Bearman, Jacob E.

Martin, Frank B.

Wiedenhofer, Hermann

University of Minnesota Digital Conservancy

* 
ON THE BIAS AND MEAN SQUARE ERROR 
OF THE RATIO OF TWO BIASED ESTIMATORS* 
BY 
Hermann Wiedenhofer, Jacob E. Bearman and 
Frank B. Martin 
Technical Report No. 192 
December 1972 
University of Minnesota 
Minneapolis, Minnesota 
Research supported in part by Project 2830, Agricultural 
Experimental Station, University of Minnesota and be PHS 
(Grant/Contract) TOI HS-00053 from the National Center for 
Health Services Research and Development. 
... 
1. Summary and Introduction. 
In the current literature of finite sampling theory, expressions 
for the bias and M.S.E. for the ratio of two unbiased estimators are 
well developed. Similar theoretical values for the ratio of two biased 
estimators, not available in the current literature, are obtained here. 
,. 
It is well known that R, the ratio of tyo unbiased estimators, is 
a biased estimator of the corresponding population ratio, R. It will 
,.. ,.. 
be shown that the bias and M.S.E. of R, when R is the ratio of two 
biased estimators has additional terms. An exact expression and a first 
and second approximation are given for the bias of R. Further, an 
,.. 
expression for the M.S.E. of R is given, utilizing the first approxi-
mation above. 
The notation and procedures used by Cochran [1] and Raj [2] for 
the ratio estimation are gnerally followed. 
2. Definitions. 
Let R be a population ratio such that 
y 
R--
- X 
where X and Y are means or totals of population characteristics. 
,.. 
Let R be an estimator of R, 
where x and y are biased estimators of X and Y respectively. Then, 
E(y) =Y+ B(y) 
E(x) = X+ B(x) 
,.. ,.. 
E(R) = R+ B(R) 
- 1 -
(2.1) 
(2.2) 
(2.3) 
(2.4) 
(2.5) 
--
i.. 
,.. 
3. The Exact Bias of R 
Theorem 1 
R[!.{tl - .fil.!.l] 1 ,.. 
"' 
- - Cov (R,x) (3.1) B(R) y . X X . = 
1 + .fil.!.l 
.X 
Proof: 
,.. 
Using the usual definition for Cov(R,x) and expression (2.2), 
it follows that 
1 ,.. 1 " 
X Cov(R,x) = X [E(y) - E(R) E(x)] (3.2) 
and by substitution from (2.3) , (2.4) and (2.5) 
1 "' Y Bf ... \ Bf x' " Bf x \ 
- Cov(R x) - - + ~ - R[l + ~] - B(R) (1 + ~X ]. X ' - X X X (3.3) 
,.. 
Solving {3.3) for B(R) yields 
R[!.{tl - .fil.!.11 - .!. Cov(R,x) 
A y X X 
B(R) = -----------------
1 .fil.!.l + X 
(3.4) 
Q. E. D. 
B¼x) and¥ are the relat!ve bias of x and y respectively, similar 
to the usual definition of B~R) in Cochran [1]. Since the exact 
,.. 
expression (3.4) for the bias of R is not always very useful, approximate 
expressions are obtained as B1(R) based on the first term, and B2(R) 
based on the first two terms in expression (4.1) of the Taylor's series 
1 
expansion of - around x. 
X 
- 2 -
,.. 
4. Approximations for the Bias of R. 
R _ R = y-Rx [l _ (x-X) + (x-x)2 _ •.. ] 
X X ~ 
x-X · If X in absolute value is rather small, then a satisfactory 
approximation can be obtained by considering the expectation of the 
R.H.s. of (4.2) 
,.. 1 
R - R ; - {y-Rx) X 
By adding and subtracting RB(x) , substituting RX for Y , and by the 
definition of E(x), the expression y-Rx in (4.2) becomes 
y - Rx= y-E{y) - R[x-E(x)] - RB(x) + B(y). 
Substituting back into (4.2) , yields 
R - R ~ i ( y-E(x) - R[x-E{x)] - RB(x) + B(y)} 
Taking expectations of both sides of (4.4), the first approximation 
,.. 
to the bias of R is 
,.. 
Using the first two terms in expression (4.1), R-R is better 
approximated by 
,.. . ( ( 1 x-x ) 
R - R = y-Rx) X -~
X 
By (4.3) and the fact that x - X = [ x-E(x) ] + B(x) , the R.H.S. of 
(4.6) may be written as 
(y-Rx) ( .! - x-X) = ! ( y-E{y) - R [ x-E(x) ] 
X X2 X 
1 
- RB(x) + B(y)} - - 2 ([x-E(x)] [y-E(y)] X 
- R[x-E(x) J2 + [x-E(x)] [B(y) - RB(x) ]} 
- .!!.Uil_ [Y - E{y) - R[x-E{x)] - RB(x) + B{y)J. 
x2 
- 3 -
(4.1) 
(4.2) 
(4.3) 
(4.4) 
(4.5) 
(4.6) 
(4.7) 
.. 
Taking expectations of the R.H.S. of (4.7) , 
(1 - !W] 
X 
1 
+ ~ [RV(x) - Cov(x,y)] (4.8) 
To obtain a form parallel to that commonly occurring in textbooks, the 
following coefficients of variation and covariation are introduced. 
1 
CV(x) = [V(x)]2 (4.9) 
X 
cc(x,y) = Cov(x,y) (4.10) 
XY 
~ 
By substitution and manipulation, B2(R) can be written as 
B2(R) = R ([CV(x)]2 - cc(x,y) + [ ¥ -B¾x) ] (1 - B¾x)]} . (4.11) 
Note that if x and y are both unbiased or have the same relative bias, then 
(4.11) reduces to the commonly current textbook expression for the bias 
of the ratio estimator. 
~ 
5. The First Approximation to the Mean Square Error of R. 
Squaring both sides of (4.4) and taking expectations on both sides, 
~ ~ 2 1 2 2 
M.S.E.(R) = E(R-R) ; - 2 (V(y} + R V(x)-2 R .Cov(x,y} + [B(y} - R B(x)] }• (5.l) X 
. 2 
Factoring out R from the R.H.S. of (5.1) 
MSE(i) ~ R2 {[cV(y}J2 + [CV(x)J2 - 2CC (x,y) + [Biy) - B¾x)J _2l 
Note that if x and y are both unbiased, then expression (5.2) reduces 
(5.2) 
to the commonly current textbook expression for the approximation of the 
MSE of the ratio estimator. 
- 4 -
i 
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; 
Iri [3] a second approximation to the MSE is obtained, however, it 
is too cumbersome to be presented here. 
- 5 -
6. References. 
1. Cochran, William G., Sampling Techniques, 2nd ed., New York: 
John Wiley and Sons, Inc., 1963. 
2. Raj, Des, Sampling Theory, New York: McGraw-Hill Book Co. Inc., 
1968. 
3. Wiedenhofer, Hermann, "Linear and Ratio Estimation Methods in 
Multistage Sampling, with Special Reference to a System of 
Voluntary Animal Disease Reporting," Ph.D. dissertation, 
University of Minnesota, 1972. 
- 6 -


On the Bias and Mean Square Error of the Ratio of Two Biased Estimators

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On the Bias and Mean Square Error of the Ratio of Two Biased Estimators

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