Page 291 - Fundamentals of Probability and Statistics for Engineers

274 Fundamentals of Probability and Statistics for Engineers
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a valid reason for choosing 2 as a better estimator, compared with 1 ,for ,
in certain cases.

9.2.3 CONSISTENCY

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

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lim P‰j j "Šˆ 0; 9:47†
n!1
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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.
2

Example 9.6. Problem: show that estimator S in Example 9.3 is a consistent
estimator for 2 .
Answer: using the Chebyshev inequality defined in Section 4.2, we
can write
1
2 2
2
2
2
PfjS j "g Ef S † g:
" 2
2
2
2
2
We have shown that EfS gˆ , and varfS gˆ 2 / n 1). Hence,
1 2 2
2
2
lim PfjS j "g lim ˆ 0:
n!1 n!1 " 2 n 1
2
Thus S 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.
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Theorem 9.3: Let be an estimator for based on a sample of size n.

Then, if
^
^
lim Ef gˆ ; and lim varf gˆ 0; 9:48†
n!1 n!1
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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.

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