Transformations for Multivariate Statistics

Abstract

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 corol-laries+ First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approx-imation+ 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

Edgeworth expansions and normalizing transforms for inequality measures

“…See also Marsh (2004) for a multivariate extension and Goncalves and Meddahi (2008) who apply the idea to transforming realized volatility . However, there are four important di¤erences.…”

“…Further improvements can be obtained by applying an additional bootstrap to the studentized transform as in Marsh (2004). 7 The idea is that the transform is more symmetrically distributed and bootstrapping converges faster for symmetric distributions.…”

Box–Cox transforms for realized volatility

“…andNiki and Konishi (1986) provide higher order expansions of the cumulants of a nonlinear transformation of an infeasible statistic in the univariate case whereasMarsh (2004) considers the multivariate case. The results in Proposition 3.1 can be obtained fromPhillips (1979) andNiki and Konishi (1986) when applied to the realized volatility context.…”

Calculating Cumulants of a Taylor Expansion of a Multivariate Function

“…The value of Edgeworth expansions in explaining the performance of the Bootstrap method is the subject matter of the entire Hall (1992b) monograph. See also Marsh (2004, ) for more recent comments on the usefulness of Edgeworth series in statistics.…”