Bounds on Normal Approximations to Student's and the Chi-Square Distributions

Abstract

Formulas closely related to $$u(t) = \lbrack n \log (1 + t^2/n)\rbrack^{\frac{1}{2}}\\ w(\chi^2) = \lbrack\chi^2 - n - n \log (\chi^2/n)\rbrack^{\frac{1}{2}}$$ are considered for converting upper tail values of Student's $t$ or chi-square variates with $n$ degrees of freedom to normal deviates. The chief object of the paper is to construct bounds on the deviation from the exact normal deviates such that the absolute deviation is bounded by $cn^{-\frac{1}{2}}$ uniformly in the entire tail. Two approximations for Student's $t$ are suggested that are remarkably accurate and an improvement over other available approximations. The bounds and approximations for Student's $t$ are given in Section 3 and those for chi-square in Section 4. Some of the methods used in obtaining bounds may be of value in other investigations. These are given in Section 2. The development of the bounds was stimulated by the work of Teichroew [3]. He obtains expansions for the normal deviates corresponding to tail values of Student's $t$ and chi-square and achieves spectacular accuracy even for small $n$. The idea and the construction of the expansion is set forth, briefly, in [4], p. 647. The first terms of these expansions are the $u(t)$ and $w(\chi^2)$ used here. The bounds of Theorems 3.1 and 4.2 show that these first approximations are correct to $O(n^{-\frac{1}{2}})$ uniformly for all $t > 0$ or $\chi^2 > n$. This fact can be used to show that the Teichroew expansions are valid asymptotic expansions.