Tie Turning Box-Cox including Quadratic Forms in Regression

In a regression model where a Box-Cox transformation is used on a positive independent variable X which appears only once in the equation, the effect of X on the dependent variable Y is either strictly increasing or decreasing over the whole range of X , since the transformation is a monotonic function of X , increasing or decreasing depending on the Box-Cox parameter ë. This paper considers the case where the variable X appears twice in the regression with two different Box-Cox parameters 1 ë and 2 ë , to allow a turning point in Y which can be a maximum or minimum. First and second-order conditions for the critical point are derived. This general specification includes as a special case the quadratic form in X where 1 ë and 2 ë are set equal to 1 and 2, respectively. If, instead of using the Box-Cox transformations, one uses simple powers of X , this form is equivalent to the Box-Cox form except that neither 1 ë nor 2 ë can be equal to zero, since in this case 1 ë X or 2 ë X reduces to a constant of value 1.