Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS

aˆ

P(ˆa|a−) P(aˆ|a+)

aˆobs

α β

Figure 31.3 An illustration of how the observed value of the estimator,aˆobs,

and the given valuesαandβdetermine the two confidence limitsa−anda+,

which are such thataˆα(a+)=aˆobs=ˆaβ(a−).

each of sizeN, were analysed then the interval [a−,a+] would contain the true

valueaon a fraction 1−α−βof the occasions.

The interval [a−,a+] is called aconfidence interval onaat theconfidence

level 1 −α−β. The valuesa−anda+themselves are called respectively the

lower confidence limitand theupper confidence limitat this confidence level. In

practice, the confidence level is often quoted as a percentage. A convenient way

of presenting our results is
∫aˆobs

−∞

P(aˆ|a+)daˆ=α, (31.28)

∫∞

aˆobs

P(aˆ|a−)daˆ=β. (31.29)

The confidence limits may then be found by solving these equations fora−and

a+either analytically or numerically. The situation is illustrated graphically in

figure 31.3.

Occasionally one might not combine the results (31.28) and (31.29) but use

either one or the other to provide aone-sidedconfidence interval ona. Whenever

the results are combined to provide atwo-sidedconfidence interval, however, the

interval isnotspecified uniquely by the confidence level 1−α−β. In other words,

there are generally an infinite number of intervals [a−,a+] for which (31.27) holds.

To specify a unique interval, one often choosesα=β, resulting in thecentral

confidence intervalona. All cases can be covered by calculating the quantities

c=aˆ−a−andd=a+−aˆand quoting the result of an estimate as

a=aˆ+−dc.

So far we have assumed that the quantitiesaother than the quantity of interest

aare known in advance. If this is not the case then the construction of confidence

limits is considerably more complicated. This is discussed in subsection 31.3.6.