\rd t = 1.\end{aligned}$$ With $S_i = (T_i,F_i)$, $i\in{\mathbb N}$, in ${\mathcal S}= {\mathbb R}^d\times C({\mathbb R}^d)$, intensity measure $\nu(\rd t \times \rd g)=\rd t\sP_F(\rd g)$ and $f((t,g), s)=g(s-t)$, $(t,g)\in{\mathcal S}$, we obtain the class of *mixed moving maxima (M3) processes* $$\begin{aligned}
  \xi(t) = \max_{i\in{\mathbb N}} U_i F_i(t- T_i), \quad t\in{\mathbb R}^d. \label{def_M3}\end{aligned}$$ These processes are max-stable and stationary on ${\mathbb R}^d$ (see for instance [@wan2010]). The function $F$ is called *shape function of $\xi$* and can also be deterministic (e.g., in case of the Smith process). In Smith’s “rainfall-storm” interpretation [@smi1990], $U_i$ and $T_i$ are the strength and center point of the $i$th storm, respectively, and $U_i F_i(t- T_i)$ represents the corresponding amount of rainfall at location $t$. In this case, $\xi(t)$ is the process of extremal precipitation.

When i.i.d. realizations $\eta_1, \ldots, \eta_n$ of $\eta$ in the MDA of a max-stable process $\xi$ are observed, a classical approach for parametric inference on $\xi$ is based on generating (approximate) realizations of $\xi$ out of the data $\eta_1, \ldots, \eta_n$ via componentwise block maxima and applying maximum likelihood (ML) estimation afterwards. A clear drawback of this method is that it ignores all information on large values that is contained in 