article · Journal of the American Statistical Association
Traditional methods for quantile regression struggle when estimating extreme quantiles where few or no observed data points exist. While extreme value theory allows for extrapolation, existing techniques such as linear or kernel regression often fail when dealing with complex relationships or high-dimensional predictor spaces. To address these issues, the extremal random forest integrates extreme value extrapolation with the flexible structure of random forests. The method estimates conditional generalized Pareto distribution parameters by maximising a local likelihood weighted through a quantile random forest, while incorporating a penalty to regularise the shape parameter across the predictor space. Theoretical analysis confirms the consistency of parameter estimates under standard domain of attraction conditions. When evaluated through simulations and an empirical demonstration predicting extreme tails in United States wage data, the approach outperforms both standard quantile regression and existing extreme value models.
Standard statistical models often fail to predict rare, high-impact events when very little historical data exists at the extremes. By providing a reliable way to model extreme quantiles across complex and multi-dimensional data, this method allows analysts to assess tails of distributions accurately, improving predictive precision in domains where extreme values are critical.
This methodology operates as an advanced statistical modelling tool, tested in simulation and demonstrated on United States wage datasets. It could enable organisations analysing heavy-tailed data to evaluate extreme scenarios with greater accuracy. Potential users include quantitative analysts and risk modellers. As an algorithmic advance demonstrated in an academic study, the technique sits at an early, applied research stage requiring software integration or internal development before broader operational use.
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Classical methods for quantile regression fail in cases where the quantile of interest is extreme and only few or no training data points exceed it. Asymptotic results from extreme value theory can be used to extrapolate beyond the range of the data, and several approaches exist that use linear regression, kernel methods or generalized additive models. Most of these methods break down if the predictor space has more than a few dimensions or if the regression function of extreme quantiles is complex. We propose a method for extreme quantile regression that combines the flexibility of random forests with the theory of extrapolation. Our extremal random forest (ERF) estimates the parameters of a generalized Pareto distribution, conditional on the predictor vector, by maximizing a local likelihood with weights extracted from a quantile random forest. We penalize the shape parameter in this likelihood to regularize its variability in the predictor space. Under general domain of attraction conditions, we show consistency of the estimated parameters in both the unpenalized and penalized case. Simulation studies show that our ERF outperforms both classical quantile regression methods and existing regression approaches from extreme value theory. We apply our methodology to extreme quantile prediction for U.S. wage data. Supplementary materials for this article are available online.
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DOI: 10.1080/01621459.2023.2300522
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