Average likelihood function and higher-order statistics

This paper deals with the approximation of the average likelihood function (ALF) in the Gaussian context. This function is obtained by averaging the likelihood function (LF) over all the random parameters for which a probability density function (PDF) is assumed to be known. We show that it is possible to express the ALF by a power series expansion which involves higher-order statistics (HOS). The obtained expression turns to be a weighted sum of the cross-correlation between some "integrated moments" of the reference signal and estimated moments of the observation. This expression leads to practical tests and allows us to solve many classification and estimation problems. As an application we derive here the fourth-order multicycle detector.