Expectile estimation limits under different data dependencies
Limit Theory for Online Expectile Estimation under Short and Long Range Dependence
Machine Learning
Summary
Estimating expectiles helps measure financial risks by focusing on data extremes, but it is hard to do this automatically when data points depend on each other. The authors analyze a simple one-pass algorithm that updates estimates gradually and show how it behaves with different types of data dependence. They find that for short-term dependent data, the estimates behave normally, but for long-memory data, the fluctuations change and follow a different pattern. Their work helps understand when and how such online algorithms give reliable expectile estimates.
What this means in practice
- •For financial risk analysts: Provide accurate and adaptive estimates of financial expectiles using online algorithms in environments with dependent market data.
- •For time series modelers: Analyze and predict tail behavior in long-memory dependent data streams via tailored expectile estimation methods.
Authors
Younghun Lee, Jake Zummo, Wei Biao Wu
Abstract
Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate $γ$ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the $\sqrtγ$ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to $γ^{β-1/2}\ell(1/γ)$, with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.