14: Turning Box-Cox Including Quadratic Forms in Regression

Our problem¹, prompted by the increasing search for flexible turning functions, for instance in the explanation of how some transport costs might vary with distance, or of how fatal road accident frequency and severity fall might rise and then fall² with increased traffic (Tegnér and Loncar-Lucassi, 1997), or of how the impact of alcohol on fatal road accident frequency might be U-shaped³, is to find the conditions under which the function y(X) has a maximum or minimum over the positive region of X in the following model:

where the positive independent variable X is transformed by two different Box-Cox parameters (λ₁ ≠ λ₂) so that the model is identified in terms of the transformed variables.