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## Efficiency

We will next make some comments on the property of efficiency of estimators. The term is frequently used in comparison of two estimators where a measure of relative efficiency is used. In particular,

Definition 8..6
Given two unbiased estimators, and of , the efficiency of relative to is defined to be

and is more efficient than if Var(Var().

Note that it is only reasonable to compare estimators on the basis of variance if they are both unbiased. To allow for cases where this is not so, we can use mse in the definition of efficiency. That is,

Definition 8..7
An estimator of is more efficient than if

with strict inequality for some . Also the relative efficiency of with respect to is
 (8.6)

Example 8..3
Let denote a random sample from U(0, ), with the corresponding ordered sample.

(i)
Show that and are unbiased estimates of .
(ii)
Find .

Solution

(i)
Now E( and Var( so

To find the mean of , first note that the probability density function of is

So

For defined by , we have

So both and are unbiased.

(ii)
Var(Var( Var( .
To find Var(), first we need to find E() from

So

Since these estimates are unbiased, we may use definition 2.5,

This is less than for so is more efficient than .

Next: Cramér-Rao Lower Bound Up: Some Properties of Estimators Previous: Consistency   Contents
Bob Murison 2000-10-31