Edgeworth expansions and rates of convergence for normalized sums: Chung’s 1946 method revisited

In this work we revisit, correct and extend Chung’s 1946 method for deriving higher order Edgeworth expansions with respect to tt-statistics and generalized self-normalized sums. Thereby we provide a set of formulas which allows the computation of the approximation of any order and specify the first four polynomials in the Edgeworth expansion, the first two of which are well known. It turns out that knowledge of the first four polynomials is necessary and sufficient for characterizing the rate of convergence of the Edgeworth expansion in terms of moments and the norming sequence appearing in generalized self-normalized sums. It will be shown that depending on the moments and the norming sequence, the rate of convergence can be O(n−i/2)O(n−i/2), i=1,…,4i=1,…,4. Finally, we study expansions and rates of convergence if the normal distribution is replaced by the tt-distribution.