14:15 - 14:40 Valid highest density intervals

Abstract

Our final project is motivated by the problem of obtaining highest density interval, or shortest prediction interval from data for a unimodal distribution. Frey, J. (2013). Data-driven nonparametric prediction intervals. Journal of Statistical Planning and Inference, 143, 1039-1048. point out that naively searching for the shortest prediction interval causes under coverage problem, and propose to apply a correction to the nominal error rate. In short, if the nominal error rate is \(\alpha\) and we have a data set of size \(n\), Frey (2013)propose to search for the shortest prediction interval with error rate \(\alpha - 1.12\sqrt{\alpha/n}\). The prediction intervals produced by this method have approximate worst case guarantee, so we call this method “conservative”.

Our approach to solving this problem is to use cross validation to determine the corrected nominal error rate \(\alpha'\), which may give tighter prediction intervals than Frey (2013). However, this method has not been shown any theoretical guarantee.

We compare the performance of the three methods of searching for the shortest prediction interval: searching without any correction, the conservative method proposed by Frey(2013), and correction using cross validation. A method that chooses a prediction interval at random is also implemented to illustrate the cause of under coverage problem in naive searching.

Our future work will be to gain a better understanding of the under coverage problem, and find methods that have (approximate) theoretical guarantee.

More information can be found on the corresponding vhdi