The Yeo-Johnson family of modified power transformations (Yeo & Johnson, 2000) is an alternative to using a start when both negative (or 0) and positive values are included in the data. The Yeo-Johnson family is defined as follows:

X ! X½p)

[ ðX þ 1Þ

ðpÞ for X ‡ 0

ð1 # XÞ

ð2#pÞ for X < 0

(

where the parenthetical superscript ðpÞ gives the Box-Cox power, as in Equation 4.2

(a) Graph the transformations X½p) in the Yeo-Johnson family for values of X between

#10 and þ10 and powers p of #1, #0:5, 0, 0.5, 1, and 2.

(b) Now consider strictly positive X-values between 0.1 and 10. Compare the Yeo-Johnson and Box-Cox transformations of X for powers p of #1, #0:5, 0, 0.5, 1, and 2.

(c) &As in Section 4.6 for Box-Cox transformations, derive the maximum-likelihood estimator of the Yeo-Johnson transformations to multivariate normality of a vector of Xsg.