  With the use of simulated supernova catalogs, we show that the statefinder
parameters turn out to be poorly and biased estimated by standard cosmography.
To this end, we compute their standard deviations and several bias statistics
on cosmologies near the concordance model, demonstrating that these are very
large, making standard cosmography unsuitable for future and wider compilations
of data. To overcome this issue, we propose a new method that consists in
introducing the series of the Hubble function into the luminosity distance,
instead of considering the usual direct Taylor expansions of the luminosity
distance. Moreover, in order to speed up the numerical computations, we
estimate the coefficients of our expansions in a hierarchical manner, in which
the order of the expansion depends on the redshift of every single piece of
data. In addition, we propose two hybrids methods that incorporates standard
cosmography at low redshifts. The methods presented here perform better than
the standard approach of cosmography both in the errors and bias of the
estimated statefinders. We further propose a one-parameter diagnostic to reject
non-viable methods in cosmography.
