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 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 √γ 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 γβ−1/2ℓ(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.
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BibTeX
@misc{lee2026limit,
title = {{Limit Theory for Online Expectile Estimation under Short and Long Range Dependence}},
author = {Younghun Lee and Jake Zummo and Wei Biao Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16030},
eprint = {2610.16030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16030},
}
RIS
TY - UNPB
TI - Limit Theory for Online Expectile Estimation under Short and Long Range Dependence
AU - Younghun Lee
AU - Jake Zummo
AU - Wei Biao Wu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16030
UR - https://arxiv.org/abs/2610.16030
AB - 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 √γ 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−1/2ℓ(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.
ER -