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

Younghun Lee; Jake Zummo; Wei Biao Wu

stat.MLOct 12, 2026

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

Authors: Younghun Lee, Jake Zummo, Wei Biao Wu

Organizations: Department of Statistics, University of Chicago

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.

Figures & tables

Figure 1: (a)–(d) Kolmogorov distance to the corresponding Gaussian limit along the expectile SGD trajectory for AR(1), GARCH $(1,1)$ , nonlinear AR, and ARFIMA $(0,0.25,0)$ processes at $\tau=0.9$ . Each curve corresponds to a different learning rate $\gamma$ . The shaded region shows the 95% Monte Carlo reference band for the estimated Kolmogorov distance under the $\mathcal{N}(0,1)$ reference, excluding calibration uncertainty. (e) Joint normalized SGD errors at $\tau=0.5,0.9$ and $\gamma=10^{-4}$ under ARFIMA $(0,0.25,0)$ with 50% and 95% contours of the joint Gaussian limit. (f) Central 90% interval widths of the normalized centered contrast between $\tau=0.5$ and $0.9$ under ARFIMA $(0,0.25,0)$ . Batch uses the last $m=\lceil 1/\gamma\rceil$ observations, with open squares for $m=10^{5},10^{6}$ . Error bars are pointwise 95% bootstrap intervals.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Model	$\tau$	$\theta_{\tau}^{*}$	s.e.	$A$	$\Omega_{g}$	$V$
AR(1)	0.5	0	—	0.50000	1.00000	1.00000
AR(1)	0.9	0.99488	—	0.25556	0.32527	0.63638
GARCH $(1,1)$	0.5	0	—	0.50000	0.25000	0.25000
GARCH $(1,1)$	0.9	0.85814	$4.5\times 10^{-5}$	0.25153	0.17481	0.34750
Nonlinear AR	0.5	1.64726	$2.7\times 10^{-4}$	0.50000	5.13856	5.13856
Nonlinear AR	0.9	3.11902	$4.3\times 10^{-4}$	0.23916	2.93354	6.13301

Appendix

Table 1: Targets and normalization constants. The s.e. column gives the Monte Carlo standard error of the calibrated target, with a dash for analytically determined targets. For short memory,

V=\Omega_{g}/(2A)

. For ARFIMA,

V

is the coefficient of

\gamma^{1-2d}

s(\gamma)^{2}

, and the score long-run variance is infinite.

		$\widehat{\operatorname{MSE}}_{\gamma}/s(\gamma)^{2}$					Slope
Model	$\tau$	$0.1$	$0.05$	$0.01$	$10^{-3}$	$10^{-4}$	Fitted	Theory
AR(1)	0.5	0.951	0.965	0.981	0.993	0.996	0.994	1.0
AR(1)	0.9	0.997	1.012	1.011	1.017	1.022	0.997	1.0
GARCH $(1,1)$	0.5	1.020	1.014	1.013	1.008	1.000	1.003	1.0
GARCH $(1,1)$	0.9	0.899	0.945	0.994	0.998	1.013	0.985	1.0
Nonlinear AR	0.5	0.837	0.907	0.983	0.998	1.003	0.977	1.0

Appendix

Table 2: Empirical MSE divided by

s(\gamma)^{2}

at the five learning rates, and fitted log–log slopes. Each MSE uses

R=30{,}000

replicates at

n=1{,}842{,}069

. The final column gives the exponent of the theoretical variance scale.

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Reinforcement Learning Risk-Sensitive RL Policy Gradient Methods

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We consider the policy evaluation and control in a finite horizon reinforcement learning (RL) setting under an expectile-based objective. First, we derive the mean-squared error (MSE) and concentration bounds for the classic estimator of expectiles based on independent and identically distributed (i.i.d.) samples. To the best of our knowledge, expectiles have not been analyzed in the non-asymptotic regime and the bounds we derive may be of independent interest. Next, we analyze a Monte Carlo type estimator of expectile of the Markov chain underlying a given policy. We derive upper bounds that hold in expectation as well as with high probability for this estimator. Further, we show the order-optimality of our estimator by deriving a lower bound on expectile-based policy evaluation. For the problem of control, we adopt a policy gradient approach and derive a policy gradient theorem for expectiles. Using this result, we propose a gradient estimator with a

O\left(1/m\right)

mean-squared error bounds, where

m

is the number of trajectories. Further, under standard assumptions for policy gradient-type algorithms, we establish smoothness of the expectile-sensitive objective, in turn leading to stationary convergence rate bounds for the overall risk-sensitive policy gradient algorithm that we propose. Finally, we conduct numerical experiments to show the utility of expectiles on popular RL benchmarks.

Abstract

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