STATISTICS
false (in which caseH 1 is true). The probabilityβ(say)thatsuchanerrorwill
occur is, in general, difficult to calculate, since the alternative hypothesisH 1 is
often composite. Nevertheless, in the case whereH 1 is a simple hypothesis, it is
straightforward (in principle) to calculateβ. Denoting the corresponding sampling
distribution oftbyP(t|H 1 ), the probabilityβis the integral ofP(t|H 1 )overthe
complementof the rejection region, called theacceptance region. For example, in
the case corresponding to (31.106) this probability is given by
β=Pr(t<tcrit|H 1 )=∫tcrit−∞P(t|H 1 )dt.This is illustrated in figure 31.10. The quantity 1−βis called thepowerof the
statistical test to reject the wrong hypothesis.
31.7.3 The Neyman–Pearson testIn the case whereH 0 andH 1 are both simple hypotheses, theNeyman–Pearson
lemma(which we shall not prove) allows one to determine the ‘best’ rejection
region and test statistic to use.
We consider first the choice of rejection region. Even in the general case, inwhich the test statistictis a multi-dimensional (vector) quantity, the Neyman–
Pearson lemma states that, for a given significance levelα, the rejection region for
H 0 giving the highest power for the test is the region oft-space for which
P(t|H 0 )
P(t|H 1 )>c, (31.107)wherecis some constant determined by the required significance level.
In the case where the test statistictis a simple scalar quantity, the Neyman–Pearson lemma is also useful in deciding which such statistic is the ‘best’ in
the sense of having the maximum power for a given significance levelα.From
(31.107), we can see that the best statistic is given by thelikelihood ratio
t(x)=P(x|H 0 )
P(x|H 1 ). (31.108)
and that the corresponding rejection region forH 0 is given byt<tcrit.Infact,
it is clear that any statisticu=f(t) will be equally good, provided thatf(t)isa
monotonically increasing function oft. The rejection region is thenu<f(tcrit).
Alternatively, one may use any test statisticv=g(t)whereg(t) is a monotonically
decreasing function oft; in this case the rejection region becomesv>g(tcrit). To
construct such statistics, however, one must knowP(x|H 0 )andP(x|H 1 ) explicitly,
and such cases are rare.