The Flexible Gumbel Distribution: A New Model for Inference about the Mode

Abstract

A new unimodal distribution family indexed via the mode and three other parameters is derived from a mixture of a Gumbel distribution for the maximum and a Gumbel distribution for the minimum. Properties of the proposed distribution are explored, including model identifiability and flexibility in capturing heavy-tailed data that exhibit different directions of skewness over a wide range. Both frequentist and Bayesian methods are developed to infer parameters in the new distribution. Simulation studies are conducted to demonstrate satisfactory performance of both methods. By fitting the proposed model to simulated data and data from an application in hydrology, it is shown that the proposed flexible distribution is especially suitable for data containing extreme values in either direction, with the mode being a location parameter of interest. Using the proposed unimodal distribution, one can easily formulate a regression model concerning the mode of a response given covariates. We apply this model to data from an application in criminology to reveal interesting data features that are obscured by outliers.

Keywords

• extreme values
• mixture distribution
• modal regression
• unimodal distribution