Abstract:

When dealing with k independent samples, it is frequently of interest to jointly assess the underlying population distributions. Usually such an
assessment is carried out by performing k independent tests; that is, we test the null hypothesis [see pdf for notation] that the population from
which the [see pdf for notaion] sample was drawn has some specified distribution. Combining the results of such independent tests may then be carried
out by Fisher's method (1950, pp. 99101). Specifically, let Ti be the test statistic associated with the [see pdf for notaion] sample. Suppose large
values of Ti are considered critical for testing H. The attained significance level (ASL) or Pvalue is denoted by Pi; that is, if a is the observed
value of the test statistic Ti , then Prob[see pdf for notaion]. Furthermore, [see pdf for notaion] has a x2 distribution with 2k degrees of freedom
when H01 ,...,HOk are true. If a null hypothesis is not true, then the corresponding Pi will tend to be small resulting in a larger S. Hence the
righttail of the distribution of S is the critical Littell and Folks (1973) have shown that Fisher's method in asymptotically optimal among
essentially all methods of combining independent tests. 