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## Consistency

A further desirable property of estimators is that of consistency, which is an asymptotic property. To understand consistency, it is necessary to think of as really being , the nth member of an infinite sequence of estimators, . Roughly speaking, an estimator is consistent if, as gets large, the probability that lies arbitrarily close to the parameter being estimated becomes itself arbitrarily close to . More formally, we have

Definition 8..4
is a consistent estimator of if

 (8.4)

This is often referred to as convergence in probability of to .

An equivalent definition (for cases where the second moment exists) is

Definition 8..5
is a consistent estimator of if

 (8.5)

That is, the mse of as an estimator of , decreases to zero as more and more observations are incorporated into its composition. Note that, using (2.3) we see that (2.5) will be satisfied if is asymptotically unbiased and if Var( as .

Asymptote means the truth. So as the sample size increases, gets closer to the true value. When , we have sampled the entire population. The idea of consistency can be gleaned from the following diagram where converges to . If it didn't, would not be a consistent estimator.

Example 8..2
Let be a random variable with mean and variance . Let be the sample mean of random observations taken on . Is a consistent estimator of ?

Now so is unbiased. Also Var( as , so is a consistent estimator of .

Next: Efficiency Up: Some Properties of Estimators Previous: Mean Square Error   Contents
Bob Murison 2000-10-31