Spring 2002 Seminar Series - March 28, 2002
University of Minnesota
School of Statistics
College of Liberal Arts
A New Family of Power Transformations To Improve Normality
Richard A. Johnson
Department of Statistics
University of Wisconsin-Madison
Thursday, March 28, 2002
4:00 PM,
B10
Ford Hall
Minneapolis, East Bank Campus
Social at 3:30 PM,
300
Ford Hall
Abstract
Many parametric techniques in statistics are based on
the assumption that the underlying distribution of the data
is normal.
We begin by commenting on existing transformations.
Then we introduce a new power transformation family which is well-defined
on the whole real line and which is appropriate for reducing skewness.
It has properties similar to those of the Box-Cox
transformation for positive variables.
In particular, there is a convexity (or concavity)
property in the parameter.
When a centering constant is included, the new family closely approximates
the
Box-Cox family where the observations must be positive.
We first investigate the large sample properties of the transformation
in the context of a single random sample. Next, we extend the
application of the new transformation to the regression setting.
Our general conditions suggest the kinds of smoothness that need
to satisfied by the sequence of design matrices and
the underlying
distribution in order for asymptotic normality to hold.
Some new applications to testing normality as well as
time series are indicated.
We then consider an alternative approach
to transforming to near symmetry. This leads us to
study the almost sure convergence of
$U$ statistics which depend on a parameter. We give a
set of sufficient conditions for uniform convergence, in
the parameter.