligible. With only a small sample in each stratum, the combined
estimate is to be recommended.
Similarly, we can show that similar conclusions hold in the case when
ranking on Y is perfect and ranking of X is not perfect.
Ratio estimation using stratified ranked set sample 85
5. Simulation study
5.1. Design of the simulation study
We did computer simulation to gain insight in the properties of the ra-
tio estimator. Bivariate random observations were generated from a bivariate
normal distribution with parameters µ
Xh
Yh
Xh
Yh
, h = 1, 2,... ,L and
correlation coefficient ρ. Also we divide the data in the sample into three
strata and in some cases into four strata. The simulation was performed with
r = 10, 20, 30 and with m = 1 for the SRSS, SSRS, RSS and SRS data sets.
The ratio of the population means were estimated for these sampling meth-
ods. Using 2000 replications, estimates of the means and mean square errors
were computed.
We considered ranking on either variable Y or X. Results of these sim-
ulations are summarized by the relative efficiencies of the estimators of the
population ratio and by the bias of estimation for different values of the cor-
relation coefficient ρ.Inorder to reduce the size of this paper we present
two tables only. Tables 1 gives efficiency when ranking is perfect on variable
X and Y respectively. Tables 2 gives the bias in estimation when ranking is
perfect on variable X and Y respectively.
5.2. Results of the simulation study
We conclude that the largest gain in efficiency is obtained by ranking the
variable X and with large values of negative ρ.(Forexample in Table 1, the
relative efficiency when ρ = 0.90 and r = 30 is 1.97 while it is 4.16 when
ρ =−.90 and r = 30. The results of simulation indicates that, when ranking
is on the variable X or Y, the efficiency will decrease with decreasing values
of ρ from 0.99 to 0, and start to increase as ρ decreases from 0 to -.99.
However, in the separate case this conclusion may be changed when r = 10
as we indicated in Section 4, when the sample size is small we cannot use
the separate ratio estimate. Moreover, the efficiency will increase when the set
size (r) is increased.
Also, there will be no change in the efficiency if the sample size is increased
by increasing the number of cycles. Also, we note that in combined case for
any values of r or ρ
MSE(
ˆ
R
SRSS
) MSE(
ˆ
R
RSS
) MSE(
ˆ
R
SSRS
) MSE(
ˆ
R
SRS
)
when R = 1.45, W
1
= 0.3, W
2
= 0.3 and W
3
= 0.4 and have equals variances
within strata. This is not completely true for different cases, e.g., when R =
1.17 or when variances within strata are not equal.
86 H. M. SAMAWI – M. I. SIAM
Table 1. Relative effeciency of ratio estimators using SRSS relative to SSRS.
W
h
:.3/.3/.4 µ
Xh
: 2/3/4 σ
Xh
: 1/1/1
R = 1.45 µ
Xh
: 3/4/6 σ
Yh
: 1/1/1
ρ r Ranking on Variable X Ranking on Variable Y
Combined Separate Combined Separate
10 1.97 13.76 1.75 258.47
.99 20 2.98 3.66 2.17 2.81
30 3.35 3.78 2.77 3.29
10 1.52 8.18 2.00 0.43
.90 20 1.81 1.92 1.17 12.34
30 1.97 2.18 1.27 1.37
10 1.33 764.91 1.07 1.28
.70 20 1.54 1.82 0.98 1.05
30 1.69 1.83 1.01 1.06
10 1.43 28.96 1.06 0.88
.50 20 1.61 1.90 1.03 1.01
30 1.76 2.13 1.03 0.99
10 1.56 1.33 1.17 14.54
.25 20 1.77 2.13 1.14 1.38
30 1.87 2.01 1.27 1.13
10 1.49 2.85 1.26 0.70
.25 20 2.32 2.66 1.42 1.29
30 2.47 2.54 1.61 1.43
10 1.97 105.26 1.56 6.02
.50 20 2.46 2.97 1.69 1.62
30 2.75 3.22 2.00 1.97
10 1.82 17.51 1.88 3.02
.70 20 2.84 3.26 2.27 2.22
30 3.38 3.61 2.58 2.70
10 2.28 106.16 1.95 4.95
.90 20 3.37 3.52 2.74 2.92
30 4.16 4.58 3.88 4.06
10 2.07 8.16 2.09 9.61
.99 20 3.29 4.04 3.11 3.62
30 4.20 4.23 4.79 5.21
From Tables 2 it appears that the bias of
ˆ
R
SRSS
is higher when ρ is negative
than when it is positive. For example, the bias when ρ = 0.99 and r = 30 is
0.0016 while the bias when ρ =−0.99 and r = 30 is 0.0065. However, in
most cases the bias is less than 0.01 but for small r the bias in separate case
exceeds 0.01.
