Ratio estimator

Assume there are two characteristics – x and y – that can be observed for each sampled element in the data set. The ratio R is

${\displaystyle R={\bar {\mu }}_{y}/{\bar {\mu }}_{x}}$

The ratio estimate of a value of the y variate (θ_y) is

${\displaystyle \theta _{y}=R\theta _{x}}$

where θ_x is the corresponding value of the x variate. θ_y is known to be asymptotically normally distributed.[1]

The sample ratio (r) is estimated from the sample

${\displaystyle r={\frac {\bar {y}}{\bar {x}}}={\frac {\sum _{i=1}^{n}y}{\sum _{i=1}^{n}x}}}$

That the ratio is biased can be shown with Jensen's inequality as follows (assuming independence between x and y):

${\displaystyle E\left({\frac {y}{x}}\right)=E\left(y{\frac {1}{x}}\right)=E(y)E\left({\frac {1}{x}}\right)\geq E(y){\frac {1}{E(x)}}={\frac {E(y)}{E(x)}}}$

Under simple random sampling the bias is of the order O( n⁻¹ ). An upper bound on the relative bias of the estimate is provided by the coefficient of variation (the ratio of the standard deviation to the mean).[2] Under simple random sampling the relative bias is O( n^−1/2 ).

A jackknife estimate of the ratio is less biased than the naive form. A jackknife estimator of the ratio is

${\displaystyle r_{\mathrm {corr} }=nr-{\frac {n-1}{n}}\sum _{i\neq j=1}^{n}r_{i}}$

where n is the size of the sample and the r_i are estimated with the omission of one pair of variates at a time.[10]

An alternative method is to divide the sample into g groups each of size p with n = pg.[11] Let r_i be the estimate of the i^th group. Then the estimator

${\displaystyle r_{\mathrm {corr} }=gr-{\frac {g-1}{g}}\sum _{i=1}^{g}r_{i}=g\left(r-{\bar {r}}\right)+{\bar {r}}}$

where ${\displaystyle {\bar {r}}}$ is the mean of the ratios r_g of the g groups, has a bias of at most O( n⁻² ).

Other estimators based on the division of the sample into g groups are:[12]

${\displaystyle r_{\mathrm {corr} }={\frac {g}{g+1}}r-{\frac {1}{g(g-1)}}\sum _{i=1}^{g}r_{i}}$

${\displaystyle r_{\mathrm {corr} }={\bar {r}}+{\frac {n}{n-1}}{\frac {m_{y}-{\bar {r}}m_{x}}{m_{x}}}}$

${\displaystyle r_{\mathrm {corr} }={\bar {r_{g}}}+{\frac {g(m_{y}-{\bar {r_{g}}}m_{x})}{m_{x}}}}$

where ${\displaystyle {\bar {r}}}$ is the mean of the ratios r_g of the g groups and

${\displaystyle {\bar {r_{g}}}=\sum {\frac {r_{i}'}{g}}}$

where r_i^' is the value of the sample ratio with the i^th group omitted.

The estimated total of the y variate ( τ_y ) is

${\displaystyle \tau _{y}=r\tau _{x}}$

where ( τ_x ) is the total of the x variate.

The variance of the sample ratio is approximately:

${\displaystyle \operatorname {var} (r)={\frac {1}{s_{x}^{2}+m_{x}^{2}}}\left[(s_{y}^{2}-s_{x^{2}[y^{2}/x^{2}]})-(s_{x[y/x]})^{2}+2m_{y}s_{x[y/x]}-{\frac {s_{x}^{2}}{m_{x}^{2}}}(m_{y}-s_{x[y/x]}^{2})\right]}$

where s_x² and s_y² are the variances of the x and y variates respectively, m_x and m_y are the means of the x and y variates respectively and s_xy is the covariance of x and y.

Although the approximate variance estimator of the ratio given below is biased, if the sample size is large, the bias in this estimator is negligible.

${\displaystyle \operatorname {var} (r)={\frac {1}{n}}{\frac {N-n}{N}}{\frac {1}{m_{x}^{2}}}{\frac {\sum _{i=1}^{n}(y_{i}-rx_{i})^{2}}{n-1}}}$

where N is the population size, n is the sample size and m_x is the mean of the x variate.

Another estimator of the variance based on the Taylor expansion is

${\displaystyle \operatorname {var} (r)={\frac {1}{n}}(1-{\frac {n-1}{N-1}}){\frac {r^{2}s_{x}^{2}+s_{y}^{2}-2rs_{xy}}{m_{x}^{2}}}}$

where n is the sample size and N is the population size and s_xy is the covariance of x and y.

