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In this paper, we extend a previous work where we presented a method based on the  N ‐point probability density function (pdf) to study the Gaussianity of the cosmic microwave background (CMB). We explore a local non‐linear perturbative model up to third order as a general characterization of the CMB anisotropies. We focus our analysis in large‐scale anisotropies  (θ &gt; 1°) . At these angular scales (the Sachs–Wolfe regime), the non‐Gaussian description proposed in this work defaults (under certain conditions) to an approximated local form of the weak non‐linear coupling inflationary model. In particular, the  quadratic  and  cubic  terms are governed by the non‐linear coupling parameters  f  NL  and  g  NL , respectively. The extension proposed in this paper allows us to directly constrain these non‐linear parameters. Applying the proposed methodology to  Wilkinson Microwave Anisotropy Probe  ( WMAP ) 5‐year data, we obtain  −5.6 × 10 5  &lt;  g  NL  &lt; 6.4 × 10 5  , at 95 per cent confidence level. This result is in agreement with previous findings obtained for equivalent non‐Gaussian models and with different non‐Gaussian estimators, although this is the first direct constraint on  g  NL  from CMB data. A model selection test is performed, indicating that a Gaussian model (i.e.   f  NL ≡ 0  and   g  NL ≡ 0 ) is preferred to the non‐Gaussian scenario. When comparing different non‐Gaussian models, we observe that a pure  f  NL  model (i.e.   g  NL ≡ 0 ) is the most favoured case and that a pure  g  NL  model (i.e.   f  NL ≡ 0 ) is more likely than a general non‐Gaussian scenario (i.e.   f  NL ≠ 0  and   g  NL ≠ 0 ). Finally, we have analysed the  WMAP  data in two independent hemispheres, in particular the ones defined by the dipolar pattern found by Hoftuft et al. We show that, whereas the  g  NL  value is compatible between both hemispheres, it is not the case for  f  NL  (with a  p ‐value of ≈0.04). However, if, as suggested by Hoftuft et al., anisotropy of the data is assumed, the distance between the likelihood distributions for each hemisphere is larger than expected from Gaussian and anisotropic simulations, not only for  f  NL  but also for   g  NL   (with a  p ‐value of ≈0.001 in the case of this latter parameter). This result is extra evidence for the CMB asymmetries previously reported in  WMAP  data.

Constraints on f NL and g NL from the analysis of the N ‐pdf of the CMB large‐scale anisotropies