# On Improving Ratio/Product Estimator by Ratio/Product-cum-Mean-per-Unit Estimator Targeting More Efficient Use of Auxiliary Information

Authors: Angela Shirley; Ashok Sahai; Isaac Dialsingh
Journal: Journal of Probability and Statistics
 (2014)

---

## Abstract

To achieve a more efficient use of auxiliary information we propose single-parameter ratio/product-cum-mean-per-unit estimators for a finite population mean in a simple random sample without replacement when the magnitude of the correlation coefficient is not very high (less than or equal to 0.7). The first order large sample approximation to the bias and the mean square error of our proposed estimators are obtained. We use simulation to compare our estimators with the well-known sample mean, ratio, and product estimators, as well as the classical linear regression estimator for efficient use of auxiliary information. The results are conforming to our motivating aim behind our proposition.


---

## Body

## 1. Introduction and Notation

This paper addresses the problem of efficiently estimating the population mean, using auxiliary information. A fairly large simple random sample of sizen is selected without replacement from, say, a large bivariate population of size N (which could, reasonably, be thought to have come from a normal superpopulation), with the sampling fraction f
=
n
/
N, N
≫
n, so that f is negligible. Quite often, we have surveys where some auxiliary variable X may be relatively less expensive to observe than the main variable Y. In order to have a survey estimate of the population mean Y
- of the main variable, assuming knowledge of the population mean X
- of the auxiliary variable, the following estimators are well known.The ratio estimator:(1)
Y
-
^
R
=
R
^
X
-
.The product estimator:(2)
Y
-
^
P
=
P
^
X
-
.
Here R
^
=
y
-
/
x
- is the estimate of the ratio of the population means and P
^
=
y
-
·
x
- is the estimate of the product of the population means, y
- and x
- being unweighted sample means of the two variables, respectively. Usually, the variability of x
- is less than that of y
-.It is straightforward to derive first order approximations to the bias and mean square error of these estimators. LetC
Y
=
S
Y
/
Y
- and C
X
=
S
X
/
X
- be the population coefficients of variation of Y and X, respectively, where
(3)
S
Y
2
=
(
N
-
1
)
-
1
∑
(
Y
i
-
Y
-
)
2
,
S
X
2
=
(
N
-
1
)
-
1
∑
(
X
i
-
X
-
)
2
are the population variances of Y and X, respectively. Let y
-
=
Y
-
(
1
+
e
) and x
-
=
X
-
(
1
+
e
′
), where the errors e and e
′ can be positive or negative, so that E
(
e
)
=
E
(
e
′
)
=
0. It is known that, for simple random sample without replacement, Var
