S p-value and combination of tests. Consider all tests with a rejection region of the form T > c for a given real-valued statistic T

S p-value and combination of tests. Consider all tests with a rejection region of the form

¹T > cº for a given real-valued statistic T . Suppose that the distribution function F of T does not depend on # on the null hypothesis ‚0, in that P#.T  c/ D F.c/ for all # 2 ‚0 and c 2 R. In particular, this implies that the test with rejection region ¹T > cº has the size 1F.c/. The p-value p.x/ of a sample x 2 X is then defined as the largest size ˛ that, given x, still leads to the acceptance of the null hypothesis, i.e., p.x/ D 1F ı T.x/. Assume that F is continuous and strictly monotone on the interval ¹0 < F < 1º and show the following:

(a) Under the null hypothesis, p./ has the distribution U0;1OE. Hint: Problem 1.18.

(b) The test with rejection region ¹p./ < ˛º is equivalent to the size-˛ test with rejection region of the form ¹T >cº.

(c) If p1./; : : : ; pn./ are the p-values for n independent studies using the test statistic T , then S D 2 Pn iD1 log pi ./ is 2 2n-distributed on the null hypothesis, and the rejection region ¹S > 2 2nI1˛

º defines a size-˛ test that combines the different studies.