An estimator is said to be a consistent estimator of the parameter , if it holds the following conditions:
is an unbiased estimator of , i.e. if is biased, it should be unbiased for large values of (in the limit sense) i.e. .
Variance of approaches to zero as becomes very large, i.e., . Consider the following example,
Example: Show that the sample mean is a consistent estimator of the population mean.
We have already seen in the previous example that is an unbiased estimator of population mean . This satisfies the first condition of consistency. The variances of is known to be . From the second condition of consistency we have,
Hence, is also a consistent estimator of .
The word BLUE stands for best linear Unbiased Estimator. An unbiased estimator which is a linear function of the random variable and possess the least variance may be called as BLUE. A BLUE estimator therefore possesses all the three properties mentioned above and in addition is a linear function of the random variable. From the last example we can conclude that the sample mean is BLUE.