Page 393 - DMTH404_STATISTICS

Unit 29: Estimation of Parameters: Criteria for Estimates

is called a consistent estimator. An estimator t (X , X , ...... X ) is said to be consistent if its Notes
n 1 2 n
probability distribution converges to  as n  .
Symbolically, we can write P(t  ) = 1 as n   Alternatively, t is said to be a consistent
n n
estimator of q if E(t )  q and Var(t )  0, as n  .
n n
We may note that X is a consistent estimator of population mean m because ( ) = and
E X
 2
Var X  0 as n  .
( ) =
n
Note: An unbiased estimator is necessarily a consistent estimator.

29.2.3 Efficiency

Let t and t be two estimators of a population parameter q such that both are either unbiased or
1 2
consistent. To select a good estimator, from t and t , we consider another property that is based
1 2
upon its variance.
If t and t are two estimators of a parameter q such that both of them are either unbiased or
1 2
consistent, then t is said to be more efficient than t if Var(t ) < Var(t ). The efficiency of an
1 2 1 2
estimator is measured by its variance.
For a normal population, we know that both the sample mean and median are unbiased estimator
 2   2
of population mean. However, their respective variances are and × , where  is
2
n 2 n
 2   2
population variance. Since < × , therefore, sample mean is said to be efficient estimator
n 2 n
of population mean.
Remarks: The precision of an estimator = 1/ S. E. of estimator.
An estimator having minimum variance among all the estimators of a population parameter is
termed as Most Efficient Estimator or Best Estimator. If an estimator is unbiased and best, then
it is termed as Best Unbiased Estimator. Further, if the best unbiased estimator is a linear
function of the sample observations, it is termed as Best Linear Unbiased Estimator (BLUE).
It may be pointed out here that sample mean is best linear unbiased estimator of population
mean.
Cramer Rao Inequality:
This inequality gives the minimum possible value of the variance of an unbiased estimator. If t
is an unbiased estimator of parameter q of a continuous population with probability density
function f(X, q), then

1
Var ( ) t ³ 2
æ ¶ log f ( ,X  )ö
nE ç è ¶ ÷ ø

29.2.4 Sufficiency

An estimator t is said to be a sufficient estimator of parameter  if it utilises all the information
given in the sample about . For example, the sample mean X is a sufficient estimator of 
because no other estimator of  can add any further information about .

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