to minimise the model degeneracy since the temperature and optical depth are generally correlated in the fitting procedure [e.g. @poptestingcompt2001; @pop2013mrk509].

We obtained a very good fit (${\chi^{2}/\textrm{dof}}=444/432$), with ${kT_{\textrm{e}}}= {80^{+ 40}_{- 20}}$ keV and $y = {1.08^{+ 0.05}_{- 0.08}}$, corresponding to an optical depth $\tau={1.7 \pm 0.7}$. The best-fitting parameters (${kT_{\textrm{e}}}, y$, and normalisation) are reported in Table \[tab:fits\_4388\], while the ${kT_{\textrm{e}}}- y$ contour plots are shown in Fig. \[fig:cont\_compps\]. The cross-normalisations, absorption and iron line parameters are consistent within the errors with the cut-off power law fit.

![\[fig:cont\_cutoffpl\] Contour plots of cut-off energy vs. photon index for NGC 4388 (black) and NGC 2110 (red). Solid, dashed, and dotted lines correspond to 68, 90, and 99 per cent confidence level, respectively.](cont_4388+2110_cutoffpl){width="\columnwidth"}

![\[fig:cont\_compps\] Contour plots of Compton parameter $y$ vs. electron temperature ${kT_{\textrm{e}}}$ for NGC 4388 (black) and NGC 2110 (red). Solid, dashed, and dotted lines correspond to 68, 90, and 99 per cent confidence level, respectively. Grey dotted lines correspond to constant values of optical depth.](cont_4388+2110_compps_new){width="\columnwidth"}

![\[fig:plot\_4388\] Upper panel: spectra of NGC 4388 with best-fitting [[compps]{}]{} model. Second panel: residuals, plotted as $\Delta \chi = $ (data-model)/error. Third panel: best-fitting model $E^2 f(E)$. The data were binned for plotting 