Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

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P(τˆ|τ)

τˆ

Figure 31.7 The sampling distributionP(τˆ|τ) for the estimatorˆτfor the case

τ=4andN= 10.

whereP(ˆτ|τ) is given by (31.80) withN= 10. The above integrals can be evaluated
analytically but the calculations arerather cumbersome. It is much simpler to evaluate
them by numerical integration, from which we find [τ−,τ+]=[2. 86 , 5 .46]. Alternatively,
we could quote the estimate and its 68% confidence interval as

τ=3. 77 +1− 0 ..^6991.

Thus we see that the 68% central confidence interval is not symmetric about the estimated
value, and differs from the standard error calculated above. This is a result of the (non-
Gaussian) shape of the sampling distributionP(τˆ|τ), apparent in figure 31.7.

In many problems, however, it is not possible to derive the full sampling

distribution of an ML estimatoraˆin order to obtain its confidence intervals.

Indeed, one may not even be able to obtain an analytic formula for its standard

errorσˆa. This is particularly true when one is estimating several parameteraˆ

simultaneously, since the joint sampling distribution will be, in general, very

complicated. Nevertheless, as we discuss below, the likelihood functionL(x;a)

itselfcan be used very simply to obtain standard errors and confidence intervals.

The justification for this has its roots in theBayesianapproach to statistics, as

opposed to the more traditionalfrequentistapproach we have adopted here. We

now give a brief discussion of the Bayesian viewpoint on parameter estimation.

31.5.5 The Bayesian interpretation of the likelihood function

As stated at the beginning of section 31.5, the likelihood functionL(x;a)is

defined by

P(x|a,H)=L(x;a),