Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSfalse (in which case H 1 is true). The probability β (say)thatsuchanerrorwilloccur is, in general, difficult to calculate, since the alternative hypothesis H 1 isoften composite. Nevertheless, in the case where H 1 is a simple hypothesis, it isstraightforward (in principle) to calculate β. Denoting the corresponding samplingdistribution of t by P (t|H 1 ), the probability β is the integral of P (t|H 1 )overthecomplement of the rejection region, called the acceptance region. For example, inthe case corresponding to (31.106) this probability is given byβ =Pr(tc, (31.107)P (t|H 1 )where c is some constant determined by the required significance level.In the case where the test statistic t is a simple scalar quantity, the Neyman–Pearson lemma is also useful in deciding which such statistic is the ‘best’ inthe sense of having the maximum power for a given significance level α. From(31.107), we can see that the best statistic is given by the likelihood ratiot(x) = P (x|H 0)P (x|H 1 ) . (31.108)and that the corresponding rejection region for H 0 is given by t