Chilean Journal of Statistics
Vol. 5, No. 1, April 2014, 49–72
Sampling Theory
Research Paper
Estimation of the ratio, product and mean using multi
auxiliary variables in the presence of non-response
Sunil Kumar
Alliance University, Bangalore, India
(Received: 05 December 2012 · Accepted in final form: 28 June 2013)
Abstract
This paper addresses the problem of estimating the population ratio, product and mean
using multi auxiliary information in presence of non-response. Some classes of estimators
have been proposed with their properties. Asymptotic optimum estimator(s) in the
class(s) have been investigated along with their mean squared error formulae. Further
the optimum value (depending upon population parameters) when replaced from sample
values gives the estimators having the mean squared errors of the asymptotic optimum
estimators. An empirical study is carried out in the support of the present study. Both
theoretical and empirical findings are encouraging and in favour of the present study.
Keywords: Study variate · Auxiliary variate · Bias · Mean squared error
· Non-response
Mathematics Subject Classification: Primary 62D05
1.
Introduction
In survey sampling, it is well recognized that the use of auxiliary information results in
substantial gain in efficiency over conventional estimators, which do not utilize such information. The problem of estimation of ratio, product and mean using single auxiliary
character has been dealt to great extent by several authors including Singh (1965), Rao
(1987), Bisht and Sisodia (1990), Naik and Gupta (1991), Upadhyaya and Singh (1999),
Singh and Tailor (2005 a,b) and Singh et al. (2007). Further the problem has been extended
by using supplementary information on additional auxiliary character by various authors
such as Chand (1975), Sahoo and Sahoo (1993), Sahoo et al. (1993), Sahoo and Sahoo
(1999) and Singh and Ruiz Espejo (2000).
Quite often information on many supplementary variables are available in the survey,
which can be utilized to increase the precision of the estimate. Olkin (1958) has considered
the use of multi auxiliary variables, positively correlated with the study variable to build
up a multi variate ratio estimator of the population mean. Singh (1967) extended Olkin’s
estimator to the case where auxiliary variables are negatively correlated with variate under
study. Later various authors including Shukla (1965, 1966), Mohanty (1967), Tujeta and
Bahl (1991) and Agrawal and Panda (1993, 1994) have used the information on several
auxiliary variables in building up estimators for population mean. Khare (1991) has suggested a generalized class of estimators for estimating the ratio of two means using multi
ISSN: 0718-7912 (print)/ISSN: 0718-7920 (online)
c Chilean Statistical Society – Sociedad Chilena de Estadística
⃝
http://www.soche.cl/chjs
50
S. Kumar
auxiliary characters with known population means.
It is well known especially in human surveys that information is generally not obtained
from all the sample units even after callbacks. The problem of estimating the parameters
such as ratio of two means, population mean and variance when some observations are
missing due to random non response has been discussed by Toutenberg and Srivastava
(1998), Singh and Joarder (1998), Singh S. et al. (2000), Singh and Tracy (2001) and
Singh H. P. et al. (2003). In case of non-random non-response, the problem of estimation
of population mean using information on single auxiliary character has been considered
by different authors such as El Badry (1956), Srinath (1971), Cochran (1977), Rao (1986,
1987), Khare and Srivastava (1993, 1995, 1997), Tabasum and Khan (2004); Tabasum and
Khan (2006), Khare and Sinha (2004, 2007), Singh and Kumar (2008 a,b, 2009 a,b, 2010,
2011), Kumar et al. (2011) and Gamrot (2011) have discussed the problem of estimating
the ratio of two means using multi auxiliary characters in the presence of non-response.
In this paper I have suggested some classes of estimators for ratio, product and mean using
multi auxiliary in different situations and their properties have been studied. Conditions
for attaining minimum mean squared error of the proposed classes of estimators have also
been obtained. Estimators based on estimated optimum values have been obtained with
their approximate mean squared error. An empirical study has been carried out in support
of the present study.
2.
Notations and sampling procedure
Let yil (i = 0, 1) and xjl (j = 1, 2, ..., p) be the non-negative values of lth unit of the study
variate yi (i = 0, 1) and the auxiliary variates xj (j = 1, 2, ..., p) for a population of size N
with population means Y i (i = 0, 1) and X j (j = 1, 2, ..., p). When non-response occurs, the
subsampling procedure of Hansen and Hurwitz (1946) is an alternative to call backs and
similar procedures. In this approach, the population of size N is assumed to be composed of
two strata of size N1 and N2 = N −N1 , of “respondents” and “non-respondents”. The initial
simple random sample of size n is drawn without replacement results in n1 respondents and
n2 non-respondents. A sub sample of size m = n2 /k, where (k > 1) is predetermined, is
drawn from the n2 non-respondents and through intensive efforts information on the study
variates yi (i = 0, 1) are assumed to be obtained from all of the m units (see, Rao (1983)).
Thus the estimator for the population mean Yi (i = 0, 1) of the finite population is
y ∗i = (n1 /n)y i(1) + (n2 /n)y i(2) , i = 0, 1
(2.1)
where y i(1) and y i(2) ;i = 0, 1 are the sample means of the characters y i (i = 0, 1) based on
n1 and m units respectively. The estimator y ∗i is unbiased and has variance
V ar (y ∗i ) =
(
1−f
n
)
Sy2i +
W2 (k − 1) 2
Syi (2)
n
(2.2)
where f = n/N , W2 = N2 /N , Sy2i and Sy2i (2) are the population mean square of the variates
yi (i = 0, 1) for the entire population and for non-responding group of the population.
Similarly the estimator x∗j (j = 1, 2, ...p) for the population mean X j is given by
x∗j = (n1 /n)xj(1) + (n2 /n)xj(2)
(2.3)
Chilean Journal of Statistics
51
The estimator x∗j (j = 1, 2, ..., p) is unbiased an has te variance
( )
V ar x∗j =
(
1−f
n
)
Sx2j +
W2 (k − 1) 2
Sxj (2)
n
(2.4)
where Sx2j and Sx2j (2) (j = 1, 2, ..., p) are the population mean square of xj for the entire
population and non responding group of the population.
ˆ ∗ = (y ∗ /y ∗α ), (y ∗ ̸= 0) denote the conventional estimator of the population
Let R
0 1
1
(α)
(
α)
parameter R(α) = Y 0 /Y 1 , Y 1 ̸= 0, α being a constant takes values (1,-1,0). For different
values of α, the following holds
∗
ˆ ∗ −→ R
ˆ ∗ = y0∗ = R
ˆ ∗ (say) is the conventional estimator of the ratio
(i) for α = 1, R
(α) (
(1) ) y 1
R(α) −→ R(1) = Y 0 /Y 1 = R (say).
∗ ∗
ˆ ∗ −→ R
ˆ∗
ˆ∗
(ii) for α = −1, R
(α)
(−1) = y 0 y 1 = P (say) is the conventional estimator of the
ratio R(α) −→ R(−1) = Y 0 Y 1 = P (say).
ˆ ∗ −→ R
ˆ ∗ = y ∗ , is the conventional estimator of the population mean
(iii) for α = 0, R
0
(α)
(0)
Y 0.
x∗
x∗
Let uj = Xj , for j = 1, 2, ..., p = Xj−p , for j = p + 1, p + 2, ..., 2p and u denotes the
j
j−p
column vector of 2p elements u1 , u2 , ..., u2p . Super fix T over a column vector denotes the
corresponding row vector. Defining
{(
)
} (
)
ˆ α∗ /Rα∗ − 1 ≈ η0 − αη1 − α2 η 2 − αη0 η1 ,
y ∗0 = Y 0 (1 + η0 ), y ∗1 = Y 1 (1 + η1 ), ε0 =
R
1
εj = (uj − 1), j = 1, 2, ..., 2p and let εT = (ε1 , ε2 , ..., ε2p ).
