Transforming grouped bivariate data to near normality (Q1099904)

Consider a sample of size n, \((X_{11},X_{21}),...,(X_{1n},X_{2n})\), from an absolutely continuous distribution with pdf g over \(R^ 2_+\) with the mean vector \(\mu_ 0\) and the covariance matrix \(\Sigma_ 0\). Assume that the usual Box and Cox transformation \[ x^{(\lambda_ 0)}=(x_ 1^{(\lambda_{10})},x_ 2^{(\lambda_{20})})'\sim N_ 2(\mu_ 0\quad,\Sigma_ 0). \] Let \(\theta =(\mu_ 0,\Sigma_ 0,\lambda_ 0)'\) and the sample be grouped into k,h cells. The large sample properties of the MLE \(\hat\theta_ n\) are given in this paper. An iterative procedure is suggested to obtain the MLE \({\hat \theta}_ n\). Regression and correlation are obtained from the transformed grouped data. Also, by transforming back to the original scale, we obtain a smoothed version of the data.