Mohammad Khoshnevisan
School of Accounting and Finance
Griffith University, Australia
Housila P. Singh
School of Studies in Statistics, Vikram University
Ujjain - 456 010 (M. P.), India
Sarjinder Singh
Departament of Mathematics and Statistics
University of Saskatchewan, Canada
Florentin Smarandache
University of New Mexico, Gallup, USA
A General Class of Estimators of
Population Median Using Two
Auxiliary Variables in Double
Sampling
Published in:
M. Khoshnevisan, S. Saxena, H. P. Singh, S. Singh, F. Smarandache (Editors)
RANDOMNESS AND OPTIMAL ESTIMATION IN DATA SAMPLING
(second edition)
American Research Press, Rehoboth, USA, 2002
ISBN: 1-931233-68-3
pp. 26 - 43
Abstract:
In this paper we have suggested two classes of estimators for population median MY of the study character
Y using information on two auxiliary characters X and Z in double sampling. It has been shown that the
suggested classes of estimators are more efficient than the one suggested by Singh et al (2001). Estimators
based on estimated optimum values have been also considered with their properties. The optimum values
of the first phase and second phase sample sizes are also obtained for the fixed cost of survey.
Keywords: Median estimation, Chain ratio and regression estimators, Study variate, Auxiliary variate,
Classes of estimators, Mean squared errors, Cost, Double sampling.
2000 MSC: 60E99
1. INTRODUCTION
In survey sampling, statisticians often come across the study of variables which have highly skewed
distributions, such as income, expenditure etc. In such situations, the estimation of median deserves special
attention. Kuk and Mak (1989) are the first to introduce the estimation of population median of the study
variate Y using auxiliary information in survey sampling. Francisco and Fuller (1991) have also
considered the problem of estimation of the median as part of the estimation of a finite population
distribution function. Later Singh et al (2001) have dealt extensively with the problem of estimation of
median using auxiliary information on an auxiliary variate in two phase sampling.
Consider a finite population U={1,2,…,i,...,N}. Let Y and X be the variable for study and auxiliary
variable, taking values Yi and Xi respectively for the i-th unit. When the two variables are strongly related
but no information is available on the population median MX of X, we seek to estimate the population
median MY of Y from a sample Sm, obtained through a two-phase selection. Permitting simple random
sampling without replacement (SRSWOR) design in each phase, the two-phase sampling scheme will be as
follows:
(i)
The first phase sample Sn(Sn⊂U) of fixed size n is drawn to observe only X in order to
furnish an estimate of MX.
(ii)
Given Sn, the second phase sample Sm(Sm⊂Sn) of fixed size m is drawn to observe Y
only.
Assuming that the median MX of the variable X is known, Kuk and Mak (1989) suggested a ratio estimator
for the population median MY of Y as
26
M
Mˆ 1 = Mˆ Y X
Mˆ X
(1.1)
where M̂ Y and M̂ X are the sample estimators of MY and MX respectively based on a sample Sm of size
m. Suppose that y(1), y(2), …, y(m) are the y values of sample units in ascending order. Further, let t be an
integer such that Y(t) ≤ MY ≤Y(t+1) and let p=t/m be the proportion of Y, values in the sample that are less
than or equal to the median value MY, an unknown population parameter. If p̂ is a predictor of p, the
ˆ ( pˆ ) where pˆ = 0.5 . Kuk and Mak
sample median M̂ Y can be written in terms of quantities as Q
Y
(1989) define a matrix of proportions (Pij(x,y)) as
Y ≤ MY
P11(x,y)
P12(x,y)
P1⋅(x,y)
X ≤ MX
X > MX
Total
Y > MY
P21(x,y)
P22(x,y)
P2⋅(x,y)
Total
P⋅1(x,y)
P⋅2(x,y)
1
and a position estimator of My given by
( p)
Mˆ Y = Qˆ Y ( pˆ Y )
where
pˆ Y =
(1.2)
1  mx pˆ 11 ( x, y ) (m − mx ) pˆ 12 ( x, y ) 


+
m  pˆ ⋅1 ( x, y )
pˆ ⋅2 ( x, y )

