Description
Traditional regression methods describe the mean function through a linear (or generalized linear) model. Focusing the attention on the mean has several advantages, such as computational simplicity or simple interpretation of estimated regression parameters. Using the words of Kneib [2], ``a common (mis-)perception of statistics in the public opinion equates statistics with means and averages, which has led to the famous quote that ‘statisticians are mean lovers’." However, the mean only captures one single aspect of the conditional response distribution. In many situations the mean may not be the actual quantity of interest (for instance, when investigating `extreme' values of the response), and focusing solely on the mean bears the risk of false conclusions about the significance/importance of covariates.Generalizing from the mean, a first step is to consider a model for the median response, given the covariate values. Computationally, this requires minimizing a `Least absolute deviation' (rather than a least squares) problem, which is typically solved through linear programming. This concept can then be easily extended from median to quantiles, by adjusting the `absolute' loss function appropriately [1]. See the image below for an example, which gives quantile-based linear regression curves for a series of different quantiles, using a simulated data set (see also [3]).
From this point, the focus of your project may drift into several directions depending on your interest. For instance, quantile regression is particularly powerful and useful in conjunction with nonparametric regression models, which additionally drop the linearity restriction of the linear predictor, and so allow for non-linear covariate effects. Further, one may consider structured additive predictors which allow for the inclusion of spatial effects, and hence enable applications such as disease mapping. Finally, one may look into the investigation of the novel concept of ``expectiles" rather than quantiles (forming the analogue of quantiles to the mean, rather than the median). See reference [2] for some introduction and examples for all of these.
Pre/Co-requisites
- Statistical Methods III
- Topics in Statistics III is useful but not necessary
Resources
- Koenker, R. (2005) Quantile Regression, Econometric Society Monograph Series, Cambridge University Press.
- Kneib, T. (2013). Beyond mean regression. Statistical Modelling 13, 275–303.
- Tutorial: Getting started with quantile regression.