X2

It can be shown that

n where a2 is the true variance. It is obvious that in a small sample S2 is biased, but as n increases indefinitely, E(S2) approaches true a2; hence it is asymptotically unbiased.

APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS 903

Consistency. 0 is said to be a consistent estimator if it approaches the true value 0 as the sample size gets larger and larger. Figure A.11 illustrates this property.

In this figure we have the distribution of 0 based on sample sizes of 25, 50, 80, and 100. As the figure shows, 0 based on n = 25 is biased since its sampling distribution is not centered on the true 0. But as n increases, the distribution of 0 not only tends to be more closely centered on 0 (i.e., 0 becomes less biased) but its variance also becomes smaller. If in the limit (i.e., when n increases indefinitely) the distribution of 0 collapses to the single point 0, that is, if the distribution of 0 has zero spread, or variance, we say that 0 is a consistent estimator of 0.

More formally, an estimator 0 is said to be a consistent estimator of 0 if the probability that the absolute value of the difference between 0 and 0 is less than 8 (an arbitrarily small positive quantity) approaches unity. Symbolically, lim P{\0 — 01 < 8}= 1 8 > 0

where P stands for probability. This is often expressed as plim 0) = 0

where plim means probability limit.

Note that the properties of unbiasedness and consistency are conceptually very much different. The property of unbiasedness can hold for any sample size, whereas consistency is strictly a large-sample property.

FIGURE A.11 The distribution of 0 as sample size increases.

904 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS

A sufficient condition for consistency is that the bias and variance both tend to zero as the sample size increases indefinitely.7 Alternatively, a sufficient condition for consistency is that the MSE(0) tends to zero as n increases indefinitely. (For MSE( 0), see the discussion presented previously.)

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