When is the inverse regression estimator MSE-superior to the standard regression estimator in multivariate controlled calibration situations?

We assume as model a standard multivariate regression of y on x, fitted to a controlled calibration sample and used to estimate unknown x's from observed y-values. The standard weighted least squares estimator ('classical', regress y on x and 'solve' for x) and the biased inverse regression estimator (regress x on y) are compared with respect to mean squared error. The regions are derived where the inverse regression estimator yields the smaller MSE. For any particular component of x this region is likely to contain 'most' future values in usual practice. For simultaneous estimation this needs not be true, however.mean squared error multivariate regression