BIVARIATE QUANTILE SMOOTHING SPLINES

It has long been recognized that the mean provides an inadequate summa ry whereas the set of quantiles can supply a more complete description of a sample. We introduce bivariate quantile smoothing splines, which belong to the space of bilinear tensor product splines, as nonparamet ric estimators for the conditional quantile functions in a two-dimensi onal design space. The estimators can be computed by using standard li near programming techniques and can further be used as building-blocks for conditional quantile estimations in higher dimensions. For modera tely large data sets, we recommend penalized bivariate B-splines as ap proximate solutions. We use real and simulated data to illustrate the methodology proposed.