Two-stage rank estimation of quantile index models.

This paper estimates a class of models which satisfy a monotonicity condition on theconditional quantile function of the response variable. This class includes as a special casethe monotonic transformation model with the error term satisfying a conditional quantilerestriction, thus allowing for very general forms of conditional heteroscedasticity.A two-stage approach is adopted to estimate the relevant parameters. In the "rst stagethe conditional quantile function is estimated nonparametrically by the local polynomialestimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246}269;Annals of Statistics 19 (1991b) 760}777) and Cavanagh (1996, Preprint). In the secondstage, the monotonicity of the quantile function is exploited to estimate the parameters ofinterest by maximizing a rank-based objective function. The proposed estimator is shownto have desirable asymptotic properties and can then also be used for dimensionalityreduction or to estimate the unknown structural function in the context of a transformationmodel.