Among a number of estimators of the same class, the estimator having the least variance is called an efficient estimator. Thus, if we have two estimators and with variances and respectively, also if then will be an efficient estimator. The ratio of the variances of two estimators denoted by is known as the efficiency of and is defined as follows
If the value of this ratio is more than 1, then will be more efficient, if it is equal to 1, both and are equally efficient and if it is less than 1, then will be less efficient. Let us consider the following worked example.
The variances of the sample mean and median are respectively.
Find the efficiency of
- median against mean
- mean against median
Using the formula , we have
e (median, mean)
Therefore, the efficiency of median against the mean is only 0.63. It means that a sample mean obtained from a sample of size 63 will be equally efficient to a sample median obtained from a sample of size 100.
Also, using the formula , we have
e (mean, median)
Therefore, the efficiency of mean against median is 1.57 or in other words mean is about 57% more efficient than the median.