An estimate accurate to O( n⁻² ) is[3]

${\displaystyle \operatorname {var} (r)={\frac {1}{n}}\left[{\frac {s_{y}^{2}}{m_{x}^{2}}}+{\frac {m_{y}^{2}s_{x}^{2}}{m_{x}^{4}}}-{\frac {2m_{y}s_{xy}}{m_{x}^{3}}}\right]}$

If the probability distribution is Poissonian, an estimator accurate to O( n⁻³ ) is[4]

${\displaystyle \operatorname {var} (r)=r^{2}\left[{\frac {1}{n}}\left({\frac {1}{m_{x}}}+{\frac {1}{m_{y}}}-{\frac {2s_{xy}}{m_{x}m_{y}}}\right)+{\frac {1}{n^{2}}}\left({\frac {6}{m_{x}^{2}}}+{\frac {3}{m_{x}m_{y}}}+s_{xy}\left[{\frac {4}{m_{y}^{2}}}-{\frac {8}{m_{x}m_{y}}}-{\frac {16}{m_{x}^{2}m_{y}}}+{\frac {5s_{xy}}{m_{x}^{2}m_{y}^{2}}}\right]+{\frac {4s_{x^{2}y}}{m_{x}^{2}m_{y}}}-{\frac {2s_{xy^{2}}}{m_{x}m_{y}^{2}}}\right)\right]}$

A jackknife estimator of the variance is

${\displaystyle \operatorname {var} (r)={\frac {(n-1)}{n}}\sum _{i=1}^{n}(r_{i}-r_{J})^{2}}$

where r_i is the ratio with the i^th pair of variates omitted and r_J is the jackknife estimate of the ratio.[10]

The skewness and the kurtosis of the ratio depend on the distributions of the x and y variates. Estimates have been made of these parameters for normally distributed x and y variates but for other distributions no expressions have yet been derived. It has been found that in general ratio variables are skewed to the right, are leptokurtic and their nonnormality is increased when magnitude of the denominator's coefficient of variation is increased.

For normally distributed x and y variates the skewness of the ratio is approximately[7]

${\displaystyle \gamma =\left({\frac {m_{y}\omega }{\sqrt {nm_{x}m_{y}\omega ^{2}+m_{x}^{2}m_{y}}}}\right)\left(6+{\frac {1}{nm_{x}}}\left[44+{\frac {1}{1+\omega ^{2}m_{y}/m_{x}}}\right]\right)}$

where

${\displaystyle \omega =1-m_{x}\operatorname {cov} (x,y)}$

An alternative method of reducing or eliminating the bias in the ratio estimator is to alter the method of sampling. The variance of the ratio using these methods differs from the estimates given previously. Note that while many applications such as those discussion in Lohr[13] are intended to be restricted to positive integers only, such as sizes of sample groups, the Midzuno-Sen method works for any sequence of positive numbers, integral or not. It's not clear what it means that Lahiri's method works since it returns a biased result.

If a linear relationship between the x and y variates exists and the regression equation passes through the origin then the estimated variance of the regression equation is always less than that of the ratio estimator. The precise relationship between the variances depends on the linearity of the relationship between the x and y variates: when the relationship is other than linear the ratio estimate may have a lower variance than that estimated by regression.

Although the ratio estimator may be of use in a number of settings it is of particular use in two cases:

• when the variates x and y are highly correlated through the origin.
• In survey methodology when estimating a weighted average in which the denominator indicates the sum of weights that reflect the total population size, but the total population size is unknown.

The first known use of the ratio estimator was by John Graunt in England who in 1662 was the first to estimate the ratio y/x where y represented the total population and x the known total number of registered births in the same areas during the preceding year.

Later Messance (~1765) and Moheau (1778) published very carefully prepared estimates for France based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample.

In 1802, Laplace wished to estimate the population of France. No population census had been carried out and Laplace lacked the resources to count every individual. Instead he sampled 30 parishes whose total number of inhabitants was 2,037,615. The parish baptismal registrations were considered to be reliable estimates of the number of live births so he used the total number of births over a three-year period. The sample estimate was 71,866.333 baptisms per year over this period giving a ratio of one registered baptism for every 28.35 persons. The total number of baptismal registrations for France was also available to him and he assumed that the ratio of live births to population was constant. He then used the ratio from his sample to estimate the population of France.

Karl Pearson said in 1897 that the ratio estimates are biased and cautioned against their use.[22]

• Mark and recapture, another way of estimating population using a ratio.
• Ratio distribution