統計学輪講（第３０回）

日時　　  2001年11月13日（火）　15時00分〜15時50分
場所　　  経済学部5階視聴覚室
講演者　　Prof. Richard Johnson(University of Wisconsin)
演題　    TRANSFORMATIONS THAT REDUCE SKEWNESS OR IMPROVE NORMALITY
	 ---SOME ASYMPTOTIC RESULTS

概要：  
We first look at the Box-Cox Transformation in a regression setting.
Then, we introduce a new power transformation family which is well-defined 
on the whole real line and which is appropriate for reducing skewness. 
We first establish properties similar to those of the Box-Cox 
transformation. In particular, there is a convexity (or concavity) 
property as the parameter varies.
  
We next investigate the large sample properties of the transformation
in the context of a single random sample. Our new transformation 
is applied to improve not only the approximation to normality 
but also the approximation to symmetry.

Finally, we consider a nonparametric setting where the goal is to
estimate a location parameter on the basis of a random sample from 
some unspecified underlying distribution. We first estimate the 
transformation parameter for which the transformed variable is 
nearly symmetrically distributed around zero. An M-estimator is 
proposed, that minimizes the integrated square of the imaginary 
part of the empirical characteristic function. As part of our 
derivation of the asymptotic properties, we develop a uniform 
strong law of large numbers for Hoeffding U-statistics.