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:
where the parenthetical superscript (p) gives the Box-Cox power, as in Equation 4.2
(a) Graph the transformations X[p]) in the Yeo-Johnson family for values of X between - 10 and + 10 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.