Power approximations for test statistics with dominant components

We consider approximating the power functions of some tests for several hypothesis testing problems in time series. The test statistics of interest are ratios of quadratic forms in normal variables and their power is related to the distributions of weighted sums of Chi-square random variables. Conventionally, power functions are evaluated from these distributions at each alternative, numerically, by Pearson's moment approximation, Imhof's procedure, Edgeworth-type expansion or the Monte Carlo method. In this study, we propose analytic approximations to the power functions when part of the weighted sum of Chi-square random variables can be well-approximated by a scaled Chi-square variable in distribution. In applications, the proposed analytic approximation may be obtained easily by evaluating the power only at a few alternative values. Several illustrative examples are presented and they show excellent agreement with the true power functions.