# Consistent Estimator

An estimator $\widehat \alpha$ is said to be a consistent estimator of the parameter $\widehat \alpha$ if it holds the following conditions:

1. $\widehat \alpha$ 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. $\mathop {\lim }\limits_{n \to \infty } E\left( {\widehat \alpha } \right) = \alpha$.
2. 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.