In this paper, we discuss the modeling of an observed time series as the output of a second-order Hammerstein model driven by an independent and identically distributed (i.i.d.) Gaussian noise. The modeling is discussed from the point of view of the hypothesis testing (HT) of statistical indexes derived from higher order statistics (HOS). Three points of view can be adopted in HT. The more general approach is to not make any assumption on the index statistics (notably on the variance) and thus to use the $T^{2}$ Hotelling test. The second point of view involves calculating a theoretical variance in order to define normalized variables whose statistics are known. The last approach is a mixed version of the others two. These three approaches are applied to statistical indexes derived from linear combination of slices of spectra from the third order up the fifth order. Some subcases need to use the derivatives of the polyspectra since general relationships are null and cannot be used for the HT. The second part of this paper is devoted to (linear and quadratic) kernel identification using the relationship defined for HT definition. As for the HT, several subcases have to be discussed using either the polyspectra or the derivatives of polyspectra.