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

Helmut Finner (1) , Thorsten Dickhaus (2)

1 German Diabetes Center
2 Department of Mathematics

In this paper we revisit, correct and extend Chung's 1946 method for deriving higher order Edgeworth expansions with respect to -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 moments and norming sequence the rate of convergence can be , . Finally, we study expansions and rates of convergence if the normal distribution is replaced by the -distribution.

Mots clés: Generalized self-normalized sum; Hermite polynomials; Normal approximation; Statistical algorithms; t-distribution; t-statistic

Domaines: Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Helmut Finner, Thorsten Dickhaus. Edgeworth expansions and rates of convergence for normalized sums: Chung's 1946 method revisited. Statistics and Probability Letters, 2010, 80 (23-24), pp.1875. ⟨10.1016/j.spl.2010.08.014⟩. ⟨hal-00691779⟩