⁡
(
e
)
=
(
1
-
f
)
C
Y
2
/
n, Var
⁡
(
e
′
)
=
(
1
-
f
)
C
X
2
/
n, and Cov
(
e
,
e
′
)
=
(
1
-
f
)
ρ
C
X
C
Y
/
n where ρ is the correlation coefficient between the variables (P. V. Sukhatme and B. V. Sukhatme [1]). Further, to validate our first order large sample approximations, we assume that the sample is large enough to make |
e
| and |
e
′
| so small that the terms involving e and/or e
′ to a degree higher than two are negligible, an assumption which is not unrealistic.Substituting the expressions fory
- and x
- in terms of e and e
′ in (1) we have
(4)
Y
-
^
R
=
(
1
+
e
)
(
1
+
e
′
)
-
1
Y
-
.
Assuming that |
e
′
|
<
1, we expand (
1
+
e
′
)
-
1 to obtain
(5)
Bias
(
Y
-
^
R
)
=
(
1
-
f
)
n
Y
-
C
X
(
C
X
-
ρ
C
Y
)
,
up
to
the
first
order
of
approximation
,
O
(
1
n
)
.
Since (
1
-
f
)
/
n
→
0 as n
→
∞, we have that the ratio estimator is asymptotically unbiased up to O
(
1
/
n
). Similarly we have that the product estimator is asymptotically unbiased (Murthy [2]). Also,
(6)
MSE
(
Y
-
^
R
)
=
E
(
Y
-
^
R
-
Y
-
)
2
≈
E
[
Y
-
^
2
(
e
2
+
e
′
2
-
2
e
e
′
)
]
,
n
up
to
the
first
order
of
approximation
=
{
Y
-
2
(
1
-
f
)
n
}
{
C
Y
2
+
C
X
2
-
2
ρ
C
X
C
Y
}
=
Var
⁡
(
y
-
)
+
(
1
-
f
)
n
Y
-
2
{
C
X
2
-
2
ρ
C
X
C
Y
}
.
Thus up to order O
(
1
/
n
) of approximation, MSE
(
Y
-
^
R
)
<
Var
⁡
(
y
-
) if and only if C
X
2
-
2
ρ
C
X
C
Y
<
0, or if and only if C
>
1
/
2, where C
=
ρ
C
Y
/
C
X. (It is worth noting here that because of long association with the experimental data, C is guessable.) Similarly we have MSE
(
Y
-
^
P
)
<
Var
⁡
(
y
-
) if and only if C
<
-
1
/
2 (Murthy [2]). Thus the ratio and product estimators are relatively more efficient than the usual unbiased estimator (u.u.e) sample mean when C
>
1
/
2 and C
<
-
1
/
2, respectively. Consequently, Y
-
^
R
/
Y
-
^
P fail to improve y
- (by using auxiliary information) when -
1
/
2
≤
C
≤
1
/
2.Also we cannot ignore the classically well-known linear regression estimator, sayY
-
^
LR:
(7)
Y
-
^
LR
=
y
-
+
β
^
(
X
-
-
x
-
)
,
where
β
^
=
∑
i
=
1
n
(
y
i
-
y
-
)
(
x
i
-
x
-
)
∑
i
=
1
n
(
x
i
-
x
-
)
2
.
If we recall the ANOVA of linear regression analysis, we must remember that the residual sum of squares for Y
-
^
LR is S
Y
2
(
1
-
ρ
2
) (Cochran [3]). Thus when |
ρ
| is high (say ρ
>
0.7 or ρ
<
-
0.7), linear regression estimator is most likely to be more efficient than ratio/product estimators in using the auxiliary information (via auxiliary variable X). We aim at improving use of auxiliary information on y
- when -
1
/
2
≤
C
≤
1
/
2; Y
-
^
R when C
>
1
/
2; Y
-
^
Pwhen C
<
-
1
/
2; and Y
-
^
LR when |
ρ
|
≤
0.7.


