Inequalities for Noncentral Chi-square Distributions

Noncentral chi-square distributions play an important role in statistics. They occur for instance as exact or limiting distributions of test statistics under alternatives. To examine the power functions of such tests we come across the problem of comparing distribution functions of noncentral chi-square variables. In this section inequalities are presented, which make such a comparison possible. They show how differences between noncentralities are carried over to differences between their distribution functions. A nontrivial application is given in Section 2, where two approximations to rhe power of Pearson’s &i-square test are investigated. In the following theorem an upper bound is given for the difference between the distribution functions of noncentral cm-square variables with the same degrees of freedom and different noncentralities. For k = 1, 2,. . . , and 6 2 0 denote by xi(S) an r.v. with a noncentral chi-square distribution with k degrees of freedom and noncentrality parameter 6.