 m pˆ ( x, y ) + (m − mx ) pˆ 12 ( x, y ) 
≈ 2 x 11

m


ˆ ij ( x, y ) being the sample analogues of the Pij(x,y) obtained from the population and mx the number
with p
of units in Sm with X ≤ MX.
~
~
Let FYA ( y ) and FYB ( y ) denote the proportion of units in the sample Sm with X ≤ MX, and X>MX,
respectively that have Y values less than or equal to y. Then for estimating MY, Kuk and Mak (1989)
suggested the 'stratification estimator' as
{
}
~ ( y)
( St )
Mˆ Y = inf y : FY ≥ 0.5
[
1 ~ ( y) ~ ( y)
FYA + FYB
where FˆY ( y ) ≅
2
(1.3)
]
It is to be noted that the estimators defined in (1.1), (1.2) and (1.3) are based on prior knowledge of the
median MX of the auxiliary character X. In many situations of practical importance the population median
MX of X may not be known. This led Singh et al (2001) to discuss the problem of estimating the
population median MY in double sampling and suggested an analogous ratio estimator as
Mˆ 1X
ˆ
ˆ
M 1d = M Y
Mˆ
X
27
(1.4)
ˆ 1 is sample median based on first phase sample Sn.
where M
X
Sometimes even if MX is unknown, information on a second auxiliary variable Z, closely related to X but
compared X remotely related to Y, is available on all units of the population. This type of situation has
been briefly discussed by, among others, Chand (1975), Kiregyera (1980, 84), Srivenkataramana and Tracy
(1989), Sahoo and Sahoo (1993) and Singh (1993). Let MZ be the known population median of Z.
Defining
 Mˆ

 Mˆ

 Mˆ 1
  Mˆ

 M̂ 1

e0 =  Y − 1, e1 =  X − 1, e2 =  X − 1, e3  Z − 1 and e 4 =  Z − 1
 MY

MX

 MX
  MZ

 MZ

1
such that E(ek)≅0 and ek<1 for k=0,1,2,3; where M̂ 2 and M̂ 2 are the sample median estimators based
on second phase sample Sm and first phase sample Sn. Let us define the following two new matrices as
X ≤ MX
X > MX
Total
Z ≤ MZ
P11(x,z)
P12(x,z)
P1⋅(x,z)
Z > MZ
P21(x,z)
P22(x,z)
P2⋅(x,z)
Total
P⋅1(x,z)
P⋅2(x,z)
1
Y ≤ MY
Y > MY
Total
Z ≤ MZ
P11(y,z)
P12(y,z)
P1⋅(y,z)
Z > MZ
P21(y,z)
P22(y,z)
P2⋅(y,z)
Total
P⋅1(y,z)
P⋅2(y,z)
1
and
Using results given in the Appendix-1, to the first order of approximation, we have
N-m
E(e02) =  N  (4m)-1{MYfY(MY)}-2,


N-m
E(e12) =  N  (4m)-1{MXfX(MX)}-2,


N-n
E(e22) =  N  (4n)-1{MXfX(MX)}-2,
 
N-m
E(e32) =  N  (4m)-1{MZfZ(MZ)}-2,


N-n
E(e42) =  N  (4n)-1{MZfZ(MZ)}-2,
 
N-m
E(e0e1) =  N  (4m)-1{4P11(x,y)-1}{MXMYfX(MX)fY(MY)}-1,


N-n
E(e0e2) =  N  (4n)-1{4P11(x,y)-1}{MXMYfX(MX)fY(MY)}-1,
 
N-m
E(e0e3) =  N  (4m)-1{4P11(y,z)-1}{MYMZfY(MY)fZ(MZ)}-1,


N-n
E(e0e4) =  N  (4n)-1{4P11(y,z)-1}{MYMZfY(MY)fZ(MZ)}-1,
 
N-n
E(e1e2) =  N  (4n)-1{MXfX(MX)}-2,
 
N-m
E(e1e3) =  N  (4m)-1{4P11(x,z)-1}{MXMZfX(MX)fZ(MZ)}-1,


28
N-n
E(e1e4) =  N  (4n)-1{4P11(x,z)-1}{MXMZfX(MX)fZ(MZ)}-1,
 
N-n
E(e2e3) =  N  (4n)-1{4P11(x,z)-1}{MXMZfX(MX)fZ(MZ)}-1,
 
N-n
E(e2e4) =  N  (4n)-1{4P11(x,z)-1}{MXMZfX(MX)fZ(MZ)}-1,
 
N-n
E(e3e4) =  N  (4n)-1(fZ(MZ)MZ)-2
 
where it is assumed that as N→∞ the distribution of the trivariate variable (X,Y,Z) approaches a continuous
distribution with marginal densities fX(x), fY(y) and fZ(z) for X, Y and Z respectively. This assumption
holds in particular under a superpopulation model framework, treating the values of (X, Y, Z) in the
population as a realization of N independent observations from a continuous distribution. We also assume
that fY(MY), fX(MX) and fZ(MZ) are positive.
Under these conditions, the sample median M̂ Y is consistent and asymptotically normal (Gross, 1980) with
mean MY and variance
 N −m
−1
−2