Ratio estimation using stratified ranked set sample 87
Table 2. Bias of ratio estimators using SRSS and SSRS.
W
h
:.3/.3/.4 µ
Xh
: 2/3/4 σ
Xh
: 1/1/1
R = 1.45 µ
Xh
: 3/4/6 σ
Yh
: 1/1/1
ρ r Ranking on Variable X Ranking on Variable Y
Combined Separate Combined Separate
SRSS SRSS SRSS SRSS SRSS SRSS SRSS SRSS
10 .0042 .0050 .0114 .0295 .0034 .0112 .0160 .0395
.99 20 .0018 .0024 .0027 .0095 .0013 .0029 .0056 .0146
30 .0016 .0019 .0016 .0062 .0020 .0041 .0052 .0133
10 .0060 .0068 .0146 .0336 .0001 .0031 .0044 .0327
.90 20 .0025 .0032 .0040 .0106 .0009 .0027 .0018 .0118
30 .0010 .0019 .0006 .0065 .0003 .0029 .0006 .0090
10 .0082 .0060 .0212 .1130 .0023 .0013 .0148 .0022
.70 20 .0037 .0048 .0071 .0171 .0045 .0044 .0137 .0056
30 .0005 .0034 .0011 .0110 .0005 .0006 .0056 .0006
10 .0061 .0100 .0223 .0668 .0053 .0099 .0331 .0304
.50 20 .0058 .0019 .0096 .0194 .0066 .0043 .0203 .0105
30 .0006 .0024 .0004 .0120 .0057 .0074 .0154 .0101
10 .0129 .0186 .0388 .0714 .0077 .0054 .0690 .0542
.25 20 .0015 .0015 .0069 .0254 .0075 .0027 .0316 .0129
30 .0007 .0004 .0023 .0119 .0088 .0104 .0214 .0203
10 .0211 .0107 .0517 .0517 .0069 .0264 .0810 .0732
.25 20 .0032 .0084 .0101 .0376 .0061 .0080 .0 .0467
30 .0021 .0046 .0061 .0220 .0045 .0083 .0238 .0357
10 .0173 .0227 .0565 .1588 .0186 .0402 .1078 .1617
.50 20 .0102 .0120 .0196 .0499 .0036 .0124 .0209 .0601
30 .0048 .0028 .0096 .0268 .0003 .0043 .0148 .0354
10 .0177 .0176 .0601 .1395 .0007 .0453 .0653 .2204
.70 20 .0097 .0145 .0206 .0526 .0023 .0166 .0255 .0706
30 .0041 .0083 .0089 .0313 .0060 .0152 .0193 .0516
10 .0183 .0265 .0630 .0733 .0088 .0442 .0772 .2112
.90 20 .0101 .0166 .0230 .0592 .0055 .0189 .0272 .0832
30 .0040 .0054 .0073 .0325 .0033 .0220 .0129 .0646
10 .0309 .0336 .0776 .1302 .0002 .0455 .0601 .2400
.99 20 .00 .0107 .0159 .0549 .0028 .0288 .0142 .1024
30 .0065 .0112 .0126 .0381 .0038 .0103 .0029 .0532
Also, the bias will decrease when the sample size is increased by increasing
r. The bias in the combined case is always less than the corresponding bias
in the separate case. Similar conclusions can be drawn when ranking on the
variable Y.However, the bias when ranking is on Y is slightly lower than
88 H. M. SAMAWI – M. I. SIAM
when ranking is on X. Moreover, it is clearly from equation (1.2) and (1.4)
that for negativethe ρ the variance of the ratio estimators, in case of separate
and combied methods, are larger than for positive ρ.However, our simulation
indicated that the efficiency of using SRSS, for ratio estimation, is higher when
ρ is negative. This may be due that the correlation between the order statistics
is always positive.
