Function compositional adjustments of conditional quantile curves

To adjust the quantile function estimated using a parametric model, the parametric function is composed with the quantile function of the probability integral transformed data. One round of bandwidth selection suffices as adjustments at all quantile levels can be obtained by smoothing the same set of probability integral transformed data. This is in contrast to the customary additive adjustment which requires the user to transform the data differently for estimating different quantiles. Another advantage of the proposed method is that it yields a diagnostic plot useful in assessing the goodness of fit of the assumed model. Compared with the additive approach, function compositional adjustment pays more attention to the fact that the quantity being estimated is a quantile function. As a result, it enjoys intrinsically some desirable properties such as range preservation and invariance to increasing transformation. It is also more amenable to the study of monotonicity as the composition of two quantile functions is a quantile function. In further support of the new adjustment method, Taylor series approximation and results from three simulation studies suggest that the adjusted estimator is robust to model misspecification, and can be more efficient than direct nonparametric estimation. We illustrate the proposed adjustment method using two examples from water resources and human biomonitoring studies.