Limit Theory for Online Expectile Estimation under Short and Long Range Dependence

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 $\gamma$ 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 $\sqrt{\gamma}$ 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 $\gamma^{\beta-1/2}\ell(1/\gamma)$, 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.

Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)

arXiv:2610.16030 [stat.ML]

(or arXiv:2610.16030v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2610.16030 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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