(4m ) { f Y (M Y )}
 N 
In this paper we have suggested a class of estimators for MY using information on two auxiliary variables X
and Z in double sampling and analyzes its properties.
2. SUGGESTED CLASS OF ESTIMATORS
Motivated by Srivastava (1971), we suggest a class of estimators of MY of Y as
{
}
(g )
(g )
g = Mˆ Y : Mˆ Y = M Y g (u , v )
where u =
Mˆ X
,v =
Mˆ 1
X
(2.1)
Mˆ Z1
and g(u,v) is a function of u and v such that g(1,1)=1 and such that it satisfies
Mˆ
Z
the following conditions.
1.
Whatever be the samples (Sn and Sm) chosen, let (u,v) assume values in a closed convex subspace, P, of the two dimensional real space containing the point (1,1).
2.
The function g(u,v) is continuous in P, such that g(1,1)=1.
3.
The first and second order partial derivatives of g(u,v) exist and are also continuous in P.
Expanding g(u,v) about the point (1,1) in a second order Taylor's series and taking expectations, it is found
that
(
)
( )
E Mˆ Y g = M Y + 0(n −1 )
so the bias is of order n−1.
Using a first order Taylor's series expansion around the point (1,1) and noting that g(1,1)=1, we have
29
( )
(g )
Mˆ Y ≅ M Y [1 + e0 + (e1 − e2 )g1 (1,1) + e4 g 2 (1,1) + 0 n −1 ]
or
(M ( ) − M ) ≅ M [e + (e
g
Y
Y
Y
0
1
− e2 )g 1 (1,1) + e4 g 2 (1,1)] (2.2)
where g1(1,1) and g2(1,1) denote first order partial derivatives of g(u,v) with respect to u and v respectively
around the point (1,1).
ˆ ( g ) to the first degree
Squaring both sides in (2.2) and then taking expectations, we get the variance of M
Y
of approximation, as
(
)
(g )
Var Mˆ Y
=
where
1
4( f Y (M Y ))
2
 1 1   1 1 
1 1  
 m − N  +  m − n  A +  n − N  B ,
 


 

(2.3)
 M f (M ) 

 M f (M ) 
A =  Y Y Y  g1 (1,1)  Y Y Y  g1 (1,1) + 2(4 P11 ( x, y ) − 1)
 M X f X (M X ) 
 M X f X (M X ) 

(2.4)
 M f (M ) 

 M f (M ) 
B =  Y Y Y  g Z (1,1) Y Y Y  g 2 (1,1) + 2(4 P11 ( y, z ) − 1)
 M Z f Z (M Z ) 
 M Z f Z (M Z ) 

(2.5)
The variance of M̂ Y
(g )
in (2.3) is minimized for
 M f (M X ) 
g1 (1,1) = − X X
(4 P11 ( x, y ) − 1)
 M Y f Y (M Y ) 
 M f (M Z ) 
g 2 (1,1) = − Z Z
(4 P11 ( y, z ) − 1)
 M Y f Y (M Y ) 
Thus the resulting (minimum) variance of M Y
(
)
min. Var Mˆ Y ( g ) =
1
4( f Y (M Y ))2
(g )
(2.6)
is given by
 1 1   1 1 

2
1 1 
 m − N  −  m − n (4 P11 ( x, y ) − 1) −  n − N (4 P11 ( y, z ) − 1)
(2.7)
Now, we proved the following theorem.
Theorem 2.1 - Up to terms of order n-1,
(
) 4( f
Var M̂ Y g ≥
1
2
y (M Y )
)
 1 1   1 1 
2
2
1 1 
 m − N  −  m − n (4 P11 ( x, y ) − 1) −  n − N (4 P11 ( y, z ) − 1) 
with equality holding if
30
 M f (M ) 
g1 (1,1) = − x x x (4 P11 ( x, y ) − 1)
 M Y f Y (M Y ) 
 M f (M z ) 
g 2 (1,1) = − z z
(4 P11 ( y, z ) − 1)
 M Y f Y (M Y ) 
It is interesting to note that the lower bound of the variance of M̂ y
(g )
at (2.1) is the variance of the linear
regression estimator
(
)
(
(l )
Mˆ Y = Mˆ Y + dˆ1 Mˆ 1X − Mˆ X + dˆ 2 M Z − Mˆ Z1
)
(2.8)
where
(
(
fˆ (Mˆ
fˆ (Mˆ
)
)
) (4 pˆ
)
fˆ Mˆ x
(4 pˆ 11 ( x, y ) − 1),
dˆ1 = X
fˆ Mˆ
dˆ 2 =
Y
y
Z
Z
Y
11
Y
( y, z ) − 1),
ˆ 11 ( x, y ) and pˆ 11 ( y, z ) being the sample analogues of the p11 ( x, y ) and p11 ( y, z ) respectively
with p
and
( )
fˆY Mˆ Y , fˆX (M X ) and fˆZ (M Z ) can be obtained by following Silverman (1986).
Any parametric function g(u,v) satisfying the conditions (1), (2) and (3) can generate an asymptotically
acceptable estimator. The class of such estimators are large. The following simple functions g(u,v) give
even estimators of the class
g (1) (u , v ) = u α v β , g ( 2 ) (u , v ) =
1 + α (u − 1)
,
1 − β (v − 1)
g (3 ) (u, v ) = 1 + α (u − 1) + β (v − 1), g (4 ) (u , v ) = {1 − α (u − 1) − β (v − 1)}
g (5 ) (u , v ) = w1u α + w2 v β , w1 + w2 = 1
g (6 ) (u , v ) = αu + (1 − α )v β , g (7 ) (u , v ) = exp{α (u − 1) + β (v − 1)}
−1
ˆ ( g ) = Mˆ g (i ) (u , v ), (i = 1 to7 ) . It is
Let the seven estimators generated by g(i)(u,v) be denoted by M
Yi
Y
easily seen that the optimum values of the parameters α,β,wi(i-1,2) are given by the right hand sides of
(2.6).
3. A WIDER CLASS OF ESTIMATORS
The class of estimators (2.1) does not include the estimator
(
)
(
)
Mˆ Yd = Mˆ Y + d1 Mˆ 1X − M X + d 2 M Z − Mˆ Z1 , (d1 , d 2 )
being constants.
31
However, it is easily shown that if we consider a class of estimators wider than (2.1), defined by
(
(G )
Mˆ Y = G1 Mˆ Y , u, v
of MY, where G(⋅) is a function of
)
(3.1)
M̂ Y , u and v such that G (M Y ,1,1) = M Y and G1 (M Y ,1,1) = 1 .
G1 (M Y ,1,1) denoting the first partial derivative of G(⋅) with respect to M̂ Y .
Proceeding as in Section 2 it is easily seen that the bias of
terms, the variance of M̂ Y
(
Var M̂ Y
(G )
) = 4( f
(G )
(G )
is of the order n−1 and up to this order of
is given by
1
Y
M̂ Y
(M Y ))
2
 1 1   1 1  f Y (M Y ) 

