Closed-form approximations for the performance upper bound of inhomogeneous quadratic tests

TL;DR

This paper introduces two closed-form approximations for the performance bounds of inhomogeneous quadratic tests involving dependent non-central chi-square and Gaussian variables, aiding analytical performance evaluation.

Contribution

It proposes novel approximations based on Edgeworth series and EVT for the distribution of complex quadratic tests, enabling better performance analysis.

Findings

The approximations closely match numerical simulations.

They are effective in GNSS integrity transient detection.

The methods improve analytical tractability of inhomogeneous quadratic tests.

Abstract

This paper focuses on inhomogeneous quadratic tests, which involve the sum of a dependent non-central chi-square with a Gaussian random variable. Unfortunately, no closed-form expression is available for the statistical distribution of the resulting random variable, thus hindering the analytical characterization of these tests in terms of probability of detection and probability of false alarm. In order to circumvent this limitation, two closed-form approximations are proposed in this work based on results from Edgeworth series expansions and Extreme Value Theory (EVT). The use of these approximations is shown through a specific case of study in the context of integrity transient detection for Global Navigation Satellite Systems (GNSS). Numerical results are provided to assess the goodness of the proposed approximations, and to highlight their interest in real life applications.