Then to the first degree of approximation, the following holds
[(
)
(
)]
W2 (k−1)
E(ε0 ) = α 1−f
C
(αC
−
ρ
C
)
+
C
αC
−
ρ
C
,
y
y
y
y
y
y
(2)
y
(2)
y
y
(2)
y
(2)
1
1
0
1
0
1
1
0
1
0
n
n
E(εj ) = 0 ∀ j = 1, 2, ..., 2p , [(
)
]
(2)
q(α)j + W2 (k−1)
E(ε0 εj ) = E [(η0 − αη1 )εj ] = 1−f
q(α)j , j = 1, 2, ..., p,
n
n
(
)
E(ε0 εj ) = 1−f
q(α)j , j = p + 1, p + 2, ..., 2p,
[( n )
]
W2 (k−1) (2)
E(εj εl ) = 1−f
a
+
a
= ejl (say), (j, l) = 1, 2, ..., p,
jl
jl
n
( n)
a
= 1−f
ajl = fjl (say), (j, l) = 1, 2, ..., p, p + 1, ..., 2p
n
where
2
2
(2)
ajl = ρxj xl Cxj Cxl , ajl = ρxj xl (2) Cxj (2) Cxl (2) , Cy2i = Sy2i /Y i , Cy2i (2) = Sy2i (2) /Y i , i = 0, 1,
2
2
Cx2j = Sx2j /X j , Cx2j (2) = Sx2j (2) /X j , j = 1, 2, ..., p,
(
) (2)
(
)
q(α)j = Cxj ρy0 xj Cy0 −αρy1 xj Cy1 , q(α)j = Cxj (2) ρy0 xj (2) Cy0 (2) −αρy1 xj (2) Cy1 (2) , j = 1, 2, ..., p,
(
)
(
)
ρy0 y1 , ρyi xj , ρxj xl , i = 0, 1; (j, l) = 1, 2, ..., p and ρy0 y1 (2) , ρyi xj (2) , ρxj xl (2) , i = 0, 1; (j, l) = 1, 2, ..., p
are the correlation coefficients between (y0 , y1 ) , (yi , xj ) and (xj , xl ) respectively for
the entire population and for the non-responding group of the population.
52
S. Kumar
bT(α)
aT(α)
(
)
(2) T
q(α)
T
C(α)
(
) (
)
T
= aT(α) , C(α)
= a(α)1 , a(α)2 , ..., a(α)p , C(α)1 , C(α)2 , ..., C(α)p
[(
)
]
(
)
1−f
W2 (k − 1) ( (2) )T
T
T
=
q(α) +
q(α)
, q(α)
= q(α)1 , q(α)2 , ..., q(α)p ,
n
n
)
]
[(
(
)
W2 (k − 1) (2)
1−f
(2)
(2)
(2)
q(α)j +
q(α)j , j = 1, 2, ..., p ,
= q(α)1 , q(α)2 , ..., q(α)p , a(α)j =
n
n
(
)
(
)
[
]
1−f
1−f
E F
T
=
q(α) , C(α)j =
q(α)j , j = 1, 2, ..., p, D =
FT F
n
n
which is assumed to be positive definite. The matrices E = (ejl )p×p and F = (fjl )p×p are
p × p matrices. Now utilizing the multi auxiliary characters with known population means,
I have suggested a class of estimators for the parameter R(α) in section 3.
3.
The class of estimators
Suppose non-response occurs on the study variables (y0 , y1 ), information on the pauxiliary variables xj , j = 1, 2, ..., p are obtained from all sample units (i.e. the
initial sample units), and the population means X j , j = 1, 2, ..., p of p-auxiliary
variables are known. In this situation we note that when suggesting the estimator for the population parameter R(α) , Khare and Sinha (2007) used only the information on the sample means xj , j = 1, 2, ..., p and on the population means
X j , j = 1, 2, ..., p of the p-auxiliary variables xj , j = 1, 2, ..., p. However one can
also obtain the unbiased estimators x∗j = (n1 /n) xj(1) + (n2 /n) xj(2) of the population
mean X j , j = 1, 2, ..., p (without any extra effort) while in the process of obtaining
y ∗i = (n1 /n) y i(1) + (n2 /n) y i(2) , i = 0, 1 the unbiased estimators of the population
means Y i , (i = 0, 1) . Thus, in the situation stated above we have two unbiased
estimators x∗j and xj of the population mean X j , j = 1, 2, ..., p of the auxiliary variate
xj , j = 1, 2, ..., p. With
this background
convinced
to suggest the class of
(
) author
(
)
ˆ ∗ , u1 , u2 , ..., u2p = G R
ˆ ∗ , uT of the population parameter
estimators, G(α) = G R
(α)
(α)
R(α) .
T denote the row vector of 2p unit elements. Whatever be the sample chosen, let
( Let e )
ˆ ∗ , uT assume values in a closed convex subset, S of the (2p + 1) dimensional real space
R
(α)
(
)
(
)
(
)
ˆ ∗ , uT be a function of R
ˆ ∗ , u1 , u2 , ..., u2p
containing the point R(α) , eT . Let G R
(α)
(α)
such that
(
)
G R(α) , eT = R(α) for all R(α)
(3.1)
and which is continuous and bounded with continuous and bounded first and second order
partial derivatives in S.
Define a class of estimators of the parameter R(α) as
(
)
(
)
ˆ ∗ , u1 , u2 , ..., u2p = G R
ˆ ∗ , uT
G(α) = G R
(α)
(α)
(3.2)
Since there are only a finite number of samples, the expectations and mean squared error
Chilean Journal of Statistics
53
of the estimator G(α) exist under the above conditions.
(
)
ˆ ∗ , uT in a second
To obtain the mean squared error of G(α) , expand the function G R
(α)
order Taylor’s series
)
(
) ( ∗
(
)
ˆ − R(α) ∂G( . )
G(α) = G R(α) , eT + R
+ (u − e)T G(1) R(α) , eT
∗
ˆ
(α)
∂ R(α)
(R(α) ,eT )
{
(
)2 2
(
)
ˆ ∗ −R(α) ∂ G(2. )
ˆ ∗ − R(α) (u − e)T ∂G(1) ( . )
+ 12
R
+
2
R
ˆ∗
ˆ∗
(α)
(α)
∂R
∂R
(α)
(α)
**
**
,u∗T )
,u∗T )
(Rˆ (α)
(Rˆ (α)
(
)
}
T
ˆ ** , u∗ (u − e) ,
+ (u − e)T G(2) R
(α)
)
(
ˆ ∗ − R(α) , u∗ = e + η (u − e) , 0 < η < 1; G(1) denotes the
ˆ ** = R(α) + η R
where R
(α)
(α)
2p elements column vector of first partial derivatives of G ( . ) and G(2) denotes 2p × 2p
ˆ ∗ and
matrix of second partial derivatives of G ( . ) with respect to u. Substituting for R
(α)
u in terms of η0 , η1 , ε0 and ε and using (3.1), one can get
{
} ∂G ( . )
G(α) =R(α) + R(α) (1 + η0 ) (1 + η1 )−α − 1
ˆ∗
∂R
(α)
(
)
+ εT G(1) R(α) , eT
+
}2 ∂ 2 G ( . )
1 {
−α
(1 + η0 ) (1 + η1 ) −1
ˆ ∗2
2
∂R
(α)
(R(α) ,eT )
**
,u∗T )
(Rˆ (α)
{
}
∂G(1) ( . )
+ 2R(α) (1 + η0 ) (1 + η1 )−α − 1 εT
ˆ∗
∂R
(α)
(
) ]
ˆ ** , u∗T ε .
