V. V. Ulianov, Y. Fujikoshi
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.