Bivariate quantile smoothing splines

It has long been recognized that the mean provides an inadequate summary 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 nonparametric estimators for the conditional quantile functions in a two‐dimensional design space. The estimators can be computed by using standard linear programming techniques and can further be used as building‐blocks for conditional quantile estimations in higher dimensions. For moderately large data sets, we recommend penalized bivariate B‐splines as approximate solutions. We use real and simulated data to illustrate the methodology proposed.

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