Power function of $${\varvec{F}}-$$ distribution: revisiting its computation and solution for geodetic studies

Abstract

The power function of \(F-\) distribution is the complementary cumulative distribution function of the non-central \(F-\) distribution. It is used to evaluate the power of the test based on the \(F\) or \({\chi }^{2}-\) distributed statistics. This paper revisits its computation and solution for the non-centrality parameter in geodetic studies and shows that the power function related to these studies can be computed efficiently and with minimal effort. To facilitate this, we introduce a novel standalone algorithm that consistently computes the power of the test, even for large non-centrality parameters (e.g., \(>{10}^{5}\)) and for \({\chi }^{2}\)-distribution. The solution of the power function for the non-centrality parameter is typically obtained using standard root finding algorithms, such as the bisection or Newton–Raphson methods. However, they may encounter convergence problems, particularly when the non-centrality parameter increases. We demonstrate that a solution can be readily obtained from a logarithmic form of the power function, ensuring convergence and removing the requirement for a precisely defined initial value. Furthermore, we utilize a few geometric relationships during the iteration to expedite the solution process. As a result, we propose a novel solution algorithm that is highly precise, stable, and at least four times faster than standard algorithms, even for the solution interval of \(<{0, 10}^{6}>\). This efficient solution is published online as a web-based application for geodetic detectability studies in addition to the given MATLAB and Python codes.