+εT G(2) R
(α)
**
,u∗T )
(Rˆ (α)
(3.3)
Taking expectation in (3.3) and noting that the second partial derivatives are bounded,
the following theorem holds.
Theorem 3.1
(
)
(
)
E G(α) = R(α) + o n−1
From theorem 3.1, it follows that the bias of the estimator G(α) is of the order n−1 , and
hence its contribution to the mean squared error of G(α) will be of the order n−2 .
Now prove the following result
Theorem 3.2 To the first degree of approximation, the mean squared error of
(
)
G(1) R(α) , eT = −R(α) D−1 b(α)
(3.4)
54
S. Kumar
and the minimum mean squared error is given by
(
)
(
)
2
T
−1
ˆ∗
min. M SE G(α) = M SE R
(α) − R(α) b(α) D b(α)
(3.5)
where
(
)
[(
)
)
1−f ( 2
∗
2
2 2
ˆ
ˆ
M SE R(α) = R(α)
n ( Cy0 + α Cy1 − 2αρCy0 Cy1
)]
2
2C 2
+ W2 (k−1)
C
+
α
−
2αρC
C
y0 (2) y1 (2)
n
y0 (2)
y1 (2)
(3.6)
ˆ ∗ to the first degree of approximation.
is the mean squared error of R
(α)
(
)
Proof From (3.3), the M SE G(α) to the first degree of approximation is given by
(
)
(
)2
M SE G(α) = E G(α) − R(α)
[
= E R(α) (η0 − αη 1 )
From (3.1) which implies that
∂G( . )
ˆ?
∂R
(α)
∂G( . )
ˆ∗
∂R
(α)
+
εT G(1)
(
R(α)
, eT
]2
)
(3.7)
(R(α) ,eT )
= 1.
(R(α) ,eT )
Thus the expression (3.7) reduces to
(
)
[
(
)]2
MSE G(α)
= E R(α) (η0 − αη 1 ) + εT G(1) R(α) , eT ,
[
(
)
2 (η − αη )2 + 2R
T (1) R
T
= E R(α)
0
(α) (η0 − αη 1 ) ε G
(α) , e
1
(
(
))T T (1) (
)]
+ G(1) R(α) , eT
εε G
R(α) , eT ,
(
)
(
(
))
(
) (
(
))
T
T
(1) R
T + G(1) R
T T D G(1) R
ˆ∗
= MSE R
(3.8)
(α) , e
(α) , e
(α) , e
(α) + 2R(α) b(α) G
which is minimized for
(
)
G(1) R(α) , eT = −R(α) D−1 b(α)
(3.9)
Substituting (3.9) in (3.8), the resulting (minimum) mean squared error of G(α)
(
)
(
)
2
T
−1
ˆ∗
min.MSE G(α) = MSE R
(α) − R(α) b(α) D b(α)
(3.10)
Thus the theorem is proved.
Remark 3.1 The class of estimators G(α) at (3.2) is very large. If the parameters in the
)
(
ˆ (α) , uT are so chosen that they satisfy (3.4), the resulting estimator will
function G R
have MSE given by (3.5). A few examples are:
{
}
[
]
ˆ ∗ exp αT log u ,
ˆ ∗ 1 + φT (u − e) ,
(i) G(1) = R
(ii)
G
=
R
(2)
(α)
(α)
{
}
ˆ ∗ exp φT (u − e)
ˆ ∗ + φT (u − e)
(iii) G(3) = R
(iv)
G
=
R
(4)
(α) {
(α)
}
∗
∗
T
ˆ
ˆ
(v) G(5) = R(α) / R(α) − φ (u − e)
where φT = (φ1 , φ2 , ..., φ2p ) is a vector of 2p constants. The optimum values of these
Chilean Journal of Statistics
55
constants are obtained from (3.4). Since (3.4) involves 2p equations, taken exactly 2p
unknown constants in defining above estimators of the class.
4.
Estimator based on estimated optimum value
To obtain the estimator based on estimated optimum, adopt the same procedure as discussed in Singh (1982) and Srivastava and Jhajj (1983).
It is to be mentioned that the proposed class of estimator G(α) at (3.2) will attained
minimum MSE given by (3.5) (or (3.10)) only when the optimum value of the derivatives
(or constants involved in the estimators) given by (3.4), which are functions of the unknown
population parameters are used. To use such estimators in practice, one has to use some
guessed values of the parameters in (3.4), either through past experience or through a
pilot sample survey. It may be noted that even if the values of the constants used in the
estimator are not exactly equal to their optimum values as given by (3.4) (or (3.9)) but are
ˆ ∗ as has been
close enough, the resulting estimator will be better than usual estimator R
(α)
demonstrated by Das and Tripathi (1978). For more discussion on this point in connection
with the estimation of population mean the reader is referred to Srivastava (1966), Murthy
(1967, p.325), Reddy (1973, 1974) and Srivenkataramana and Tracy (1980).
However there are situations where the exact optimum values of the derivative given by
(3.4) or its guessed value may be rarely known in practice, hence it is advisable to replace
it by its estimate from sample values. We suppose that the equation (3.4) can be solved
uniquely for the 2p unknown constants in the estimator (3.2). The optimum values of these
constants will involve D−1 b(α) or may be both D−1 b(α) and R(α) , which are unknown.
When these optimum values are inserted in (3.2), it no longer remains an estimator since
it involves unknown ψ = D−1 b(α) , and may be also R(α) . Let ψˆ be a consistent estimator
ˆ∗
of ψ computed from the sample data at hand. Then replace ψ by ψˆ and also R(α) by R
(α)
if necessary, in the optimum G(α) resulting in the estimator G∗(α) say, which will now be a
ˆ Define
ˆ ∗ , u and ψ.
function of R
(α)
(
)
ˆ ∗ , uT , ψˆT
G∗(α) = G∗ R
(α)
(4.1)
(
)
(
)
ˆ ∗ , uT , ψˆT is derived from the function G R
ˆ (α) , uT cited at
where the function G∗ R
(α)
(3.2) by replacing the unknown constants in it by the consistent estimates. The condition
(3.1) will then imply that
(
)
ˆ ∗ , eT , bT = R(α) for all R(α) ,
G∗ R
(α)
(4.2)
which in turns implies
∂G∗ ( . )
ˆ∗
∂R
(α)
=1
(4.3)
(R(α) ,eT , ψ T )
We further assume that
∂G∗ ( . )
∂G ( . )
T
T = −R(α) ψ
|(R(α) ,eT , ψT ) =
|
∂u
∂u (R(α) ,e , ψ )
(4.4)
56
S. Kumar
and
∂G∗ ( . )
∂ ψˆ
(4.5)
=0
(R(α) ,eT , ψ T )
(
(
)
)
Expanding the function G∗ R(α) , uT , ψ T about the point R(α) , eT , ψ T in a Taylor’s
series and using (4.1) to (4.5), one get
)
(
) ( ∗
ˆ − R(α) ∂G∗ ( . )
G∗(α) = G∗ R(α) , eT , ψ T + R
+ (u − e)T
ˆ∗
(α)
∂R
(α)
T
T
(R(α) ,e ,ψ )
(
)T
∂G∗ ( . )
ˆ
+ ψ−ψ
+ second order terms,
ˆ
T
T
∂ψ
∂G∗ ( . )
∂u
(R(α) ,eT ,ψ T )
(R(α) ,e ,ψ )
(
)
= R(α) + R(α) ε0 + εT −R(α) ψ + second order terms.
(4.6)
Since ψˆ( is a )consistent estimator of ψ, the expectation of the second order terms in (4.6)
will be o n−1 and hence
(
)
(
)
E G∗(α) = R(α) + o n−1
From (4.6) one obtain
(
)
G∗(α) − R(α) = R(α) (η0 − αη 1 ) − R(α) εT ψ + second order terms.