[  −  +  − 
 m N   m n  M X f X (M X ) 
 f Y (M Y ) 

G2 (M Y ,1,1) + 2(4 P11 ( x, y ) − 1)
G2 (M Y ,1,1)
 M X f X (M X ) 

 1 1  f Y (M Y )
+ − 
 n N  f Z (M Z )M Z
 f Y (M Y ) 

G3 (M Y ,1,1) + 2(4 P11 ( y, z ) − 1) ]

 M Z f Z (M Z ) 

(3.2)
where G2(MY1,1) and G3(MY1,1) denote the first partial derivatives of u and v respectively around the point
(MY,(1,1).
The variance of M̂ Y
(G )
is minimized for
 M f (M X ) 
(4 P11 ( x, y ) − 1)
G2 (M Y ,1,1) = − X X
 f Y (M Y ) 
 M f (M Z ) 
(4 P11 ( y, z ) − 1)
G3 (M Y ,1,1) = − Z Z
 f Y (M Y ) 
Substitution of (3.3) in (3.2) yields the minimum variance of
(
min. Var M̂ Y
(G )
) = 4( f
M̂ Y
(G )
(3.3)
as
1
 1 1   1 1
1 1 
2
2
[
−  −  − (4 P11 ( x, y ) − 1) −  − (4 P11 ( y, z ) − 1) ]
2 
m N  m n
n N 
Y (M Y ))
(
= min.Var M̂ Y
(g)
)
(3.4)
Thus we established the following theorem. Theorem 3.1 - Up to terms of order n-1,
32
(
Var M̂ Y
(G )
) ≥ 4( f
1
2
Y (M Y ))
 1 1   1 1 
1 1 
2
2
 m − N  −  m − n (4 P11 ( x, y ) − 1) −  n − N (4 P11 ( y, z ) − 1) 
 





with equality holding if
 f (M X )M X 
(4 P11 ( x, y ) − 1)
G2 (M Y ,1,1) = − x
 f Y (M Y ) 
 M f (M Z ) 
(4 P11 ( y, z ) − 1)
G3 (M Y ,1,1) = − Z Z
(
)
f
M
Y
Y


If the information on second auxiliary variable z is not used, then the class of estimators M̂ Y
the class of estimators of MY as
(
(H )
Mˆ Y = H Mˆ Y , u
(
)
(
(G )
)
reduces to
(3.5)
)
ˆ , u is a function of Mˆ , u such that H (M ,1) = M and H (M ,1) = 1,
where H M
Y
Y
Y
Y
1
Y
∂H (⋅) 
(H )
. The estimator M̂ Y
is reported by Singh et al (2001).

∂Mˆ Y  ( M ,1)
Y
H 1 (M Y ,1) =
The minimum variance of
(
min.Var M̂ Y
(H )
M̂ Y
) = 4( f
(H )
to the first degree of approximation is given by
 1 1   1 1 
2
 m − N  −  m − n (4 P11 ( x, y ) − 1) 

 


1
2
Y (M Y ))
(3.6)
From (3.4) and (3.6) we have
(
minVar M̂ Y
(H )
) − min.Var(M̂ ( ) ) =  1n − N1  4( f
1
G
Y


