Consistent Estimator
An estimator $$\widehat \alpha $$ is said to be a consistent estimator of the parameter $$\widehat \alpha $$ if it holds the following conditions:
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$$\widehat \alpha $$ 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. $$\mathop {\lim }\limits_{n \to \infty } E\left( {\widehat \alpha } \right) = \alpha $$.
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The variance of $$\widehat \alpha $$ approaches zero as $$n$$ becomes very large, i.e., $$\mathop {\lim }\limits_{n \to \infty } Var\left( {\widehat \alpha } \right) = 0$$. 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 $$\frac{{{\sigma ^2}}}{n}$$. From the second condition of consistency we have,
\[\begin{gathered} \mathop {\lim }\limits_{n \to \infty } Var\left( {\overline X } \right) = \mathop {\lim }\limits_{n \to \infty } \frac{{{\sigma ^2}}}{n} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\sigma ^2}\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\sigma ^2}\left( 0 \right) = 0 \\ \end{gathered} \]
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.