(4.7)
Squaring both sides of (4.7) and neglecting terms of ε′ s having power greater than two
(
)2
2
2
2
G∗(α) − R(α) = R(α)
ψ T εεT ψ − 2R(α)
(η0 − αη 1 ) εT ψ
(η0 − αη 1 )2 + R(α)
or
(
)2
[
]
2
G∗(α) − R(α) = R(α)
(η0 − αη 1 )2 + ψ T εεT ψ − 2 (η0 − αη1 ) εT ψ
(4.8)
Taking expectations of both sides in (4.8) one get the mean squared error of G∗(α) , to the
first degree of approximation as
(
)
(
)
2
T
−1
ˆ∗
MSE G∗(α) = MSE R
(α) − R(α) b(α) D b(α)
which is same as given in (3.5) (or (3.10)) i.e.
(
)
(
)
(
)
2
T
−1
ˆ∗
MSE G∗(α) = min.MSE G(α) = MSE R
(α) − R(α) b(α) D b(α)
It may be noted that the following estimators:
(4.9)
Chilean Journal of Statistics
{
}
ˆ ∗ exp ψˆT log u
(i) d∗(1) = R
(α)
{
}
ˆ ∗ exp −ψˆT (u − e) ,
(iii) d∗(3) = R
(α)
}
{
∗
∗2 / R
ˆ ∗ +ψ T (u − e) , etc.
ˆ
(v) d(5) = R(α)
(α)
57
[
]
ˆ ∗ 1 − ψˆT (u − e) ,
(ii) d∗(2) = R
(α)
∗
ˆ
(iv) d = R∗ − ψˆT (u − e),
(4)
(α)
are the members of the suggested class of estimators G∗(α) . It can be shown to the first
degree of approximation that the mean squared errors of the estimators G∗(j) , j = 1 to 5
(
)
(
)
are same and equals to the MSE G∗(α) = min.MSE G(α) given by (4.9).
For different values of α one can obtain a class of estimators for ratio, product and
population mean for G∗(α) , H(α) , F(α) and J(α) respectively. The results are explained in
the Appendix I, II, III and IV, respectively.
Remark 4.1 Population means X 1 , X 2 , ..., X p are known, incomplete information on
the study variates (y0 , y1 ) and on the auxiliary variates xj (j = 1, 2, ..., p).
In this case we use information on (n1 + m) responding units on the study variates (y0 , y1 )
and the auxiliary variate xj (j = 1, 2, ..., p) from the sample of size n along with known
population means X 1 , X 2 , ..., X p . Thus propose a class of estimators for R(α) as
(
)
ˆ∗ , νT
H(α) = H R
(α)
(4.10)
where ν denotes the column vector of p elements ν1 , ν2 , ..., νp with νj = x∗j /X j ,
(
(
)
)
ˆ ∗ , ν T such that
ˆ ∗ , ν T is a function of R
j = 1, 2, ..., p; H R
(α)
(α)
(
)
H R(α) , eT = R(α) for all R(α)
⇒
∂H ( . )
ˆ∗
∂R
(α)
=1
(4.11)
(4.12)
(R(α) ,eT )
and also satisfies certain conditions similar to those given for the class of estimators G(α)
at (3.2) and eT denote the row vector of p unit elements.
It can be shown that
(
(
)
)
E H(α) = R(α) + o n−1 ,
and to the first degree of approximation the MSE of H(α) is given by
(
)
(
)
(
)
T
(1) R
T
ˆ∗
MSE H(α) = MSE R
(α) , e
(α) + 2R(α) a(α) H
(
(
))T ( (1) (
))
+ H (1) R(α) , eT
E H
R(α) , eT
which is minimized for
(
)
H (1) R(α) , eT = −R(α) E −1 a(α) ,
(4.14)
58
S. Kumar
and thus the resulting minimum mean squared error
)
(
(
)
2
T
−1
ˆ∗
min.MSE H(α) = MSE R
(α) − R(α) a(α) E a(α)
(4.14)
[{(
(
)
)
}T
W2 (k−1) (2)
1−f
∗
2
ˆ
= MSE R(α) −R(α)
q(α) +
q(α)
n
n
{(
)
}]
W2 (k−1) (2)
1−f
+E −1
q
,
q
+
(α)
n
n
(α)
(
)
where H (1) R(α) , eT denote the p elements column vector of first partial derivatives of
H ( . ).
Thus state the following theorem:
Theorem 4.1 Up to terms of order n−1 ,
(
)
]
(
) [
2
T
−1
ˆ∗
MSE H(α) ≥ MSE R
−
R
a
E
a
(α) ,
(α)
(α) (α)
with equality holding if
(
)
H (1) R(α) , eT = −R(α) E −1 a(α) .
Remark 4.2 Population means X 1 , X 2 , ..., X p are known, incomplete information on the
study variates (y0 , y1 ) and complete information on the auxiliary variates xj (j = 1, 2, ..., p).
In this case observe that n1 units respond on the study variates (y0 , y1 ) but there is complete information on the auxiliary variate xj (j = 1, 2, ..., p) and the population means
X 1 , X 2 , ..., X p are known. In such a situation define a class of estimators for population
parameter R(α) as
(
)
ˆ ∗ , wT
F(α) = F R
(α)
(4.15)
where w denotes the column vector of p elements w1 , w2 , ..., wp with wj = x∗j /X j ,
(
(
)
)
ˆ ∗ , wT such that
ˆ ∗ , wT is a function of R
j = 1, 2, ..., p; F R
(α)
(α)
(
)
F R(α) , eT = R(α) for all R(α) ,
⇒
∂F ( . )
ˆ∗
∂R
(α)
=1
(4.16)
(4.17)
T
(R(α) ,e )
and also satisfies certain conditions similar to those given for the class of estimators G(α)
at (3.2) and eT denote the row vector of p unit elements.
It can be shown that
Chilean Journal of Statistics
59
(
)
(
)
E F(α) = R(α) + o n−1 ,
and to the first degree of approximation, the MSE of F(α) is given by
(
)
(
)
(
)
T
(1) R
T
ˆ∗
MSE F(α) = MSE R
(α) , e
(α) + 2R(α) C(α) F
(
(
))T ( (1) (
))
+ F (1) R(α) , eT
F F
R(α) , eT
(4.18)
(
)
where F (1) R(α) , eT is the p elements column vector of the partial derivatives of F ( . ),
(
(
)
)
(
)
C(α) = C(α)1 , C(α)2 , ..., C(α)p , C(α)j = 1−f
q
=
C
ρ
C
−αρ
C
q
,
xj
y0 xj y0
(α)j
(α)j
y1 xj xj ,
n
)
(
ρxj xl Cxj Cxl .
F = (fjl )p×p , fjl = 1−f
n
The MSE of F(α) at (4.18) is minimized for
(
)
F (1) R(α) , eT = −R(α) F −1 C(α)
(4.19)
and thus the resulting minimum mean squared error of F(α) is given by
(
)
(
)
2
T
−1
ˆ∗
min.MSE F(α) = MSE R
C(α)
(α) − R(α) C(α) F
)
(
(
)
1−f
T
∗
2
ˆ
q(α)
F0−1 q(α)
= MSE R(α) − R(α)
n
(4.20)
where F0 = (ajl )p×p and ajl = ρxj xl Cxj Cxl .
Thus the following theorem holds.
Theorem 4.2 Up to terms of order n−1 ,
(
)
]
(
) [
∗
2
T
−1
ˆ
MSE F(α) ≥ MSE R(α) − R(α) C(α) F C(α) ,
with equality holding if
(
)
F (1) R(α) , eT = −R(α) F −1 C(α) .
Remark 4.3 Population means of auxiliary characters are unknown, incomplete information on the study variates (y0 , y1 ) and complete information on the auxiliary variates
xj (j = 1, 2, ..., p).