## 2. Our Proposed Estimators

BecauseY
-
^
R and Y
-
^
P are relatively more efficient than y
- when C
>
1
/
2 and C
<
-
1
/
2, respectively, we try the following single parameter linear combinations of Y
-
^
R and -
y
-, as well as Y
-
^
P and -
y
- to propose the estimators:(i)
Shirley-Sahai-Dialsingh-ratio-cum-mean, sayY
-
^
SSDR:
(8)
Y
-
^
SSDR
=
(
1
+
θ
)
Y
-
^
R
-
θ
y
-
.
(ii)
Shirley-Sahai-Dialsingh-product-cum-mean, sayY
-
^
SSDP:
(9)
Y
-
^
SSDP
=
(
1
+
θ
)
Y
-
^
P
-
θ
y
-
.In (8) and (9), θ is the design parameter for our proposed estimators to be assigned an optimal value, for example, so as to minimize the first order of MSE, M
1
(
∘
), as in our case. Note that when θ
=
0, Y
-
^
SSDR
=
Y
-
^
R and Y
-
^
SSDP
=
Y
-
^
P. As remarked earlier, quite often a good guess of C is available from which we can give a suitable value to θ.


## 3. Sampling Bias and Mean Square Error of the Proposed Estimators

We derive the first order approximation,B
1
(
Y
-
^
SSDR
), to the bias of Y
-
^
SSDR. Using the notation introduced in Section 1 and substituting the expressions for y
- and x
- in terms of e and e
′ in (8) we have
(10)
Y
-
^
SSDR
=
Y
-
(
1
+
e
)
(
1
-
θ
e
′
)
(
1
+
e
′
)
-
1
.
It is realistic practically to suppose that |
e
′
|
<
1 so that (
1
+
e
′
)
-
1 is expandable. Then to the first order of approximation {
O
(
1
/
n
)
}, the bias of Y
-
^
SSDR is given by
(11)
B
1
(
Y
-
^
SSDR
)
=
E
(
Y
-
^
SSDR
-
Y
-
)
≈
Y
-
{
(
1
-
θ
)
E
(
e
′
2
)
-
(
1
+
θ
)
E
(
e
e
′
)
}
=
B
0
{
a
(
2
-
F
)
-
ρ
F
}
,
where a
=
C
X
/
C
Y, F
=
1
+
θ, and B
0
=
a
(
1
-
f
)
Y
-
C
Y
2
/
n. B
0
→
0, as n
→
∞; therefore, Y
-
^
SSDR is asymptotically unbiased up to O
(
1
/
n
).To compute the MSE ofY
-
^
SSDR we have
(12)
M
1
(
Y
-
^
SSDR
)
=
E
(
Y
-
^
SSDR
-
Y
-
)
2
≈
Y
-
2
E
{
(
e
-
F
e
′
)
2
}
,
up
to
the
first
order
of
approximation
{
O
(
1
n
)
}
M
1
(
Y
-
SSDR
)
=
{
Y
-
2
(
1
-
f
)
n
}
{
C
Y
2
+
F
2
C
X
2
-
2
F
ρ
C
X
C
Y
}
M
1
(
Y
-
SSDR
)
=
V
0
{
1
+
a
2
F
2
-
2
a
F
ρ
}
,
where V
0
=
Var
⁡
(
y
-
)
=
(
(
1
-
f
)
/
n
)
S
Y
2 and S
Y
=
Y
-
C
Y.For large sample size,M
1
(
Y
-
^
SSDR
) is minimum for a
F
=
ρ. The optimal value of F is thus ρ
/
a
=
C. If a good guess of C, say C
*, is available, we use θ
=
C
*
-
1 in our proposed estimator (8), so that
(13)
Y
-
^
SSDR
≈
C
*
y
-
X
-
x
-
-
(
C
*
-
1
)
y
-
.
We deduce the large sample approximation for bias of Y
-
^
SSDP in a similar manner:
(14)
Y
-
^
SSDP
=
Y
-
(
1
+
e
)
(
1
+
(
1
+
θ
)
e
′
)
,
B
1
(
Y
-
^
SSDP
)
=
E
(
Y
-
^
SSDP
-
Y
-
)
=
Y
-
(
1
+
θ
)
E
(
e
e
′
)
=
B
0
ρ
F
,
where F and B
0 are as before. B
0
→
0, as n
→
∞; therefore Y
-
^
SSDP is asymptotically unbiased.To compute the MSE ofY
-
^
S
S
D
P we have
(15)
M
1
(
Y
-
^
S
S
D
P
)
=
E
(
Y
-
^
S
S
D
P
-
Y
-
)
2
≈
Y
-
2
E
{
(
e
+
F
e
′
)
2
}
,
n
up
to
the
first
order
of
approximation
{
O
(
1
n
)
}
M
1
(
Y
-
S
S
D
P
)
=
{
Y
-
2
(
1
-
f
)
n
}
{
C
Y
2
+
F
2
C
X
2
+
2
F
ρ
C
X
C
Y
}
M
1
(
Y
-
S
S
D
P
)
=
V
0
{
1
+
a
2
F
2
+
2
a
F
ρ
}
,
WWWWWWW
where
V
0
=
Var
⁡
(
y
-
)
.
Up to O
(
1
/
n
), M
1
(
Y
-
^
SSDP
) is a minimum for a
F
=
-
ρ. The optimal value of F is thus -
ρ
/
a
=
-
C.We useθ
=
-
C
*
-
1 in our proposed estimator (9), where C
* is the guess of C. Thus,
(16)
Y
-
^
SSDP
≈
-
C
*
x
-
y
-
X
-
+
(
C
*
+
1
)
y
-
.


