Exact vs Approximated ML Estimation for the Box-Cox Transformation

The Box-Cox (BC) transformation is widely used in data analysis for achieving approximate normality in the transformed scale. The transformation is only possible for non-negative data. This positiveness requirement implies a truncation to the distribution on the transformed scale and the distribution in the transformed scale is truncated normal. This fact has consequences for the estimation of the parameters specially if the truncated probability is high. In the seminal paper Box and Cox proposed to estimate parameters using the normal distribution which in practice means to ignore any consequences of the truncation on the estimation process. In this work we present the framework for exact likelihood estimation on the PN distribution to which we call method m(1) and how to calculate the parameters estimates using consistent estimators. We also present a pseudo-Likelihood function for the same model not taking into account truncation and allowing to replace parameters mu and sigma for their estimates. We call m(2) to this estimation method. We conclude that for cases where the truncated probability is low both methods give good estimation results. However for larger values of the truncated probability the m(2) method does not present the same efficiency.