Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

aˆ

P(aˆ|a)

aˆα(a) aˆβ(a)

α β

Figure 31.2 The sampling distributionP(aˆ|a)ofsomeestimatorˆafor a given

value ofa. The shaded regions indicate the two probabilities Pr(a<ˆ aˆα(a)) =α

and Pr(a>ˆ ˆaβ(a)) =β.

ofaˆbyP(aˆ|a). For any particular value ofa, one can determine the two values

aˆα(a)andaˆβ(a) such that

Pr(a<ˆ aˆα(a)) =

∫ˆaα(a)

−∞

P(aˆ|a)dˆa=α, (31.25)

Pr(a>ˆ aˆβ(a)) =

∫∞

aˆβ(a)

P(ˆa|a)daˆ=β. (31.26)

This is illustrated in figure 31.2. Thus, for any particular value ofa, the probability

that the estimatoraˆlies within the limitsaˆα(a)andaˆβ(a) is given by

Pr(ˆaα(a)<a<ˆ aˆβ(a)) =

∫ˆaβ(a)

ˆaα(a)

P(aˆ|a)daˆ=1−α−β.

Now, let us suppose that from our samplex 1 ,x 2 ,...,xN, we actually obtain the

valueaˆobsfor our estimator. Ifˆais a good estimator ofathen we would expect

aˆα(a)andˆaβ(a) to be monotonically increasing functions ofa(i.e.ˆaαandaˆβboth

change in thesamesense asawhen the latter is varied). Assuming this to be the

case, we can uniquely define the two numbersa−anda+by the relationships

aˆα(a+)=aˆobs and aˆβ(a−)=ˆaobs.

From (31.25) and (31.26) it follows that

Pr(a+<a)=α and Pr(a−>a)=β,

which when taken together imply

Pr(a−<a<a+)=1−α−β. (31.27)

Thus, from our estimateaˆobs, we have determined two valuesa−anda+such that

this interval contains the true value ofawith probability 1−α−β. It should be

emphasised thata−anda+are random variables. If a large number of samples,