6. Ratio of bilirubin level to weight at birth
We give a real life example about Bilirubin level in jaundiced babies
who stay in neonatal intensive care. Most birth surveys on live newborns
showed that jaundice is common. Jaundice in new born can be pathological
and physiological. It start on second day of life and it has relationship with
race, method of feeding and gestational age.
On the other hand if the total serum bilirubin in blood is above 1.5 mg/dl
then we classify it as hyper-bilirubin. Neonatal jaundice is define as yellowish
discoloration of skin and sclera and it occurs if bilirubin level is more than
5mg/dl. (see Nelson et al., 1994).
Neonatal jaundice usually appears on the second day of life. Most of
normal newborn babies leave the hospital after 24 hours of life. Therefore,
our primary concern will be on babyies staying in neonatal intensive care.
Physicians are interested in the jaundice, because of the risk on the hearing,
brain and death. We will focus on the ratio of the level of bilirubin to the
weight at birth for the newborn babies.
The data was collected from five hospitals in Jordan. The jaundice is
measured by the level of bilirubin in the blood. This level is determined
according to a blood test (TSB), which takes nearly 30 minutes. Moreover,
ranking on the level of bilirubin in the blood can be done visually by observing
the following:
(i) Color of the face.
(ii) Color of the chest.
(iii) Color of the lower parts of the body and
(iv) the color of the terminal parts of the whole body.
Then as the yellowish goes from (i) to (iv) the bilirubin level in the blood
goes higher. We present below the analysis of the collected data for 120 babies
according to their weight at birth, sex and bilirubin level.
For illustration, assume that the collected data of 120 babies from the five
hospitals is the study population. Denote the bilirubin level by Y and the weight
at birth by X. Since there are two strata, L = 2, m = 2 and r = 10, then
n = r.m = 20, W
1
=
72
120
= 0.6 and W
2
=
48
120
= 0.4. Therefore, for male babies
Ratio estimation using stratified ranked set sample 89
n
1
= mr
1
= 0.6 × 20 = 12 and or for female babies n
1
= mr
2
= 0.4 × 20 = 8.
Also, the parameter of interest to be estimated is R = 3.90.
Using the sampling schemes of SRSS and SSRS, Table 3 contains the two
selected samples. Note that the ranking was on variable X (weight).
Table 3. The selected samples.
SRSS sample SSRS sample
Female Male Female Male
XY X Y XYXY
kg mg/dl kg Mg/dl kg mg/dl kg mg/dl
2.80 9.30 2.43 10.80 3.00 5.90 3.60 9.50
3.00 5.50 2.60 7.70 2.85 13.10 3.15 1.41
cycle 1 2.85 13.10 3.20 6.12 3.15 7.80 2.60 10.94
3.15 7.80 2.95 9.41 1.55 8.82 3.10 23.41
3.85 15.76 3.70 12.82
4.15 21.29 2.70 15.47
1.55 8.82 1.40 10.94 2.60 9.24 2.45 8.71
2.10 20.41 1.90 11.88 1.50 8.51 3.65 16.20
cycle 2 2.60 9.24 2.50 13.60 2.53 11.50 1.85 9.20
3.00 12.55 3.15 29.24 2.65 5.40 2.80 7.06
3.10 12.30 3.10 12.30
3.70 5.50 2.20 7.60
Based on the SRSS and SSRS, Table 4 contains the results of the illustration.
Table 4. Summary of the results of the illustration using Bilirubin data.
ˆ
R
SSRS(s)
ˆ
R
SSRS(C)
ˆ
R
SRSS(s)
ˆ
R
SRSS(C)
Estimate 3.63.68 4.32 4.32
Estimated Variance 0.14 0.13 0.13 0.12
Finally, we get eff(
ˆ
R
SRSS2(s)
,
ˆ
R
SRSS2(c)
) = 1.08, eff(
ˆ
R
SSRS2(s)
,
ˆ
R
SSRS2(c)
) =
1.07, eff(
ˆ
R
SSRS(c)
,
ˆ
R
SRSS2(c)
) = 1.1, eff(
ˆ
R
SSRS(s)
,
ˆ
R
SRSS2(s)
) = 1.1.
