4 - Asymptotic Theory of Quantile Regression

Summary

Although the finite-sample distribution theory of regression quantiles can be represented explicitly as has been illustrated in Chapter 3, the practical application of this theory would entail a host of hazardous assumptions and an exhausting computational effort. It is generally conceded throughout statistics that approximation methods involving local linearization and the central limit theorem play an indispensable role in the analysis of the performance of statistical procedures and in rendering such procedures practical tools of statistical inference. The zealous pursuit of these objectives is inevitably met with accusations that we live in a cloud-cuckoo land of “asymptopia,” but life is full of necessary compromises and approximations. And it is fair to say that those who try to live in the world of “exact results” in finite-sample statistical distribution theory are exiled to an even more remote and exotic territory.

Fortunately, there are many tools available to help us evaluate the adequacy of our asymptotic approximations. Higher order expansions, although particularly challenging in the present context, may offer useful assessments of the accuracy of simpler approximations and possible refinement strategies. Monte Carlo simulation can be an extremely valuable tool, and the rapid development of resampling methods for statistical inference offers many new options for inference.

The fundamental task of asymptotic theory is to impose some discipline and rigor on the process of developing statistical procedures. The natural enthusiasm that arises from the first few “successful” applications of a new technique can be effectively tempered by some precisely cast questions of the following form: suppose data arose according to the conditions A, does the procedure produce a result that converges in some appropriate sense to object B?