In this case, I use information on (n1 + m) responding units on the study variates
(y0 , y1 ) and complete information on the auxiliary variate xj (j = 1, 2, ..., p). Here in formulation of the estimator, in addition to xj (j = 1, 2, ..., p) I also use the information on
x∗j (j = 1, 2, ..., p) which can be easily computed while computing y ∗i (i = 0, 1). The population means xj (j = 1, 2, ..., p) of the auxiliary characters xj (j = 1, 2, ..., p) are not known.
With this background define a class of estimators for the parameter R(α) as
(
)
ˆ∗ , zT
J(α) = J R
(α)
(4.21)
60
S. Kumar
where z denotes the column vector of p elements z1 , z2 , ..., zp , super fix T( over a column
)
ˆ ∗ , z T is a
vector denotes the corresponding row vector, zj = x∗j /xj , j = 1, 2, ..., p; J R
(α)
(
)
∗
T
ˆ
such that
function of R(α) , z
(
)
J R(α) , eT = R(α) for all R(α)
⇒
∂J ( . )
ˆ∗
∂R
(α)
=1
(4.22)
(4.23)
T
(R(α) ,e )
and also satisfies certain conditions similar to those given for the class of estimators G(α)
at (3.2) and eT denote the row vector of p unit elements. It can be shown that
(
)
(
)
E J(α) = R(α) + o n−1 ,
and to the first degree of approximation the MSE of J(α) is given by
(
)
(
)
(
)
)
(2) T (1) (
ˆ∗
MSE J(α) = MSE R
−
2R
a
J
R(α) , eT
(α)
(α)
(α)
( (1) (
(
))T
(
))
2
+R(α)
J
R(α) , eT
M J (1) R(α) , eT
(4.24)
)
(1) (
where J(α) R(α) , eT denote the p elements column vector of the first partial derivatives
(
)
(
)
ˆ ∗ about the point R(α) , eT ;
ˆ ∗ , z T with respect to R
of J R
(α)
(α)
(
)
(
)
(2) T
(2)
(2)
(2)
(2)
(2)
a(α)
= a(α)1 , a(α)2 , ..., a(α)p , M = (mjl )p×p , a(α)j = W2 (k−1)
q(α)j , j = 1, 2, ..., p,
n
(
)
(
)
(
)
(2) T
(2) T
(2)
q(α) , q(α)j = Cxj (2) ρy0 xj (2) Cy0 (2) −αρy1 xj (2) Cy1 (2) , j = 1, 2, ..., p,
a(α)
= W2 (k−1)
n
mjl =
(2)
W2 (k−1)
ρxj xl (2) Cxj (2) Cxl (2) = W2 (k−1)
ajl ,
n
n
(j, l) = 1, 2, ..., p.
The MSE of J(α) at (4.24) is minimized for
(
)
(2)
(2)
J (1) R(α) , eT = −R(α) M −1 a(α) = −R(α) M0−1 q(α)
(
)
(2)
where M0 = ajl
(4.25)
(2)
p×p
, ajl = ρxj xl (2) Cxj (2) Cxl (2) .
Thus the resulting minimum mean squared error of J(α) is given by
(
)
(
)
(
)
(
)
(2) T
(2)
2
−1
ˆ∗
min.MSE J(α) = MSE R
−
R
a
M
a
(α)
(α)
(α)
(α)
(
)
(
)
(
)
(2) T
(2)
−1
2 W2 (k − 1)
ˆ∗
q
= MSE R
−
R
M
q
0
(α)
(α)
(α)
(α)
n
Thus we state the following theorem.
Theorem 4.3 Up to terms of order n−1 ,
(4.26)
Chilean Journal of Statistics
61
[
(
)
(
)
(
)]
(
)
(2) T
(2)
∗
2
−1
ˆ
MSE J(α) ≥ MSE R(α) − R(α) a(α) M
a(α)
with equality holding if
(
)
(2)
J (1) R(α) , eT = −R(α) M −1 a(α) .
5.
Efficiency comparisons
Note that
(
)T
(
)
bT(α) D−1 b(α) = aT(α) E −1 a(α) + F T E −1 a(α) − C(α) A−1 F T E −1 a(α) − C(α)
(5.1)
and
(
)
(
)
(2) T
(2)
T
bT(α) D−1 b(α) = C(α)
F −1 C(α) + a(α) M −1 a(α)
(5.2)
(
)
where A = F − F T E −1 F .
Thus from (3.10), the result follows
(
)
[
(
)
2
T
−1
ˆ∗
min.MSE G(α) = MSE R
−
R
(α)
(α) a(α) E a(α)
(
)T
(
)]
+ F T E −1 a(α) − C(α) A−1 F T E −1 a(α) − C(α)
[
(
)
(
)
(
)]
(2) T
(2)
2
T
−1
−1
ˆ∗
= MSE R
−
R
C
F
C
+
a
M
a
(α)
(α)
(α)
(α)
(α)
(α)
(5.3)
(5.4)
From (4.23), (4.37), (4.52) and (5.4), one obtain
(
)
(
)
min.MSE H(α) − min.MSE G(α) =
( T −1
)T
(
)
2
R(α)
F E a(α) − C(α) A−1 F T E −1 a(α) − C(α) ≥ 0
)
(
)
(
(
)
(
)
(2) T
(2)
2
min.MSE F(α) − min.MSE G(α) = R(α)
a(α) M −1 a(α) ≥ 0
(
)
(
)
T
2
min.MSE J(α) − min.MSE G(α) = R(α)
C(α)
F −1 C(α) ≥ 0
(5.5)
(5.6)
(5.7)
Thus from (5.5), (5.6) and (5.7), the following inequalities holds
(
)
(
)
min.MSE G(α) ≤ min.MSE H(α)
(
)
(
)
min.MSE G(α) ≤ min.MSE F(α)
(
)
(
)
min.MSE G(α) ≤ min.MSE J(α)
(5.8)
(5.9)
(5.10)
From (5.8), (5.9) and (5.10) it follows that the proposed class of estimators G(α) given
by (3.2) is the best (in the sense of having least minimum MSE) among the classes of
estimators G(α) , H(α) , F(α) and J(α) .
62
S. Kumar
6.
Empirical study
To demonstrate the performance of the suggested estimator relative to usual estimator
ˆ ∗ with α = 1, consider a natural population data earlier considered by Khare and Sinha
R
(α)
(2007). The description of the population is given below:
The data on the physical growth of upper-socio- economic group of 95 school going
children of Varanasi under an ICMR study, Department of Pediatrics, BHU during 198384 has been taken under study. The first 25% (i.e. 24 children) units have been considered
as non-response units. Denote by
y0 : Height (in cm) of the children, y1 : Weight (in kg) of the children,
x1 : Skull circumference (in cm) of the children, x2 : Chest circumference (in cm) of the
children.
The required values of the parameters are:
Y 0 = 115.9526,
Cy1 = 0.15613,
Cx1 (2) = 0.02478,
ρy1 x2 = 0.846,
ρx1 x1 (2) = 0.570,
Y 1 = 19.4968,
Cx1 = 0.03006,
Cx2 (2) = 0.054,
ρy0 x1 (2) = 0.571,
ρy0 y1 = 0.713,
X 1 = 51.1726,
Cx2 = 0.05860,
ρy0 x1 = 0.374
ρy0 x2 (2) = 0.401,
ρy0 y1 (2) = 0.678
X 2 = 55.8611,
Cy0 (2) = 0.044,
ρy0 x2 = 0.620,
ρy1 x1 (2) = 0.477,
Cy0 = 0.0515,
Cy1 (2) = 0.121,
ρy1 x1 = 0.328,
ρx1 x1 = 0.297,
To illustrate results, consider the difference type estimator using two auxiliary variables:
ˆ ∗ + α1 (u∗1 − 1) + α2 (u∗2 − 1) + φ1 (u1 − 1) + φ2 (u2 − 1)
td = R
(α)
(6.1)
(
)
where αi′ s and φ′i s, (i = 1, 2) are suitably chosen constants, u∗i = x∗i /X i and
(
)
ui = xi /X i , (i = 1, 2).
