Restoring Convergence in Heavy-Tailed Risk Models: A Weighted Kolmogorov Approach for Robust Backtesting

TL;DR

This paper introduces a weighted Kolmogorov metric that improves convergence rates in risk model validation for heavy-tailed financial data, enabling more reliable backtesting.

Contribution

It proposes a novel weighted Kolmogorov approach with a specific exhaustion function to restore optimal convergence rates for heavy-tailed distributions.

Findings

Restores Gaussian convergence rate of O(n^{-1/2}) for heavy-tailed data.

Effectively downweights tail noise to improve model validation.

Applicable to Pareto and Student-t distributions in financial markets.

Abstract

Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central tendency accuracy. In this paper, we introduce a Weighted Kolmogorov Metric tailored for financial time series with sub-cubic moments ($\mathbb{E}|X|^{2+\delta}<\infty$). By incorporating an exhaustion function $h(x)$ that mechanically downweights extreme tail noise, we prove that we can restore the optimal Gaussian convergence rate of $O(n^{-1/2})$ even for Pareto and Student-t distributions common in Crypto and FX markets. We provide a complete proof using a core/tail truncation scheme and establish the optimal tuning of the weight parameter $q$.