Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSa region ˆR in â-space, such that∫∫P (â|a) d M â =1− α.ˆRA common choice for such a region is that bounded by the ‘surface’ P (â|a) =constant. By considering all possible values a and the values of â lying withinthe region ˆR, one can construct a 2M-dimensional region in the combined space(â, a). Suppose now that, from our sample x, the values of the estimators areâ i,obs , i =1, 2,...,M. The intersection of the M ‘hyperplanes’ â i = â i,obs withthe 2M-dimensional region will determine an M-dimensional region which, whenprojected onto a-space, will determine a confidence limit R at the confidencelevel 1 − α. It is usually the case that this confidence region has to be evaluatednumerically.The above procedure is clearly rather complicated in general and a simplerapproximate method that uses the likelihood function is discussed in subsection31.5.5. As a consequence of the central limit theorem, however, in thelarge-sample limit, N →∞, the joint sampling distribution P (â|a) will tend, ingeneral, towards the multivariate GaussianP (â|a) =1(2π) M/2 |V| 1/2 exp [ − 1 2 Q(â, a)] , (31.38)where V is the covariance matrix of the estimators and the quadratic form Q isgiven byQ(â, a) =(â − a) T V −1 (â − a).Moreover, in the limit of large N, the inverse covariance matrix tends to theFisher matrix F given in (31.36), i.e. V −1 → F.For the Gaussian sampling distribution (31.38), the process of obtaining confidenceintervals is greatly simplified. The surfaces of constant P (â|a) correspondto surfaces of constant Q(â, a), which have the shape of M-dimensional ellipsoidsin â-space, centred on the true values a. In particular, let us suppose that theellipsoid Q(â, a) =c (where c is some constant) contains a fraction 1 − α of thetotal probability. Now suppose that, from our sample x, we obtain the values â obsfor our estimators. Because of the obvious symmetry of the quadratic form Qwith respect to a and â, it is clear that the ellipsoid Q(a, â obs )=c in a-space thatis centred on â obs should contain the true values a with probability 1 − α. ThusQ(a, â obs )=c defines our required confidence region R at this confidence level.This is illustrated in figure 31.4 for the two-dimensional case.It remains only to determine the constant c corresponding to the confidencelevel 1 − α. As discussed in subsection 30.15.2, the quantity Q(â, a) is distributedas a χ 2 variable of order M. Thus, the confidence region corresponding to the1242