Transformations For Multivariate Statistics

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi (1986, Annals of the Institute of Statistical Mathematics 38, 371–383). Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order. Application of these results yields two useful corollaries. First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approximation. Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran (1988, Journal of the American Statistical Association 83, 687–697), applied to the original statistic, implying a significant computational saving.The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood. The numerical properties are examined within a class of nonlinear regression models (logit, probit, Poisson, and exponential regressions), where the adequacy of the limiting normal and of the bootstrap (utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119–162) as distributional approximations is assessed.This paper is derived from my Ph.D. thesis, “Higher-Order Asymptotics for Econometric Estimators and Tests,” for which thanks for patient and helpful supervision go to Grant Hillier. Comments by Karim Abadir, Francesco Bravo, Giovanni Forchini, Soren Johansen, Paul Marriott, Mark Salmon, and Steve Satchell, participants at the conference “Differential Geometric Methods in Econometrics,” held at EUI, Florence, and by two anonymous referees proved most helpful. In particular, I thank Peter Phillips for showing interest in the paper, helping with improving the exposition, and providing me with copies of two unpublished research notes. Financial support in the form of a Leverhulme Special Research Fellowship in Economics and Mathematics is gratefully acknowledged.