Box-Cox transformed linear models: A parameter-based asymptotic approach

A Box-Cox transformed linear model usually has the form y(λ) = μ + β₁x₁ +… + β_px_p + oe, where y(λ) is the power transform of y. Although widely used in practice, the Fisher information matrix for the unknown parameters and, in particular, its inverse have not been studied seriously in the literature. We obtain those two important matrices to put the Box-Cox transformed linear model on a firmer ground. The question of how to make inference on β = (β₁,…,β_p)^T when λ; is estimated from the data is then discussed for large but finite sample size by studying some parameter-based asymptotics. Both unconditional and conditional inference are studied from the frequentist point of view.