## 4. Comparison of the Estimators

Apparently, no algebraic comparison of mean square errors is feasible. We, therefore, have a numerical setup under simulation to do so. KnowingC exactly is seldom tenable in practice. Consequently, we have to assume the availability of a guess value of C, which we have called C
*, defined by C
*
=
C
(
1
+
R
), where R designates the quantum of relative under guess/overguess. We have taken the following R values: 0, ±
0.02, ±
0.04, ±
0.06, ±
0.08, and ±
0.10. We have also assumed that the parent population is very large, envisaged to have come from a superpopulation which is bivariate normal with the following parameters, therefore having the same parametric values:
(17)
Y
-
=
20.00
,
X
-
=
15.00
,
S
Y
=
3.00
,
S
X
=
1.00
.
Consequently we have used the softwareR[4] to calculate the MSEs of each of the following estimators:
(18)
Y
-
^
R
=
R
^
X
-
;
Y
-
^
P
=
P
^
X
-
;
Y
-
^
SSDR
=
(
1
+
θ
)
Y
-
^
R
-
θ
y
-
;
Y
-
^
SSDP
=
(
1
+
θ
)
Y
-
^
P
-
θ
y
-
,
Y
-
^
LR
=
y
-
+
β
^
(
X
-
-
x
-
)
.
We use 10,000 replications of simulated sample sizes n
=
30, 40, 60, 80, and 100. Hence we have compared the efficiencies of these estimators relative to y
- by using
(19)
REff
(
∘
)
=
MSE
(
y
-
)
MSE
(
∘
)
×
100
%
.
Motivated by our desire to beat ratio/product estimators (implicitly, therefore Y
-
^
LR also), we have, therefore, taken up the numerical simulation comparisons, for example, values of |
ρ
|: 0.1 (0.1) 0.7. For ρ positive, we compare Y
-
^
R, Y
-
^
SSDR, and Y
-
^
LR, while for ρ negative we compare Y
-
^
P, Y
-
^
SSDP, and Y
-
^
LR.


## 5. Results and Discussion

The results of our simulations are tabulated in the Appendix. For a given value ofn the relative efficiencies of Y
-
^
R, Y
-
^
P, and Y
-
^
LR do not depend on R; they are, therefore, not included in the main body of the tables but are stated at the top for each value of ρ. For n
= 30, 40, 60, 80, and 100, for each value of R
=
0, ±
0.02, ±
0.04, ±
0.06, ±
0.08, and ±
0.10, we have given the values of REff
(
Y
-
^
SSDR
) for ρ
= 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7 and of REff
(
Y
-
^
SSDP
) for ρ
= −0.1, −0.2, −0.3, −0.4, −0.5, −0.6, and −0.7.As apparent, our proposed estimatorsY
-
^
SSDR and Y
-
^
SSDP are consistently significantly better than Y
-
^
LR and Y
-
^
R (or Y
-
^
P, as the case may be).For illustrative purposes, we highlight below the relative efficiency values for the various values ofρ, for the cases when n
=
30 and R
=
0. To lessen the obscurity in the results, we have rounded these values to two decimal places. We also include a column for the value of C
=
ρ
C
Y
/
C
X.Tables1 and 2 illustrate very well the relative betterment achieved by our proposed estimators vis-à-vis Y
-
^
LR and Y
-
^
R (or Y
-
^
P, as the case may be). Notably, when |
C
| isnot greater than 1/2, our estimators are more efficient than y
- even though Y
-
^
R or Y
-
^
P (as the case may be) is worse than y
- which does not even use auxiliary information. Also when |
C
| is significantly less than 1/2, our estimators are more efficient than y
- even though Y
-
^
LR is worse thany
- (i.e., it fails to use auxiliary information rightly)!Table 1
Relative efficiencies (in %) of the estimators whenn
=
30 and R
=
0.
ρ
C
Y
-
^
LR
Y
-
^
R
Y
-
^
SSDR
0.1
0.225
97.00
89.71
100.88
0.2
0.45
100.35
98.09
104.22
0.3
0.675
105.26
106.32
109.10
0.4
0.90
114.29
117.98
118.34
0.5
1.125
128.73
133.07
133.58
0.6
1.35
152.63
152.08
159.40
0.7
1.575
186.79
172.17
193.33Table 2
Relative efficiencies (in %) of the estimators whenn
=
30 and R
=
0.
ρ
C
Y
-
^
LR
Y
-
^
P
Y
-
^
SSDP
−0.1
−0.225
97.66
89.14
100.73
−0.2
−0..45
100.42
98.05
104.14
−0.3
−0.675
106.48
108.23
110.39
−0.4
−0.90
114.87
119.11
119.43
−0.5
−1.125
127.32
131.54
132.05
−0.6
−1.35
150.37
150.36
155.88
−0.7
−1.575
186.53
172.80
194.68


## 6. Conclusion

Our results conform to our motivating aim of achieving more efficient use of auxiliary information. Many other authors, such as Sahai [5] and Chami et al. [6], have suggested efficient variants of ratio and product estimators. In future work we are engaged in comparing these estimators and in trying even better estimators, like the proposed ones, which will be not only more efficient relatively, but also, possibly, more robust against the possible over/underguess of the key-population parameter C.



---

*Source: 360549-2014-09-23.xml*