Y
(M Y ))
which is always positive. Thus the proposed class of estimators M̂ Y
M̂ Y
(H )
(G )
2
(4 P11 ( y, z ) − 1)2
(3.7)
is more efficient than the estimator
considered by Singh et al (2001).
4. ESTIMATOR BASED ON ESTIMATED OPTIMUM VALUES
We denote
α1 =
M X f X (M X )
(4 P11 ( x, y ) − 1)
M Y f Y (M Y )
M f (M Z )
(4 P11 ( y, z ) − 1)
α2 = Z Z
M Y f Y (M Y )
33
(4.1)
In practice the optimum values of g1(1,1)(=-α1) and g2(1,1)(=-α2) are not known. Then we use to find out
their sample estimates from the data at hand. Estimators of optimum value of g1(1,1) and g2(1,1) are given
as
gˆ 1 (1,1) = −αˆ 1
gˆ 2 (1,1) = −αˆ 2
(4.2)
where
(
(
(
(
)
)
)
)
fˆX Mˆ X
(4 pˆ 11 ( x, y ) − 1)
ˆ Mˆ
f
Y Y
Y
ˆ
Mˆ f Mˆ
αˆ 2 = Z Z Z (4 p11 ( y, z ) − 1)
Mˆ Y fˆY Mˆ Y
αˆ 1 =
Mˆ X
Mˆ
(4.3)
Now following the procedure discussed in Singh and Singh (19xx) and Srivastava and Jhajj (1983), we
define the following class of estimators of MY (based on estimated optimum) as
( g *)
= Mˆ Y g * (u , v, αˆ 1 , αˆ 2 )
Mˆ Y
(4.4)
where g*(⋅) is a function of (u , v, αˆ 1 , αˆ 2 ) such that
g * (1,1,α 1α 2 ) = 1
g1* (1,1,α 1 , α 2 ) =
∂g * (⋅)
= −α 1
∂u (1,1,α1 ,α 2 )
g 2* (1,1,α 1 , α 2 ) =
∂g * (⋅)
= −α 2
∂v (1,1,α1 ,α 2 )
g 3* (1,1,α 1 , α 2 ) =
∂g * (⋅)
=0
∂αˆ 1 (1,1,α ,α )
1
g 4* (1,1,α 1 , α 2 ) =
2
∂g * (⋅)
=0
∂αˆ 2 (1,1,α ,α )
1
2
and such that it satisfies the following conditions:
1.
2.
3.
Whatever be the samples (Sn and Sm) chosen, let u , v, αˆ 1αˆ 2 assume values in a closed convex subspace, S, of the four dimensional real space containing the point (1,1,α1,α2).
The function g*(u,v, α1, α2) continuous in S.
The first and second order partial derivatives of g * (u , v, αˆ 1 , αˆ 2 ) exst. and are also continuous in
S.
Under the above conditions, it can be shown that
(
)
( )
( g *)
E Mˆ Y
= M Y + 0 n −1
34
and to the first degree of approximation, the variance of
(
( g *)
is given by
Mˆ Y
)
(
( g *)
g
Var Mˆ Y
= min.Var M̂ Y
(
(g )
where min.Var M̂ Y
)
(4.5)
) is given in (2.7).
A wider class of estimators of MY based on estimated optimum values is defined by
(
(G* )
Mˆ Y
= G * Mˆ Y , u , v, αˆ 1* , αˆ 2*
)
(4.6)
where
( )
( )
Mˆ fˆ (Mˆ )
(4 pˆ
=
fˆ (Mˆ )
αˆ 1* =
Mˆ X fˆX Mˆ X
(4 pˆ 11 ( x, y ) − 1)
fˆ Mˆ
Y
αˆ 2*
Z
Y
Z
Z
11
Y
(4.7)
( y, z ) − 1)
Y
are the estimates of
α 1* =
M x f X (M X )
(4 P11 ( x, y ) − 1)
f Y (M Y )
M f (M Z )
(4 P11 ( y, z ) − 1)
α = Z Z
f Y (M Y )
*
2
(
)
ˆ , u , v, α , αˆ such that
and G*(⋅) is a function of M
Y
1
2
(
)
(
)
*
*
G * M Y ,1,1,α1* ,α 2* = M Y
G1* M Y ,1,1,α 1* ,α 2* =
∂G * (⋅)
∂Mˆ Y (M
* *
Y ,1,1,α1 ,α 2
)
=1
(
)
∂G * (⋅)
= −α 1*
∂u (M Y ,1,1,α1* ,α 2* )
(
)
∂G * (⋅)
= −α 2*
∂v (M Y 1,1,∂1*,α 2* )
(
)
∂G * (⋅)
∂αˆ 1* (M
G2* M Y 1,1,α 1* ,α 2* =
G3* M Y 1,1, α 1* ,α 2* =
G4* M Y 1,1, α 1* ,α 2* =
*
*
Y ,1,1,α1 ,α 2
)
=0
35
(4.8)
(
)
G5* M Y 1,1, α1* , α 2* =
∂G * (⋅)
∂αˆ 2* (M
* *
Y ,1,1,α1 ,α 2
)
=0
Under these conditions it can be easily shown that
(
)
( )
(G* )
E Mˆ Y
= M Y + 0 n −1
ˆ (G*) is given by
and to the first degree of approximation, the variance of M
Y
(
)
(
(G )
G*
Var Mˆ Y = min.Var Mˆ Y
where
(
min.Var M̂ Y
G
)
(4.9)
) is given in (3.4).
ˆ ( g *) and
It is to be mentioned that a large number of estimators can be generated from the classes M
Y
(G * )
based on estimated optimum values.
Mˆ Y
5. EFFICIENCY OF THE SUGGESTED CLASS OF ESTIMATORS FOR FIXED COST
The appropriate estimator based on on single-phase sampling without using any auxiliary variable is M̂ Y ,
whose variance is given by
( )
1
1 1
Var Mˆ Y =  − 
2
 m N  4( f Y (M Y ))
(5.1)
In case when we do not use any auxiliary character then the cost function is of the form C0-mC1, where C0
and C1 are total cost and cost per unit of collecting information on the character Y.
The optimum value of the variance for the fixed cost C0 is given by