For the sake of convenience, the MSE of td to the first degree of approximation is given
by
[
(
) ∑
) ∑
(
2
2
2
∑
ˆ∗
MSE (td ) = MSE R
αj2 ej + 2α1 α2 e12 + 2R(α) αj a(α)j + 1−f
φ2j Cx2j +2φ1 φ2 a12
(α) +
n
j=1
j=1
2
∑
{
}
+ 2R(α) φj q(α)j + 2 α1 φ1 Cx21 +α2 φ1 a12 +α1 φ2 a12 +α2 φ2 Cx22
j=1
(6.2)
j=1
where
{(
)
}
{(
)
}
W2 (k−1) (2)
1−f
1−f
2 + W2 (k−1) C 2
ej =
C
,
j
=
1,
2;
e
=
a
+
a
;
12
12
x
12
n
n
n
n
xj (2)
( j
)
(2)
q(α)j = Cxj ρy0 xj Cy0 −αρy1 xj Cy1 , j = 1, 2; a12 = ρx1 x2 Cx1 Cx2 ; a12 = ρx1 x2 (2) Cx1 (2) Cx2 (2) ;
(
)
q(α)j(2) = Cxj (2) ρy0 xj (2) Cy0 (2) −αρy1 xj (2) Cy1 (2) , j = 1, 2;
)
}
{(
W2 (k−1)
1−f
q
+
q
a(α)j =
(α)j
(α)j(2) , j = 1, 2 .
n
n
Expression (6.2) can also be obtained from (3.8) just by putting the suitable values of
the derivatives. The MSE at (6.2) is minimized for
Chilean Journal of Statistics
63
α10 = R(α) d∗(2)
(6.3)
α20 = R(α) d∗1(2)
(
)
φ10 = R(α) d∗ − d∗(2)
(
)
φ20 = R(α) d∗1 − d∗1(2)
(6.4)
(6.5)
(6.6)
where
[q(α)2 ρx1 x2 Cx1 −q(α)1 Cx2 ] ∗ [q(α)1 ρx1 x2 Cx2 −q(α)2 Cx1 ]
d∗ =
, d1 =
,
[Cx21 Cx2 (1−ρ2x1 x2 )]
[Cx1 Cx22 (1−ρ2x1 x2 )]
[q(α)2(2) ρx1 x2 (2) Cx1 (2) −q(α)1(2) Cx2 (2) ] ∗
[q(α)1(2) ρx1 x2 (2) Cx2 (2) −q(α)2(2) Cx1 (2) ]
d∗(2) =
, d1(2) =
.
[Cx21 (2) Cx2 (2) (1−ρ2x1 x2 (2) )]
[Cx1 (2) Cx22 (2) (1−ρ2x1 x2 (2) )]
Putting (6.2)-(6.6) in (6.1), we get the asymptotic optimum estimator (AOE) in the class
of estimators td as
[
]
(0)
ˆ ∗ + R(α) d∗ (u∗1 − u1 ) + d∗ (u∗2 − u2 ) + d∗ (u1 − 1) + d∗1 (u2 − 1)
(6.7)
td = R
(α)
(2)
1(2)
(0)
The MSE of td to the first degree of approximation is given by
{
}
) (a C − a C )2 +2a a C C (1 − ρ
(
( )
(
)
)
1
x
2
x
1
2
x
x
x
x
1
2
1
2
1
2
(0)
2 1−f
ˆ∗
(
)
MSE td
= MSE R
(α) − R(α)
n
1 − ρ2x1 x2
{(
)2
(
)}
W2 (k − 1) a1(2) Cx1 (2) − a2(2) Cx2 (2) +2a1(2) a2(2) Cx1 (2) Cx2 (2) 1 − ρx1 x2 (2)
(
)
+
2
1 − ρ2x1 x2 (2)
= min.MSE (td )
(6.8)
where
C
C
C
C
C
C
a1 = ρy0 x1 Cxy0 −ρy1 x1 Cxy1 , a2 = ρy0 x2 Cxy0 −ρy1 x2 Cxy1 , a1(1) = ρy0 x1 (1) Cxy0 (2)
−ρy1 x1 (2) Cxy1 (2)
,
(2)
(2)
1
1
C
2
C
2
1
1
−ρy1 x2 (2) Cxy1 (2)
.
a2(2) = ρy0 x2 (2) Cxy0 (2)
(2)
(2)
2
2
In practice the optimum values of α1 , α2 , φ1 and φ2 given by (6.2)-(6.6) are not known.
In such a case it is worth advisable to replace them by their consistent estimators in (6.8)
and thus one get an estimator based on “estimated optimum values” as
[
]
(0)
ˆ ∗ 1 + dˆ∗ (u∗1 − u1 ) + dˆ∗ (u∗2 − u2 ) + dˆ∗ (u1 − 1) + dˆ∗1 (u2 − 1)
tˆd = R
(6.9)
(α)
(2)
1(2)
where dˆ∗ , dˆ∗1 , dˆ∗(2) and dˆ∗1(2) are the consistent estimators of d∗ , d∗1 , d∗(2) and d∗1(2) based
on the available data under the given sampling design. It can be easily shown to the first
degree of approximation that
( )
(0)
MSE tˆd
= min.MSE (td ) = min.MSE (td )
(6.10)
64
S. Kumar
( )
(0)
where MSE td
isgiven by (6.8).
Further consider the following difference type estimators:
ˆ ∗ + α1 (u∗ − 1) + α2 (u∗ − 1)
td1 = R
1
2
(α)
(6.11)
ˆ ∗ + φ1 (u1 − 1) + φ2 (u2 − 1)
td2 = R
(α)
(6.12)
ˆ ∗ + λ1 (z1 − 1) + λ2 (z2 − 1)
td3 = R
(α)
(6.13)
where z1 = x∗1 /x1 , z2 = x∗2 /x2 , αi′ s, φ′i s and λi′ s , (i = 1, 2) are suitably chosen constants.
To the first degree of approximation, the minimum MSE of td1 , td2 and td3 are respectively
given by
2
(
)
[(
]
R(α)
)2
(
)2
∗
ˆ
(
)
min.MSE (td1 ) = MSE R(α) −
a
e
+
a
e
−
2a
a
e
2
1
12
(α)1
(α)2
(α)1
(α)2
e1 e2 − e212
(6.14)
for optimum values of α1 and α2 given by
∗ =
α10
∗ =
α20
R(α) [a(α)2 e12 −a(α)1 e2 ]
(e1 e2 −e212 )
R(α) [a(α)1 e12 −a(α)2 e1 ]
(e1 e2 −e212 )
)
(
) (
1−f [
ˆ∗
min.MSE (td2 ) = MSE R
−
n
(α)
}
2
R(α)
2
x1 (2)
C
C
2
x2 (2)
−(a
(2)
12
(6.15)
,
)
]
2
[(
q(α)1(2)
)2
Cx22 (2)
]
(
)2 2
(2)
+ q(α)2(2) Cx1 (2) − 2q(α)1(2) q(α)2(2) a12 ,
(6.16)
for optimum values of φ1 and φ2 given by
φ∗10 =
(
)
ˆ∗
min.MSE (td3 ) = MSE R
(α) −
(2)
R(α) [q(α)1(2) a12 −q(α)2(2) Cx21 (2) ]
]
[
(2) 2
Cx21 (2) Cx22 (2) −(a12 )
(2)
φ∗20 =
R(α) [q(α)2(2) a12 −q(α)1(2) Cx22 (2) ]
]
[
(2) 2
Cx21 (2) Cx22 (2) −(a12 )
2
R(α)
W2 (k−1) [
]
(2) 2
n
Cx21 (2) Cx22 (2) −(a12 )
[(
(6.17)
,
q(α)1(2)
)2
Cx22 (2)
]
(
)2
(2)
+ q(α)2(2) Cx21 (2) − 2q(α)1(2) q(α)2(2) a12 ,
(6.18)
for optimum values of λ1 and λ2 given by
(2)
λ10 =
R(α) [q(α)1(2) a12 −q(α)2(2) Cx21 (2) ]
[
]
(2) 2
Cx21 (2) Cx22 (2) −(a12 )
(2)
λ20 =
R(α) [q(α)2(2) a12 −q(α)1(2) Cx22 (2) ]
[
]
(2) 2
Cx21 (2) Cx22 (2) −(a12 )
.
