Fundamentals of Probability and Statistics for Engineers

274 Fundamentals of Probability and Statistics for Engineersa valid reason for choosing ^ 2 as a better estimator, compared with ^ 1, for ,in certain cases.9.2.3 CONSISTENCYAn estimator ^ is said to be a consistent estimator for if, as sample size nincreases,lim P‰j ^ j "Š ˆ0; …9:47†n!1for all "> 0. The consistency condition states that estimator ^ converges in thesense above to the true value as sample size increases. It is thus a large-sampleconcept 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 consistentestimator for 2 .Answer: using the Chebyshev inequality defined in Section 4.2, wecan writePfjS 2 2 j"g 1 " 2 Ef…S2 2 † 2 g:We have shown that EfS 2 gˆ 2 , and varfS 2 gˆ2 2 /n 1). Hence,limn!1 PfjS2 2 1 2 2 j"g limn!1 " 2 ˆ 0:n 1Thus S 2 is a consistent estimator for 2 .Example 9.6 gives an expedient procedure for checking whether an estimatoris consistent. We shall state this procedure as a theorem below (Theorem 9.3). Itis 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, iflim Ef ^g ˆ; and lim varf ^g ˆ0; …9:48†n!1 n!1estimator ^ is a consistent estimator for .The proof of Theorem 9.3 is essentially given in Example 9.6 and will not berepeated here.TLFeBOOK