 G 1 
Opt.Var Mˆ Y = V0 
− 
C
N 
0


( )
(5.2)
where
V0
1
4( f Y (M Y ))
(5.3)
2
When we use one auxiliary character X then the cost function is given by
C 0 = Gm + C 2 n,
(5.4)
where C2 is the cost per unit of collecting information on the auxiliary character Z.
The optimum sample sizes under (5.4) for which the minimum variance of M̂ Y
36
(H )
is optimum, are
m opt =
[
(V0 − V1 ) / C1
(V0 − V1 )C1 + V1C 2 ]
C0
n opt =
[
C 0 V1 / C 2
(V0 − V1 )C1 +
V1C 2
(5.5)
]
where V1=V0(4P11(x,y)-1)2.
Putting these optimum values of m and n in the minimum variance expression of M̂ Y
the optimum
(
min.Var M̂ Y
(H )
) as
)] (

(H )
Opt. min.Var Mˆ Y
=


[
(H )
(
(V0 − V1 )C1 +
V1C 2
)
2
C0
−
in (3.6), we get
V0 

N

(5.7)
Similarly, when we use an additional character Z then the cost function is given by
C 0 = C1 m + (C 2 + C 3 )n
(5.8)
where C3 is the cost per unit of collecting information on character Z.
It is assumed that C1>C2>C3. The optimum values of m and n for fixed cost C0 which minimizes the
ˆ
minimum variance of M
Y
(g )
(orMˆ ) (2.7) (or (3.4)) are given by
(G )
Y
m opt =
n opt =
[
[
(V0 − V1 ) C1
(V0 − V1 )C1 + (C 2 + C3 )(V1 − V2 ) ]
C0
(5.9)
(V1 − V2 ) C 2 + C3
(V0 − V1 )C1 + (C 2 + C3 )(V1 − V2 ) ]
C0
(5.10)
where V2=V0(4P11(y,z)-1)2.
The optimum variance of
(
)
(g )
(G )
corresponding to optimal two-phase sampling strategy is
Mˆ Y orMˆ Y
[ (V0 − V1 )C1 + (C 2 + C 3 )(V1 − V2 )] 2 V2 
(g )
(G )
ˆ
ˆ
Opt min.Var M Y or min.Var M Y
=
− 
C0
N 

[
(
)
(
)]
(5.11)
Assuming large N, the proposed two phase sampling strategy would be profitable over single phase
sampling so long as
[Opt.Var(Mˆ )] > Opt.[min.Var(Mˆ ( ) )or min.Var(Mˆ ( ) )]
g
Y
Y
37
G
Y
i.e.
C 2 + C 3  V0 − V0 − V1 
<

C1
V1 − V2


(5.12)
When N is large, the proposed two phase sampling is more efficient than that Singh et al (2001) strategy if
[
(
)
(
)]
[
(
(g )
(G )
(H )
Opt min.Var Mˆ Y or min.Var Mˆ Y
< Opt min.Var Mˆ Y
i.e.
)]
C 2 + C3
V1
<
C1
V1 − V2
(5.13)
6. GENERALIZED CLASS OF ESTIMATORS
We suggest a class of estimators of MY as
{
(
(F )
(F )
ℑ = Mˆ Y : Mˆ Y = F Mˆ Y , u, v, w
)}
(6.1)
ˆ / Mˆ ′ , v = Mˆ ′ / M , w = Mˆ / M and the function F(⋅) assumes a value in a
where u = M
X
X
Z
Z
Z
Z
bounded closed convex subset W⊂ℜ4, which contains the point (MY,1,1,1)=T and is such that
F(T)=MY⇒F1(T)=1, F1(T) denoting the first order partial derivative of F(⋅) with respect to M̂ Y around the
point T=(MY,1,1,1). Using a first order Taylor's series expansion around the point T, we get
(
)
(F )
Mˆ Y = F (T ) + Mˆ Y = M Y F1 (T ) + (u − 1) F2 (T ) + (v − 1) F3 (T ) + ( w − 1) F4 (T ) + 0(n −1 )
(6.2)
(
)
ˆ , u, v, w with respect to u,
where F2(T), F3(T) and F4(T) denote the first order partial derivatives of F M
Y
v and w around the point T respectively. Under the assumption that F(T)=MY and F1(T)=1, we have the
following theorem.
Theorem 6.1. Any estimator in ℑ is asymptotically unbiased and normal.
Proof: Following Kuk and Mak (1989), let PY, PX and PZ denote the proportion of Y, X and Z values
respectively for which Y≤MY, X≤MX and Z≤MZ; then we have
Mˆ Y − M Y =
Mˆ X − M X =
1
1
(1 − 2 PY ) + 0 p  n − 2 ,


