Fundamentals of Probability and Statistics for Engineers

a valid reason for choosing 2 as a better estimator, compared with 1 ,for ,
in certain cases.

9.2.3 Consistency

An estimator is said to be a consistent estimator for if, as sample size n
increases,

for all 0. The consistency condition states that estimator converges in the
sense above to the true value as sample size increases. It is thus a large-sample
concept and is a good quality for an estimator to have.

Ex ample 9. 6. Problem: show that estimator S^2 in Example 9.3 is a consistent
estimator for^2.
Answer: using the Chebyshev inequality defined in Section 4.2, we
can write

We have shown that and var H ence,

Thus S^2 is a consistent estimator for^2.

Example 9.6 gives an expedient procedure for checking whether an estimator
is consistent. We shall state this procedure as a theorem below (Theorem 9.3). It
is important to note that this theorem gives a sufficient, but not necessary,
condition for consistency.

Theorem 9. 3: Let be an estimator for based on a sample of size n.
Then, if

estimator is a consistent estimator for.
The proof of Theorem 9.3 is essentially given in Example 9.6 and will not be
repeated here.

274 Fundamentals of Probability and Statistics for Engineers

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