RANK ESTIMATORS FOR A TRANSFORMATION MODEL

We establish [square root]n-consistency and asymptotic normality of Han's (1987a, Journal of Econometrics 35, 191–209) estimator of the parameters characterizing the transformation function in a semiparametric transformation model. We verify a Vapnik–Cervonenkis (VC) condition for the parameterizations of Box and Cox (1964, Journal of the Royal Statistical Society, Series B 34, 187–200) and Bickel and Doksum (1981, Journal of the American Statistical Association 76, 296–311). The verification establishes the VC property for a class of sets generated by a nonlinear function of the transformation parameters. We also introduce a new class of rank estimators for these parameters. These estimators require only O(n2 logn) computations to evaluate the criterion function, compared to O(n4) computations for Han's estimator. We prove that these estimators are also [square root]n-consistent and asymptotically normal. A simulation study compares two of the new estimators to Han's estimator, the fully parametric estimator of Bickel and Doksum (1981), and the nonlinear two-stage least squares estimator of Amemiya and Powell (1981, Journal of Econometrics 17, 351–381).