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 , i.e. 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. Variance of  \widehat  \alpha approaches to 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 variances 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:

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 \overline X is BLUE.     

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