  The PLANCK collaboration has determined values for the spectral parameters of
the CMB radiation, namely the spectral index $n_s$, its running $\alpha_s$, the
running of the running $\beta_s$, using a growing body of measurements of CMB
anisotropies by the Planck satellite and other missions. These values do not
follow the hierarchy of sizes predicted by single field, slow roll inflationary
theory, and are thus difficult to fit for such inflation models.
  In this work we present first a study of 49 single field, slow roll
inflationary potentials in which we assess the likelyhood of these models
fitting the spectral parameters to their currently most accurate determination
given by the PLANCK collaboration. We check numerically with a MATLAB program
the spectral parameters that each model can yield for a very broad,
comprehensive list of possible parameter and field values. The comparison of
spectral parameter values supported by the models with their determinations by
the PLANCK collaboration leads to the conclusion that the data provided by
PLANCK2015 TT+lowP and PLANCK2015 TT,TE,EE+lowP taking into account the running
of the running disfavours 40 of the 49 models with confidence level at least
92.8\%.
  Next, we discuss the reliability of the current computations of these
spectral parameters. We identify a bias in the method of determination of the
spectral parameters by least residue parameter fitting (using MCMC or any other
scheme) currently used to reconstruct the power spectrum of scalar
perturbations. This bias can explain the observed contradiction between theory
and observations. Its removal is computationally costly, but necessary in order
to compare the forecasts of single field, slow roll theories with observations.
