Forum des Jeunes Mathématicien.ne.s 2021
Date:
Talk in French for the Forum des Jeunes Mathématicien.ne.s 2021, organized in Besançon (France).
Title. Statistical Learning Framework for Distributional Regression using Continuous Ranked Probability Score
Abstract. The distributional regression fulfills a fundamental need of statistical analysis : being able to make forecasts and quantify their uncertainty. This approach overcomes the limits of classical regression which estimates only the conditional mean by estimation the whole distribution law. This methodology, called probabilistic forecast, is widely used in numerous fields such as meteorology and energy production, but its theoretical aspects have not been studied. By analogy with the classical theory of statistical learning, we define a framework where the predictor is a law of probability, called prediction law, and where the loss function is given by a strictly proper scoring rule in the sense of Gneiting and Raftery (2007). Bayes predictor is then the conditional law. In the case of the Continuous Ranked Probability Score, we study then the minimax rate of convergence and show that, in particular in dimension higher or equal to 2, the k nearest neighbor algorithm for the distributional regression reaches the optimal rate of convergence.
Long summary in French : here Associated article : Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk, Pic et al. (2022) link