Bayesian transformation family selection: moving towards a transformed Gaussian universe

The problem of transformation selection is thoroughly treated from a Bayesian perspective. Several families of transformations are considered with a view to achieving normality: the Box-Cox, the Modulus, the Yeo & Johnson and the Dual transformation. Markov chain Monte Carlo algorithms have been constructed in order to sample from the posterior distribution of the transformation parameter $\lambda_T$ associated with each competing family $T$. We investigate different approaches to constructing compatible prior distributions for $\lambda_T$ over alternative transformation families, using a unit-information power-prior approach and an alternative normal prior with approximate unit-information interpretation. Selection and discrimination between different transformation families is attained via posterior model probabilities. We demonstrate the efficiency of our approach using a variety of simulated datasets. Although there is no choice of transformation family that can be universally applied to all problems, empirical evidence suggests that some particular data structures are best treated by specific transformation families. For example, skewness is associated with the Box-Cox family while fat-tailed distributions are efficiently treated using the Modulus transformation.