Developments on Quantile Regression

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Francis Galton, the progenitor of modern regression, chided those of his statistical colleagues who "limit their inquiries to Averages and do not revel in more comprehensive views". Arguing that any complete analysis of the full variety of experience requires the entire distribution of a trait, not just a measure of its central tendency, he introduced the empirical quantile function as a convenient graphical device for this purpose. Unfortunately, the very success of least squares methods throughout applied statistics has obscured the need for a more complete analysis of the statistical relationship among variables. The least squares regression limits its inquiries to the conditional mean function and thus can fail to find when structural relationships in the data may depend on the size of the response. For example, patients with long survival times may respond to treatment differently from those with average survival times; or persons with long periods of unemployment may respond to training differently from those with shorter unemployment periods. Such differences could not be seen in standard analyses that model only the mean response. The investigators propose to extend conditional quantile functions to more complex situations, specifically to parametric and semiparametric regression quantiles for correlated or censored response variables (which are common in both examples mentioned above). The computation of the conditional quantile functions is facilitated by modern linear programming algorithms, and appropriate statistical inference can be developed through traditional large sample theory or Markov Chain Marginal Bootstrap being developed by the PI and his colleagues. Conditional quantile functions help data analysts understand general heterogeneity in the population. They are often of direct interest in applications ranging from biomedical research, economic and business analyses to infrastructure studies. The proposed research is to establish a firm statistical theory for regression quantiles and provide a complete toolkit for their applications in complex problems with correlated and/or censored data.