Chapter 7
Sufficiency
7.1 MeasuresofQualityofEstimators
In Chapters 4 and 6 we presented procedures for finding point estimates, interval
estimates, and tests of statistical hypotheses based on likelihood theory. In this
and the next chapter, we present some optimal point estimates and tests for certain
situations. We first consider point estimation.
In this chapter, as in Chapters 4 and 6, we find it convenient to use the letter
fto denote a pmf as well as a pdf. It is clear from the context whether we are
discussing the distributions of discrete or continuous random variables.
Supposef(x;θ)forθ∈Ω is the pdf (pmf) of a continuous (discrete) random
variableX. Consider a point estimatorYn=u(X 1 ,...,Xn) based on a sample
X 1 ,...,Xn. In Chapters 4 and 5, we discussed several properties of point estimators.
Recall thatYnis a consistent estimator (Definition 5.1.2) ofθifYnconverges to
θin probability; i.e.,Ynis close toθfor large sample sizes. This is definitely a
desirable property of a point estimator. Under suitable conditions, Theorem 6.1.3
shows that the maximum likelihood estimator is consistent. Another property was
unbiasedness (Definition 4.1.3), which says thatYnis an unbiased estimator ofθ
ifE(Yn)=θ. Recall that maximum likelihood estimators may not be unbiased,
although generally they are asymptotically unbiased (see Theorem 6.2.2).
If two estimators ofθare unbiased, it would seem that we would choose the one
with the smaller variance. This would be especially true if they were both approx-
imately normal because the one with the smaller asymptotic variance (and hence
asymptotic standard error) would tend to produce shorter asymptotic confidence
intervals forθ. This leads to the following definition:
Definition 7.1.1.For a given positive integern,Y=u(X 1 ,X 2 ,...,Xn)is called
aminimum variance unbiased estimator(MVUE) of the parameterθifYis
unbiased, that is,E(Y)=θ, and if the variance ofY is less than or equal to the
variance of every other unbiased estimator ofθ.
Example 7.1.1. As an illustration, letX 1 ,X 2 ,...,X 9 denote a random sample
from a distribution that isN(θ, σ^2 ),where −∞<θ<∞. Because the statistic
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