The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical so symmetrical tests such as the t test should not be used to generate confidence intervals. The bias is of the order O(1/n) (see big O notation) so as the sample size (n) increases, the bias will asymptotically approach 0. Therefore, the estimator is approximately unbiased for large sample sizes.
Definition Assume there are two characteristics – x and y – that can be observed for each sampled element in the data set. The ratio R is
R = μ ¯ y / μ ¯ x {\displaystyle R={\bar {\mu }}_{y}/{\bar {\mu }}_{x}}
The ratio estimate of a value of the y variate (θy) is
θ y = R θ x {\displaystyle \theta _{y}=R\theta _{x}}
where θx is the corresponding value of the x variate. θy is known to be asymptotically normally distributed.
Statistical properties
The sample ratio (r) is estimated from the sample
r = y ¯ x ¯ = ∑ i = 1 n y i ∑ i = 1 n x i {\displaystyle r={\frac {\bar {y}}{\bar {x}}}={\frac {\sum _{i=1}^{n}y_{i}}{\sum _{i=1}^{n}x_{i}}}}
That the ratio is biased can be shown with Jensen's inequality as follows (assuming independence between x ¯ {\displaystyle {\bar {x}}} and y ¯ {\displaystyle {\bar {y}}} ):
E ( y ¯ x ¯ ) = E ( y ¯ 1 x ¯ ) = E ( y ¯ ) E ( 1 x ¯ ) ≥ E ( y ¯ ) 1 E ( x ¯ ) = E ( y ¯ ) E ( x ¯ ) = E ( y ) E ( x ) = m y m x {\displaystyle E\left({\frac {\bar {y}}{\bar {x}}}\right)=E\left({\bar {y}}{\frac {1}{\bar {x}}}\right)=E({\bar {y}})E\left({\frac {1}{\bar {x}}}\right)\geq E({\bar {y}}){\frac {1}{E({\bar {x}})}}={\frac {E({\bar {y}})}{E({\bar {x}})}}={\frac {E(y)}{E(x)}}={\frac {m_{y}}{m_{x}}}}
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