Asymmetric statistics quantiles expectiles and data depth Seminar
- Date:
- 10 May 2012
- Venue:
- Building 54 Room 10037
For more information regarding this seminar, please email Mrs Jane Revell at j.revell@southampton.ac.uk .
Event details
Statistics research seminars
Abstract
We all are familiar with means and medians. The mean minimizes the sum of squares of residuals and the median the sum of their absolute values. Negative and positive residuals are treated equally. What happens if we give them different weights? This question leads us into the fascinating world of asymmetric statistics.
Quantiles can be computed by minimizing an symmetrically weighted sum of absolute values. For example, we get the 75-th percentile if we give weight 0.25 to negative residuals, and weight 0.75 to the positive ones. This approach can be directly extended to regression on predictors, so-called quantile regression. With suitable basis functions and a penalty we get quantile smoothing.
It is not generally known that asymmetric weighting of residuals can also be used in a least squares setting. This leads to expectile regression or smoothing. Least asymmetrically weighted squares (LAWS) has many interesting and useful properties, of which I will give examples. Expectile computations are easy.
One can use either quantiles or expectiles to define {\em data depth}, which is a measure of how close individual data points are to the center of a sample. In two dimensions one can compute convex quantile or expectile contours for a cloud of points. They are a valuable enhancement of a scatterplot.
Speaker information
Dr Paul Eilers , Erasmus Medical Center, Rotterdam. Department of Biostatistics The Netherlands