(6.19)
∗ , α∗ ), (φ∗ , φ∗ ) and (λ , λ ) are respecEstimators based on estimated values of (α10
10
20
20
10
20
tively given by
Chilean Journal of Statistics
65
(0)
∗
∗
ˆ∗ + α
tˆd1 = R
ˆ 10
(u∗1 − 1) + α
ˆ 20
(u∗2 − 1)
(α)
(6.20)
(0)
ˆ ∗ + φˆ∗ (u1 − 1) + φˆ∗ (u2 − 1)
tˆd2 = R
10
20
(α)
(6.21)
(0)
ˆ 10 (z1 − 1) + λ
ˆ 20 (z2 − 1)
ˆ∗ + λ
tˆd3 = R
(α)
(6.22)
∗ , α
∗ , φ
ˆ 10 and λ
ˆ 20 are the consistent estimates of the optimum value
where α
ˆ 10
ˆ 20
ˆ∗10 , φˆ∗20 , λ
∗
∗
∗
∗
α10 , α20 , φ10 , φ20 , λ10 and λ20 respectively based on the data available under the given
sampling design. To the first degree of approximation, it can be shown that
( )
(0)
MSE tˆd1 = min.MSE (td1 )
( )
(0)
MSE tˆ
= min.MSE (td2 )
(6.23)
(6.24)
d2
( )
(0)
MSE tˆd3 = min.MSE (td3 )
(6.25)
where min.MSE (td1 ), min.MSE (td2 ) and min.MSE (td3 ) are respectively given by (6.14),
(6.16) and (6.18).
)
(
)
(
(0)
(0)
(0)
(0)
, td1 or tˆd1 ,
I have computed the percent relative efficiencies (PREs) of td or tˆd
)
(
)
(
(0)
(0)
(0)
(0)
ˆ ∗ with α = 1 where
with respect to usual estimator R
and td3 or tˆd3
td2 or tˆd2
(α)
(0)
(0)
(0)
td1 , td2 and td3 are respectively the optimum estimators in td1 , td2 and td3 .
The findings are given in Table 1.
ˆ∗
Table 1. Percent relative efficiencies of the estimators with respect to R
with α = 1 for fixed n and different
(α)
values of k.
Estimator
ˆ∗
)
(R
(0)
(0)
td or tˆd
)
(
(0)
(0)
td1 or tˆd1
(
)
(0)
(0)
td2 or tˆd2
(
)
(0)
(0)
td3 or tˆd3
(1/5)
100.00
(1/k)
(1/4)
(1/3)
100.00 100.00
(1/2)
100.00
368.22
352.24
332.44
309.50
240.12
248.64
261.16
282.91
147.86
158.73
175.89
207.44
117.68
114.78
111.13
106.39
(
)
(0)
(0)
It is observed from Table 1 that the percent relative efficiencies of td or tˆd
)
)
(
(
(0)
(0)
(0)
(0)
decrease while the percent relative efficiencies of td1 or tˆd1
and
and td3 or tˆd3
(
)
(0)
(0)
ˆ ∗ as the sub-sampling fraction increases. It has
td2 or tˆd2
increase with respect to R
(
)
(
)
(
)
(0)
(0)
ˆ ∗ , t(0) or tˆ(0) , t(0) or tˆ(0)
is the best among R
also been perceived that td or tˆd
d1
d2
d2
(
)
(
) d1
(0)
(0)
(0)
(0)
ˆ
ˆ
and td3 or td3 . Thus, the suggested estimator td or td
is to be preferred for its use
in practice, when the difference type estimator using two auxiliary variables is used.
66
S. Kumar
7.
Conclusion
In the present problem, some classes of estimators for ratio, product and mean are discussed
by using multi auxiliary in different situations in the presence of non-response and their
properties have been studied. Conditions for attaining minimum mean squared error of
the proposed classes of estimators have also been obtained. Estimators based on estimated
optimum values have been obtained with their approximate mean squared error. Due to the
non-availability of the data, I have tried to show the performance of the suggested estimator
ˆ ∗ with α = 1 for two auxiliary variables. The performance
relative to usual estimator R
(α)
of the suggested estimator is preferable when the non-response occurs on the study as well
as auxiliary variables.
Acknowledgements
Author wish to thank the learned referees for their critical and constructive comments
regarding improvement of the paper
Appendix I
Putting α = 1, −1, 0 in (3.2) we get
(i) a class of estimators for ratio R as
(
)
ˆ ∗ , uT
G(1) = G R
(I-1)
(ii) a class of estimators for product P as
(
)
G(−1) = G Pˆ ∗ , uT
(I-2)
(iii) a class of estimators for population mean Y 0 as
(
)
G(0) = G y ∗0 , uT
(I-3)
The minimum mean squared errors of the estimators G(1) , G(−1) and G(0) are respectively
given by
( )
(
)
ˆ ∗ − R2 bT D−1 b(1)
min.MSE G(1) = MSE R
(1)
( )
(
)
min.MSE G(−1) = MSE Pˆ ∗ − P 2 bT(−1) D−1 b(−1)
(
)
2
min.MSE G(0) = Var (y ∗0 ) − Y (0) bT(0) D−1 b(0)
where
[(
)(
( )
)
ˆ ∗ = R2 1−f Cy2 +Cy2 −2ρy y Cy Cy
MSE R
0 1
0
1
n (
0
1
)],
2
2
+ W2 (k−1)
C
+C
−2ρ
C
C
y0 y1 (2) y0 (2) y1 (2)
y0 (2)
y1 (2)
n
(I-4)
(I-5)
(I-6)
(I-7)
Chilean Journal of Statistics
67
( )
[(
)(
)
2
2
MSE Pˆ ∗ = P 2 1−f
n ( Cy0 +Cy1 +2ρy0 y1 Cy0 Cy1
)],
Cy20 (2) +Cy21 (2) +2ρy0 y1 (2) Cy0 (2) Cy1 (2)
+ W2 (k−1)
n
(I-8)
ˆ ∗ and Pˆ ∗ to the first degree of approximation, respecare the mean squared errors of R
tively, and
Var (y ∗0 )
[(
=
1−f
n
)
2
S0∗
W2 (k − 1) ∗2
+
S0(2)
n
]
(I-9)
where
(
)
bT(1) = a(1)1 , a(1)2 , ..., a(1)p , C(1)1 , C(1)2 , ..., C(1)p ,
(
)
bT(−1) = a(−1)1 , a(−1)2 , ..., a(−1)p , C(−1)1 , C(−1)2 , ..., C(−1)p ,
(
)
bT(0) = a(0)1 , a(0)2 , ..., a(0)p , C(0)1 , C(0)2 , ..., C(0)p ,
[(
)
]
(2)
a(1)j = 1−f
q(1)j + W2 (k−1)
q(1)j , j = 1, 2, ..., p;
n
n
[(
)
]
W2 (k−1) (2)
a(−1)j = 1−f
q
+
q
(−1)j
n
(−1)j , j = 1, 2, ..., p;
[( n)
]
(2)
W2 (k−1)
q
+
a(0)j = 1−f
q
(0)j
n
n
(0)j , j = 1, 2, ..., p;
(
) (2)
(
)
q(1)j = Cxj ρy0 xj Cy0 −ρy1 xj Cy1 , q(1)j = Cxj (2) ρy0 xj (2) Cy0 (2) −ρy1 xj (2) Cy1 (2) ,
(
) (2)
(
)
q(−1)j = Cxj ρy0 xj Cy0 +ρy1 xj Cy1 , q(−1)j = Cxj (2) ρy0 xj (2) Cy0 (2) +ρy1 xj (2) Cy1 (2) ,
)
(
(2)
q(1)j ,
q(0)j = ρy0 xj Cy0 Cxj ,
q(0)j = ρy0 xj (2) Cy0 (2) Cxj (2) ,
C(1)j = 1−f
n
(
)
(
)
C(−1)j = 1−f
q(−1)j , C(0)j = 1−f
q(0)j .
