Ratio estimator

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 and 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. Assume there are two characteristics – x and y – that can be observed for each sampled element in the data set. The ratio R is The ratio estimate of a value of the y variate (θy) is where θx is the corresponding value of the x variate. θy is known to be asymptotically normally distributed. ratio distribution The sample ratio (r) is estimated from the sample That the ratio is biased can be shown with Jensen's inequality as follows (assuming independence between x and y): 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). Under simple random sampling the relative bias is O( n−1/2 ). The correction methods, depending on the distributions of the x and y variates, differ in their efficiency making it difficult to recommend an overall best method. Because the estimates of r are biased a corrected version should be used in all subsequent calculations. A correction of the bias accurate to the first order is where mx is the mean of the variate x and sxy is the covariance between x and y. To simplify the notation sxy will be used subsequently to denote the covariance between the variates x and y. Another estimator based on the Taylor expansion is where n is the sample size, N is the population size, mx is the mean of the x variate and sx2 and sy2 are the sample variances of the x and y variates respectively.