2 f Y (M Y )
1
1
(1 − 2 PX ) + 0 p  n − 2 ,


2 f X (M X )
Mˆ x′ − M X =
1
1
(1 − 2 PX ) + 0 p  n − 2 


2 f X (M X )
Mˆ z − M Z =
1
1
(1 − 2 PZ ) + 0 p  n − 2 


2 f Z (M Z )
38
and
Mˆ Z′ − M Z =
1
1
(1 − 2 PZ ) + 0 p  n − 2 


2 f Z (M z )
Using these expressions in (6.2), we get the required results.
Expression (6.2) can be rewritten as
(
)
(F )
Mˆ Y − M Y ≅ Mˆ Y − M Y + (u − 1)F2 (T ) + (v − 1) F3 (T ) + ( w − 1) F4 (T )
or
(F )
Mˆ Y − M Y ≅ M Y e0 + (e1 − e2 )F2 (T ) + e4 F3 (T ) + e3 F4 (T )
Squaring both sides of (6.3) and then taking expectation, we get the variance of M̂ Y
of approximation, as
(
)
(F )
Var Mˆ Y
=
1
2
4( f Y (M Y ))
(6.3)
(F )
 1 1 
 1 1
1 1  
 m − N  A1 +  m − n  A2 +  n − N  A3 ,
 





where
  f (M )  2 2

 f Y (M Y ) 
Y
Y
 F4 (T ) + 2(4 P11 ( y, z ) − 1)
 F4 (T )
A1 = 1 + 
  M Z f Z (M Z ) 

 M Z f Z (M Z ) 
 f Y (M Y )  2

 F2 (T ) + 2(4 P11 ( x, y ) − 1) F2 (T )

 f Y (M Y )   M X f X (M X ) 


A2 = 


(
)
M
f
M


X 
 X X
 + 2(4 P11 ( x, z ) − 1 f Y (M Y )  F2 (T ) F4 (T ) 
 M f (M ) 


Z 
 z Z
 f Y (M Y )  2

 F3 (T ) + 2(4 P11 ( y, z ) − 1) F3 (T )

 f Y (M Y )   M Z f Z (M Z ) 


A3 = 


 f Y (M Y ) 
 M Z f Z (M Z )  
 F3 (T ) F4 (T ) 
+ 2


 M Z f Z (M Z ) 
The
(
Var M̂ Y
(F )
) at (6.4) is minimized for
39
to the first degree
(6.4)
F2 (T ) = −
[(4 P11 ( x, y ) − 1) − (4 P11 ( x, z ) − 1)(4 P11 ( y, z ) − 1)] M X f X (M X )
⋅
⋅
2
f Y (M Y )
[1 − (4 P11 ( x, z ) − 1) ]
= − a 2 (say)
(6.5)
F3 (T ) = −
(4 P11 ( x, z ) − 1)[(4 P11 ( x, y ) − 1) − (4 P11 ( y, z ) − 1)(4 P11 ( x, z ) − 1)] ⋅ M Z f Z (M Z ) ⋅
2
f Y (M Y )
[1 − (4 P11 ( x, z ) − 1) ]
= −a 2 (say)
F4 (T ) = −
[(4 P11 ( y, z ) − 1) − (4 P11 ( x, y ) − 1)(4 P11 ( x, z ) − 1)] M Z f Z (M Z )
⋅
⋅
2
f Y (M Y )
[1 − (4 P11 ( x, z ) − 1) ]
= − a3 (say)
Thus the resulting (minimum) variance of
(
(F )
minVar Mˆ Y
)
M̂ Y
(F )
is given by
2
 1 1
 
D2
1 1 



 −  −  − 
+ (4 P11 ( x, y ) − 1) 
1
 m N   m n 1 − (4 P11 ( x, z ) − 1)2
 
=
2 

4(f Y (M Y ))
1 1 
2


−  − (4 P11 ( y, z ) − 1)