n
n
8.
Appendix II
Putting α = 1, −1, 0 in (4.10) we get the class of estimators
(i) for ratio R as
(
)
ˆ∗, ν T
H(1) = H R
(II-1)
(ii) for product P as
(
)
H(−1) = H Pˆ ∗ , ν T
(II-2)
(iii) for population mean Y 0 as
(
)
H(0) = H y ∗0 , ν T
(II-3)
The minimum mean squared errors of the estimators H(1) , H(−1) and H(0) can be obtained from (4.13) by putting α = 1, −1, 0 and are respectively given by
68
S. Kumar
( )
(
)
ˆ ∗ − R2 aT E −1 a(1)
min.MSE H(1) = MSE R
(1)
(
)
(
)
min.MSE H(−1) = MSE Pˆ ∗ − P 2 aT(−1) E −1 a(−1)
(II-5)
(
)
2
min.MSE H(0) = Var (y ∗0 ) − Y (0) aT(0) E −1 a(0)
(II-6)
(II-4)
where
(
)
a(1) = a(1)1 , a(1)2 , ..., a(1)p ,
(
)
a(0) = a(0)1 , a(0)2 , ..., a(0)p ,
(
)
a(−1) = a(−1)1 , a(−1)2 , ..., a(−1)p ,
[(
)
]
W2 (k−1) (2)
a(1)j = 1−f
q
+
q
j
, j = 1, 2, ..., p;
(1)j
n
(1)
[(n )
]
(2)
W
(k−1)
2
a(−1)j = 1−f
q
j
, j = 1, 2, ..., p;
q
+
(−1)j
n
(−1)
[( n)
]
(2)
a(0)j = 1−f
q(0)j + W2 (k−1)
q(0) j , j = 1, 2, ..., p;
n
n
where
(2)
(2)
(2)
q(1)j , q(−1)j , q(0)j , q(1)j , q(−1)j , q(0)j are same as defined earlier.
It is to be mentioned that the class of estimators
( )
ˆ∗h ν T
t1 = R
(II-7)
of the ratio R reported by Khare and Sinha (2007) is a member of the proposed class of
estimator H(1) . To the first degree of approximation,
(
)
min.MSE (t1 ) = min.MSE H(1)
(II-8)
(
)
where min.MSE H(1) is given by (II-2).
9.
Appendix III
Putting α = 1, −1, 0 in (4.15) we get the class of estimators
(i) for ratio R as
(
)
ˆ ∗ , wT
F(1) = F R
(III-1)
(ii) for product P as
(
)
F(−1) = F Pˆ ∗ , wT
(III-2)
(iii) for population mean Y 0 as
)
(
F(0) = F y ∗0 , wT
(III-3)
The minimum mean squared errors of the estimators F(1) , F(−1) and F(0) are respectively
given by
Chilean Journal of Statistics
( )
(
)
ˆ ∗ − R2 C T F −1 C(1)
min.MSE F(1) = MSE R
(1)
(
)
(
)
T
min.MSE F(−1) = MSE Pˆ ∗ − P 2 C(−1)
F −1 C(−1)
(
)
2
T
min.MSE F(0) = Var (y ∗0 ) − Y (0) C(0)
F −1 C(0)
69
(III-4)
(III-5)
(III-6)
where
(
)
(
)
(
)
C(1) = (
C(1)1 ,) C(1)2 , ..., C(1)p , C(0) = C(0)1 , C(0)2 , ..., C(0)p , C(−1) =
( C(−1)1
) , C(−1)2 , ..., C(−1)p
C(1)j = 1−f
q(1)j , j = 1, 2, ..., p,
C(−1)j = 1−f
q(−1)j , j = 1, 2, ..., p,
n
( n )
C(0)j = 1−f
q(0)j , j = 1, 2, ..., p,
n
q(1)j , q(−1)j and q(0)j , j = 1, 2, ..., p are same as defined earlier.
Khare and Sinha (2007) suggested a class of estimators for ratio R as
( )
ˆ ∗ f wT
t2 = R
(III-7)
( )
( )
where f wT is a function of wT = (w1 , w2 , ..., wp ) such that f eT = 1. The estimator
t2 is due to Khare and Sinha (2007) a member of the class F(1) defined at (III-7). The
minimum MSE of t2 is given by
( )
(
)
ˆ ∗ − R2 C T F −1 C(1)
min.MSE (t2 ) = min.MSE F(1) = MSE R
(1)
(III-8)
Appendix IV
Putting α = 1, −1, 0 in (4.21) we get the class of estimators
(i) for ratio R as
(
)
ˆ∗, zT
J(1) = J R
(IV-1)
(ii) for product P as
(
)
J(−1) = J Pˆ ∗ , z T
(IV-2)
(iii) for population mean Y 0 as
(
)
J(0) = J y¯0∗ , z T
(IV-3)
The minimum mean squared errors of the estimators J(1) , J(−1) and J(0) are respectively
given by
70
S. Kumar
)
( )
(
)T
(
(
)
ˆ ∗ − R2 a(2) M −1 a(2)
min.MSE J(1) = MSE R
(1)
(1)
(IV-4)
( )
(
)T
(
)
(
)
(2)
(2)
min.MSE J(−1) = MSE Pˆ ∗ − P 2 a(−1) M −1 a(−1)
(IV-5)
)
(
)
(
(
)
(2)
(2) T
min.MSE J(0) = MSE (¯
y0∗ ) − Y¯02 a(0) M −1 a(0)
(IV-6)
where
(
)
(2)
(2)
(2)
(2)
(2)
a(1) = a(1)1 , a(1)2 , a(1)3 , ..., a(1)p ,
(
)
(2)
(2)
(2)
(2)
(2)
a(0) = a(0)1 , a(0)2 , a(0)3 , ..., a(0)p ,
(2)
a(0)j =
W2 (k−1) (2)
q(0)j ,
n
(
)
(2)
(2)
(2)
(2)
(2)
a(−1) = a(−1)1 , a(−1)2 , a(−1)3 , ..., a(−1)p ,
(2)
a(1)j =
(2)
(2)
W2 (k−1) (2)
a(−1)j = W2 (k−1)
q(1)j ,
q(−1)j ,
n
n
(
)
(2)
q(1)j = Cxj (2) ρy0 xj (2) Cy0 −ρy1 xj (2) Cy1 (2) ,
(
) (2)
(2)
q(−1)j = Cxj (2) ρy0 xj (2) Cy0 +ρy1 xj (2) Cy1 (2) , q(0)j = ρy0 xj (2) Cy0 Cxj (2) , j = 1, 2, ..., p.
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