V. V. Ulianov, Y. Fujikoshi
abstract:
For a statistic $S$ whose distribution can be approximated by
$\chi^2$-dis\-tri\-bu\-ti\-ons, there is a considerable interest in
constructing improved
$\chi ^2$-approximations. A typical approach is to consider a
transformation
$T=T(S)$ based on the Bartlett correction or the Bartlett type correction. In
this paper we consider two cases in which $S$ is expressed as a scale mixture
of a $\chi ^2$-variate or the distribution of $S$ allows an asymptotic
expansion in terms of $\chi^2$-distributions. For these statistics, we give
sufficient conditions for $T$ to have an improved $\chi ^2$-approximation.
Furthermore, we present a method for obtaining its error bound.