On estimating a transformation correlation coefficient

We consider a semiparametric and a parametric transformation-to-normality model for bivariate data. After an unstructured or structured monotone transformation of the measurement scales, the measurements are assumed to have a bivariate normal distribution with correlation coefficient „ , here termed the 'transformation correlation coefficient'. Under the semiparametric model with unstructured transformation, the principle of invariance leads to basing inference on the marginal ranks. The resulting rank-based likelihood function of „ is maximized via a Monte Carlo procedure. Under the parametric model, we consider Box-Cox type transformations and maximize the likelihood of „ along with the nuisance parameters. Efficiencies of competing methods are reported, both theoretically and by simulations. The methods are illustrated on a real-data example.