  We consider approaches to cosmological parameter estimation in the
inflationary cosmology, focussing on the required accuracy of the initial power
spectra. Parametrizing the spectra, for example by power-laws, is well suited
to testing the inflationary paradigm but will only correctly estimate
cosmological parameters if the parametrization is sufficiently accurate, and we
investigate conditions under which this is achieved both for present data and
for upcoming satellite data. If inflation is favoured, reliable estimation of
its physical parameters requires an alternative approach adopting its detailed
predictions. For slow-roll inflation, we investigate the accuracy of the
predicted spectra at first and second order in the slow-roll expansion
(presenting the complete second-order corrections for the tensors for the first
time). We find that within the presently-allowed parameter space, there are
regions where it will be necessary to include second-order corrections to reach
the accuracy requirements of MAP and Planck satellite data. We end by proposing
a data analysis pipeline appropriate for testing inflation and for cosmological
parameter estimation from high-precision data.
