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Transformations for multivariate statistics. (English) Zbl 1071.62049

Summary: This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending results of N. Niki and S. Konishi [Ann. Inst. Stat. Math. 38, 371–383 (1986; Zbl 0609.62075)]. Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformations 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 R. Beran [J. Am. Stat. Assoc. 83, No. 403, 687–697 (1988; Zbl 0662.62024)], 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 D. W. K. Andrews, Econometrica 70, 119–162 (2002)] as distributional approximations is assessed.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
65C60 Computational problems in statistics (MSC2010)
62B05 Sufficient statistics and fields
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