Acknowledgments
Theauthorswouldliketothanktherefereesfortheir comments which were helpful in improving
the paper.
90 H. M. SAMAWI – M. I. SIAM
REFERENCES
Cochran, W. G. (1977) Sampling Techniques, Third edition, John Wiley & Sons.
Dell, T. R. and Clutter, J. L. (1972) Ranked set sample theory with order statistics background,
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Hansen, M. H., Hurwitz., W. N., and Madow, W. G. (1953) Sampling survey methods and
theory,Vol. 2. John Wiley & Sons, New York.
Levy P. S. and Lemeshow S. (1991) Sampling ofpopulations methods and applications, John Wiley
& Sons, New York.
McIntyre, G. A. (1952) A method of unbiased selective sampling using ranked sets. Australian, J.
Agricultural Research,3,385-390.
Nelson, W. E., Behrman, R. E., Kliegman, R. M., and Vaughan, V. C. (1994) Textbook of
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Samawi, H. M. (1996) Stratified ranked set sample, Pakistan J. of Stat.,Vol. 12 (1), 9 - 16.
Samawi, H. M. and Muttlak, H. A. (1996) Estimation of ratio using rank set sampling, Biom.
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Stokes S. L. (1977) Ranked set sampling with concomitant variables, Comm. Statist. -Theor. Meth.,
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Stokes S. L. (1980) Estimation of the varianceusing judgment order ranked set samples, Biometrics,
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HANI M. SAMAWI
Department of Mathematics & Statistics
Sultan Qaboos University
P.O.Box 36
Al-khod 123, Sultanate of Oman
hsamawi@squ.edu.om
MAHMOUD I. SIAM
Department of Mathematics & Statistics
Sultan Qaboos University
P.O.Box 36
Al-khod 123, Sultanate of Oman
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Ranked set sampling has been shown by Dell and Clutter (1972, Biometrics 28, 545-553) to be a useful technique for improving estimates of the mean when actual measurement of the observations is difficult but ranking of the elements in a sample is relatively easy. The technique is extended here to show an estimator of variance, which is asymptotically unbiased regardless of the presence of errors in ranking. Furthermore, the asymptotic efficiency of these estimators, relative to those based on the same number of quantified observations from a random sample, is greater than unity for any underlying distribution, even if ranking errors occur.
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Ranked set sampling employs judgment ordering to obtain an estimate of a population mean. The method is most useful when the measurement or quantification of an element is difficult but the elements of a set of given size are easily drawn and ranked with reasonable success by judgment. In each set all elements are ranked but only one is quantified. Sufficient sets are processed to yield a specified number of quantified elements and the mean for each rank. The average of these means is an unbiased estimate of the population mean regardless of the errors in the ranking. Precision relative to random sampling, with the same number of units quantified, depends upon properties of the population and success in ranking. The ranked set concept is reviewed with particular consideration of the errors in judgment ordering.
Article
Ranked set sampling is a procedure which may be used to improve the precision of the estimator of the mean. It is useful in cases where the variable of interest is much more difficult to measure than to order. However, even if ordering is difficult, but there is an easily ranked concomitant variable available, then it may be used to “judgment order” the original variable. The amount of increase in the precision of the estimator is dependent upon the correlation between the 2 variables.
Article
Ranked set sampling (RSS) as suggested by McIntyre (1952) and developed by Takahasi and Wakimoto (1968) is used to estimate the ratio. It is proved that by using RSS method the efficiency of the estimator relative to the simple random sampling (SRS) method has increased. Computer simulated results are given. An example using real data is presented to illustrate the computations.
Textbook of Pediatrics, 4-th edn
  • W E Nelson
  • R E Behrman
  • R M Kliegman
  • V C Vaughan
Nelson, W. E., Behrman, R. E., Kliegman, R. M., and Vaughan, V. C. (1994) Textbook of Pediatrics, 4-th edn, W.B. Saunders Company Harcourt Barace Jovanovich, Inc.
Sultanate of Oman hsamawi@squ.edu.om MAHMOUD I
  • Al-Khod
Al-khod 123, Sultanate of Oman hsamawi@squ.edu.om MAHMOUD I. SIAM Department of Mathematics & Statistics Sultan Qaboos University P.O.Box 36