  This paper concerns the development of Stein's method for chi-square
approximation and its application to problems in statistics. New bounds for the
derivatives of the solution of the gamma Stein equation are obtained. These
bounds involve both the shape parameter and the order of the derivative.
Subsequently Stein's method for chi-square approximation is applied to bound
the distributional distance between Pearson's statistic and its limiting
chi-square distribution, measured using smooth test functions. In combination
with the use of symmetry arguments, Stein' method yields explicit bounds on
this distributional distance of order $n^{-1}$.
