Reduced-bias Moment Approximations in Original Units When Using Multivariate Box‐Cox Transformations

Motivated by A‐10 single engine aircraft climb experiments, we demonstrate use of the multivariate Box–Cox power transformations in fitting normal‐theory linear models to a q‐variate response vector Y. As predictions in the original units of the response are often desired, a retransformation of the fitted model back to the original units can be performed. Unfortunately, Jensen's inequality suggests that for a nonlinear transformation, such an approach can induce significant bias into the retransformed values. Further, although the retransformation offers a direct estimate of E(Y), it offers no direct estimates for the variances and covariances of Y. An estimate for the variance–covariance matrix can be useful when constructing approximate joint prediction regions on Y. To address these concerns, we consider the class of multivariate Box–Cox transformations and derive a closed‐form approximation to EYikiYjkj(ki,kj∈[0,1,...]; i,j = 1,…,q), which can then be used to provide reduced‐bias estimates of elements of the mean vector and covariance matrix of the original response Y, given parameter estimates obtained from fitting the model in the transformed domain. Using our approximation, we then construct an approximate 100(1 − α)% joint prediction ellipsoid on Y. Unlike the prediction ellipsoid offered by ordinary least squares analysis, the proposed prediction ellipsoid can change in both size and orientation, depending on the levels of the experimental factors.