A Density—Quantile Function Perspective on Robust Estimation

Abstract

This chapter presents a perspective on robust estimation by regarding it as one of the components of a unified theory of statistical science, in which various approaches are applied simultaneously, such as parametric statistical inference, goodness of fit procedures, robust statistical inference, exploratory data analysis, and non-parametric statistical inference. A key to accomplishing such unification might be to think in terms of quantile functions and density-quantile functions. Parametric statistical inference may be concerned with statistical inference of parameters of a model from data assumed to satisfy the model. Goodness of fit procedures is concerned with testing whether a specified parametric model adequately fits the observed sample probabilities. Robust statistical inference may be concerned with statistical inference of parameters of a model from data assumed to satisfy the model only approximately. Exploratory data analysis may be concerned with statistical inference from data that is nonideal in the sense that it is not assumed to obey a specified model.