Ratio Estimations

3. Ratio Estimation
3.1 Introduction
Most often, the characteristic or variable to be estimated is highly correlated with other
variables in the same population. To obtain a better estimate of a variable under study ( y ),
we find a ratio estimate of y to an auxiliary variable ( x ) which correlates with y . The two
variables vary from unit to unit in a given random sample. In practice, the value xi may

become the value of y i at some previous time since they correlate with each other. Thus, y

may be the population of a town in a current survey and , the population at a previous
census. We may also consider as the output of production and , the number of workers in
establishments in a survey of industries. In both cases and are highly believed to be
closely related. The aim of ratio estimation is to obtain increased precision by taking
advantage of the correlation between the two variables X and Y .


3.2 Properties of Estimates
Let y i be the variable of interest and xi , the auxiliary variable correlating with y i . Then we
define the population totals of X and Y , and the corresponding sample totals as follows;
N n
 X   xi and x   xi
i 1 i 1

N n
 Y   yi and y   yi
i 1 i 1


The ratio of Y to X , R and its estimate, are given by

Y Y
 R  , from which we have Y  RX or Y  RX
X X
y y
 Rˆ  
x x
Theorem
 The ratio of estimates of the population total Y , the population mean, Y and the
population ratio R  Y X are respectively:
y y
(i) YˆR  X  ˆ
X  RX
x x
YˆR  X 
y y ˆ
(ii) X  RX
x x
y y
(iii) Rˆ  
x x

,  If and are measured on each unit of a simple random sample of large sample size
, then the mean square error (MSE) and variance of the R̂ are approximately the
same:
(1  f ) 1 N
MSE ( Rˆ )  Var ( Rˆ )   ( yi  Rxi )2
nX 2 N  1 i 1

 Corollary
The following are deduced form the above theorem:
(i) Var (YˆR )  Var ( RX
ˆ )
 Var ( NXRˆ )
 N 2 X 2Var ( Rˆ )
N 2 (1  f ) N
 
n( N  1) i 1
( yi  Rxi ) 2

ˆ )
(ii) Var (YR )  Var ( RX
 X 2Var ( Rˆ )
(1  f ) N
  ( yi  Rxi )2
n( N  1) i 1

(iii) Var ( Rˆ )  Var ( RX
ˆ )
(1  f ) N
 ( yi  Y )  R( xi  X )
2

nX ( N  1) i 1
2


(1  f )  N N N

  i        ( xi  X ) 2 
2 2
( y Y ) 2 R ( y Y )( x X ) R
nX ( N  1)  i 1
2 i i
i 1 i 1 
We define the correlation coefficient,
N

E[( yi  Y )( xi  X )]  ( y  Y )( x  X )
i i
  i 1

E[( yi  Y ) 2 ]E[( xi  X ) 2 ] ( N  1) S y S x


E[( yi  Y )( xi  X )]   ( N  1) S y S x  Cov( xi , yi ),
1 N 1 N
where S y  
N  1 i 1
( yi  Y ) 2 and S x  
N  1 i 1
( xi  X ) 2

Hence,
(1  f ) 2
Var ( Rˆ ) 
nX 2 
S y  2 R S y S x  R 2 S x2 