## Box-Cox Transformation

Box and Cox (1964) proposed the model3

z,(A) = x# + u„

where, forj>, > 0,

— 1

Z,(A)== )T if ХФ0

= log y, if A = 0.

Note that because lim^oCy? — 1 )/A = log y,, z,(A) is continuous at A = 0. It is assumed that (и,) are i. i.d. with Eu, = 0 and Vu, = a2.

The transformation z,(A) is attractive because it contains y, and log y, as special cases, and therefore the choice between y, and log y, can be made within the framework of classical statistical inference.

Box and Cox proposed estimating A, fi, and a1 by the method of maximum likelihood assuming the normality of ut. However, u, cannot be normally distributed unless A = 0 because z,(A) is subject to the following bounds:

z,(A)S-I if A>0 (8.1.14)

if A <0.

A

Later we shall discuss their implications on the properties of the Box – Cox M...

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