Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

a regionRˆinˆa-space, such that

∫∫

Rˆ

P(ˆa|a)dMaˆ=1−α.

A common choice for such a region is that bounded by the ‘surface’P(aˆ|a)=

constant. By considering all possible valuesaand the values ofaˆlying within

the regionRˆ, one can construct a 2M-dimensional region in the combined space

(aˆ,a). Suppose now that, from our samplex, the values of the estimators are

aˆi,obs,i=1, 2 ,...,M. The intersection of theM‘hyperplanes’ˆai=aˆi,obswith

the 2M-dimensional region will determine anM-dimensional region which, when

projected ontoa-space, will determine a confidence limitRat the confidence

level 1−α. It is usually the case that this confidence region has to be evaluated

numerically.

The above procedure is clearly rather complicated in general and a simpler

approximate method that uses the likelihood function is discussed in subsec-

tion 31.5.5. As a consequence of the central limit theorem, however, in the

large-sample limit,N→∞, the joint sampling distributionP(ˆa|a) will tend, in

general, towards the multivariate Gaussian

P(ˆa|a)=

1

(2π)M/^2 |V|^1 /^2

exp

[

−^12 Q(aˆ,a)

]

, (31.38)

whereVis the covariance matrix of the estimators and the quadratic formQis

given by

Q(aˆ,a)=(aˆ−a)TV−^1 (aˆ−a).

Moreover, in the limit of largeN, the inverse covariance matrix tends to the

Fisher matrixFgiven in (31.36), i.e.V−^1 →F.

For the Gaussian sampling distribution (31.38), the process of obtaining confi-

dence intervals is greatly simplified. The surfaces of constantP(ˆa|a) correspond

to surfaces of constantQ(aˆ,a), which have the shape ofM-dimensional ellipsoids

inaˆ-space, centred on the true valuesa. In particular, let us suppose that the

ellipsoidQ(ˆa,a)=c(wherecis some constant) contains a fraction 1−αof the

total probability. Now suppose that, from our samplex, we obtain the valuesaˆobs

for our estimators. Because of the obvious symmetry of the quadratic formQ

with respect toaandˆa, it is clear that the ellipsoidQ(a,aˆobs)=cina-space that

is centred onaˆobsshould contain the true valuesawith probability 1−α. Thus

Q(a,aˆobs)=cdefines our required confidence regionRat this confidence level.

This is illustrated in figure 31.4 for the two-dimensional case.

It remains only to determine the constantccorresponding to the confidence

level 1−α. As discussed in subsection 30.15.2, the quantityQ(aˆ,a) is distributed

as aχ^2 variable of orderM. Thus, the confidence region corresponding to the