McCullagh Index Notation Revisited, with Special Reference to Cumulants of Nonlinear Transformations in Linear Regression Models

• Activity Number: 207
• Type: Contributed
• Date/Time: Tuesday, August 5, 2003 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Kamanzi Binyavanga*+
• Companies: University of Transkei
• Keywords: MINO ; cumulants ; nonlinear transformations ; regression models ; Edgeworth series

Abstract

Recently , McCullagh (1984,1987) has made monumental efforts to demonstrate that many complicated multivariate statistical calculations can be made comparatively simpler and more transparent through the systematic use of appropriately chosen index notation and special arrays called tensors. In this paper, we consider three aspects of the author's notation MINO (for McCullagh Index Notation). First, we review its original account. In the review, we take the opportunity to propose remedies for several shortcomings that we have observed. We also simplify and repackage the notation in a more understable fashion, thus making it more accessible to other researchers. Second, the application of MINO is extended in several respects with regard to the theory of cumulants of nonlinear transformations in linear regression models. Last, a further application of the notation is proposed as a natural alternative to the delta method of Hall (1992a,1992b) for skewness-reduction in an Edgeworth series expansion.