n N 
D2
1
 1 1
(G )
ˆ
= min.Var M Y
− − 
2
2
 m n  4( f Y (M Y )) 1 − 4 P11 ( x, z ) − 1
(
)
[
]
(6.6)
where
D = [(4 P11 ( y, z ) − 1) − (4 P11 ( x, y ) − 1)(4 P11 ( x.z ) − 1)]
(
and min.Var M̂ Y
(G )
) is given in (3.4)
Expression (6.6) clearly indicates that the proposed class of estimators M̂ Y
ˆ
class of estimator M
Y
(6.7)
(G )
(2001) and the estimator
(
or Mˆ Y
(g )
) and hence the class of estimators M̂
(F )
is more efficient than the
(H )
Y
suggested by Singh et al
M̂ Y at its optimum conditions.
The estimator based on estimated optimum values is defined by
{
(
F*
( F *)
: Mˆ Y = F * Mˆ Y , u , v, w, aˆ1 , aˆ 2 , aˆ 3
p* = Mˆ Y
where
40
)}
(6.8)
aˆ1 =
[(4 pˆ 11 (x, y ) − 1) − (4 pˆ 11 (x, z ) − 1)(4 pˆ 11 ( y, z ) − 1)] Mˆ x fˆx (Mˆ x ) ⋅
2
[1 − (4 pˆ 11 ( x, z ) − 1) ]
fˆY (Mˆ Y )
aˆ 2 =
(4 pˆ 11 ( x, z ) − 1)[(4 pˆ 11 ( x, y ) − 1) − (4 pˆ 11 ( y, z ) − 1)(4 pˆ 11 ( x, z ) − 1)] Mˆ Z fˆZ (Mˆ Z )
⋅
[1 − (4 pˆ 11 (x, z ) − 1)2 ]
fˆY (Mˆ Y )
a3 =
[(4 pˆ 11 ( y, z ) − 1) − (4 pˆ 11 (x, y ) − 1)(4 pˆ 11 (x, z ) − 1)] Mˆ Z fˆZ (Mˆ Z ) ⋅
[1 − (4 pˆ 11 (x, z ) − 1)2 ]
fˆY (Mˆ Y )
(6.9)
are the sample estimates of a1, a2 and a3 given in (6.5) respectively, F*(⋅) is a function of
(Mˆ
Y
)
, u , v, w, aˆ1 , aˆ 2 , aˆ 3 such that
F * (T *) = M Y
⇒ F1 * (T *) =
∂F * (⋅)
∂Mˆ
Y
=1
T*
F2 * (T *) =
∂F * (⋅)
= − a1
∂u T *
F3 * (T *) =
∂F * (⋅)
= −a 2
∂v T *
F4 * (T *) =
∂F * (⋅)
= −a3
∂w T *
F5 * (T *) =
∂F * (⋅)
=0
∂aˆ1 T *
F6 * (T *) =
∂F * (⋅)
=0
∂aˆ 2 T *
F7 * (T *) =
∂F * (⋅)
=0
∂aˆ 3 T *
where T* = (MY,1,1,1,a1,a2,a3)
Under these conditions it can easily be shown that
(
)
( )
( F *)
E Mˆ Y
= M Y + 0 n −1
41
ˆ ( F *) is given by
and to the first degree of approximation, the variance of M
Y
(
)
(
( )
) is given in (6.6).
where min.Var (M̂
F
( F *)
= min.Var Mˆ Y
Var Mˆ Y
)
(6.10)
F
Y
Under the cost function (5.8), the optimum values of m and n which minimizes the minimum variance of
M̂ Y
(F )
is (6.6) are given by
m opt =
(V0 − V1 − V3 ) / C1
(V0 − V1 − V3 )C1 + (V1 − V2 − V3 )(C 2 + C3 )]
C0
[
(6.11)
(V1 − V2 − V3 ) / C 2
(V0 − V1 − V3 )C1 + (V1 − V2 + V3 )(C 2 + C3 )]
C0
n opt =
[
where
D 2V0
V3 =
[1 − (4P (x, z ) − 1) ]
( )
) is given by
for large N, the optimum value of min.Var (M̂
( )
)] = [ (V − V − V )C + C(V − V
Opt.[min.Var(Mˆ
(6.12)
2
11
F
Y
F
0
1
3
1
1
2
+ V3 )(C 2 + C 3 )
]
Y
(6.13)
0
The proposed two-phase sampling strategy would be profitable over single phase-sampling so long as
[ ( )]
[
(
(F )
Opt. Var Mˆ Y > Opt. min.Var M̂ Y
)]
C 2 + C 3  V0 − V0 − V1 − V3 
i.e.
<

c1
V1 − V2 + V3


2
(6.14)
It follows from (5.7) and (6.13) that
[
(
Opt. min.Var Mˆ Y
if
 V0 − V1 − V0 − V1 − V3


V1 − V2 + V3

(F )
)] < Opt.[min.Var(Mˆ )]
H
Y
  C 2 + C3
>
−
 
C1
 
for large N.
Further we note from (5.11) and (6.13) that
42
V1
C2 
(V1 − V2 +V 3 )C1 C1 
(6.15)
[
(
)]
[
(
G
(F )
(g )
Opt. min.Var Mˆ Y
< Opt. min.Var Mˆ Y orMˆ Y
if
C 2 + C 3  (V0 − V1 ) − (V0 − V1 − V3 ) 
<

C1
 (V1 − V2 + V3 ) − V1 − V2 
)]
2
(6.16)
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Kiregyera, B. (1984): Regression-type estimators using two auxiliary variables and the model of double
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43