An estimator $$\widehat \alpha $$ is said to be a consistent estimator of the parameter $$\widehat \alpha $$ if it holds the following conditions:
$$\widehat \alpha $$ is an unbiased estimator of $$\alpha $$, so if $$\widehat \alpha $$ is biased, it should be unbiased for large values of $$n$$ (in the limit sense), i.e. $$\mathop <\lim >\limits_ E\left( \right) = \alpha $$.
The variance of $$\widehat \alpha $$ approaches zero as $$n$$ becomes very large, i.e., $$\mathop <\lim >\limits_ Var\left( \right) = 0$$. Consider the following example.
Example: Show that the sample mean is a consistent estimator of the population mean.
Solution:
We have already seen in the previous example that $$\overline X $$ is an unbiased estimator of population mean $$\mu $$. This satisfies the first condition of consistency. The variance of $$\overline X $$ is known to be $$\frac>>$$. From the second condition of consistency we have,
Hence, $$\overline X $$ is also a consistent estimator of $$\mu $$.
BLUE
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 a BLUE. A BLUE therefore possesses all the three properties mentioned above, and is also a linear function of the random variable. From the last example we can conclude that the sample mean $$\overline X $$ is a BLUE.