A linear approximation to the power function of a test

In this paper we obtain a linear approximation to the power function of a test that is very accurate for small sample sizes. This is especially useful for robust tests where not many power functions are available. The approximation is based on the von Mises expansion of the tail probability functional and on the Tail Area Influence Function (TAIF). The goals of the paper are, first to extend the definition of the TAIF to the case of non identically distributed random variables, defining the Partial Tail Area Influence Functions and the Vectorial Tail Area Influence Function; second, to obtain exact expressions for computing these new influence functions; and, finally, to find accurate approximations to the power function, that can be used in the case of non identically distributed random variables. We include some examples of the application of this linear approximation to tests that involve the Huber statistic and also saddlepoint tests, so proving that the approximations apply not only to simple problems but also to complex ones.