  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.
