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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.
Solution:
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,
&n bsp; 
Hence,  is also a consistent estimator of .
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 possess 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.
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