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arxiv: 2606.26270 · v1 · pith:VNHYNVO6new · submitted 2026-06-24 · 🧮 math.ST · stat.ME· stat.TH

On the optimal prediction of extreme events

Benjamin Bobbia , Stilian Stoev This is my paper

Pith reviewed 2026-06-26 00:56 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords multivariate regular variationangular measureextreme value theorypeaks-over-thresholdoptimal predictiontail dependence coefficientconditional quantilehomogeneous predictor
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The pith

The asymptotically optimal positive homogeneous predictor of extreme Y given X is the non-extreme conditional quantile of a tilted distribution derived from the angular measure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the best positive homogeneous predictors for extreme values of a response Y in terms of covariates X when observations are scarce in the tail. It works under the assumption that the joint distribution is multivariate regularly varying, so the tail dependence is summarized by an angular measure on the unit sphere. The asymptotic performance of any such predictor reduces to the tail dependence coefficient, an integral functional of that angular measure, which turns the search for the optimum into a variational problem. Solving the variational problem produces an explicit form for the optimal predictor as a conditional quantile taken from a tilted version of the angular distribution. This form immediately yields peaks-over-threshold estimators that the paper proves are consistent across wide classes of angular measures.

Core claim

Under multivariate regular variation of (Y,X), the asymptotic prediction precision of any positive homogeneous predictor h(X) equals the tail dependence coefficient λ(Y,h(X)), expressed as an integral over the angular measure of the pair. The h that maximizes this coefficient is the non-extreme conditional quantile of the distribution obtained by tilting the angular measure; the associated peaks-over-threshold estimators are universally consistent over large classes of angular measures.

What carries the argument

The angular measure of the multivariate regular variation of (Y,X), which encodes all tail dependence; the optimal predictor is recovered as the conditional quantile of the distribution tilted by this measure.

If this is right

  • Any positive homogeneous predictor's performance is exactly the tail dependence coefficient expressed as an integral functional of the angular measure.
  • The variational problem for the optimal predictor admits an explicit solution via the tilted conditional quantile for arbitrary angular measures.
  • Peaks-over-threshold estimators constructed from the tilted quantile are consistent without further parametric restrictions on the angular measure.
  • The resulting procedure matches or exceeds oracle performance on the solar flare prediction task.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tilting construction could be used to derive optimal predictors when several response variables are extreme simultaneously.
  • Replacing the homogeneous restriction with a more flexible class of functions might improve finite-sample performance while retaining the regular-variation tail analysis.
  • The method supplies a natural benchmark against which existing extreme regression techniques can be compared on data sets with verifiable angular measures.

Load-bearing premise

The pair (Y,X) must be multivariate regularly varying so that its tail dependence structure is fully captured by a finite angular measure on the unit sphere.

What would settle it

A simulation with a known angular measure in which the tail dependence coefficient achieved by the estimated predictor is strictly less than the theoretical maximum computed directly from the angular measure.

Figures

Figures reproduced from arXiv: 2606.26270 by Benjamin Bobbia, Stilian Stoev.

Figure 1
Figure 1. Figure 1: Pairs of boxplots of the empirical tail dependence coefficient [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pairs of boxplots of the empirical tail dependence coefficient [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: Optimal extremal precision λ (opt) G for predicting one component of the Pareto￾Dirichlet model via a homogeneous function of the rest (C.9). Right panel: The empirical tail￾dependence coefficients λˆ p for: (i) the asymptotically optimal Oracle predictor h (opt)(X) ∝ ∥X∥ (black solid line) (ii) a non-parametric estimator of the optimal predictor Yb := bh(X) based on a training and testing samp… view at source ↗
Figure 4
Figure 4. Figure 4: Time series of maximum daily X-ray flux (in Watts per square meter). The data is based [PITH_FULL_IMAGE:figures/full_fig_p068_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evaluation metrics for the prediction of M-class (left panel) and X-class (right panel) solar [PITH_FULL_IMAGE:figures/full_fig_p070_5.png] view at source ↗
read the original abstract

The prediction of the extremely large values of a response variable $Y$ in terms of a vector of covariates $X=(X_i)_{i=1}^d$ is a fundamental problem arising in many scientific and engineering domains. The scarcity of data in the extremes makes the optimal solution of this problem of particular importance. The optimal predictors of such events can be explicitly characterized in just a few cases and it is of fundamental practical and theoretical interest to develop optimal estimators over large classes of models and predictors. In this work, the focus is on the case where $(Y,X)$ have a multivariate regularly varying distribution and one seeks an optimal predictor expressed as a positive homogeneous function $h(X)$ of the covariates. The asymptotic prediction precision in this setting coincides with the tail-dependence coefficient $\lambda(Y,h(X))$ and it can be expressed as an integral functional of the associated angular measure of $(Y,X)$. Thus, finding asymptotically optimal homogeneous predictors amounts to solving a variational problem. We obtain a general solution to this problem, which is expressed in terms of a non-extreme conditional quantile of a tilted distribution derived from the angular measure. This leads to a general inference methodology for the optimal predictors in the peaks-over-threshold framework form extreme value theory. We establish the universal consistency for these estimators over large classes of angular measures. A general-purpose implementation of the resulting inference procedure is shown to work remarkably well against optimal oracle estimators, as well as in the challenging problem of extreme solar flare prediction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper considers prediction of extreme values of a response Y given covariates X under the multivariate regular variation assumption on the joint distribution. It shows that the asymptotically optimal positive homogeneous predictor h(X) is obtained by solving a variational problem whose objective is the tail-dependence coefficient expressed as an integral functional of the angular measure; the solution is characterized as a non-extreme conditional quantile of a tilted distribution derived from that measure. Peaks-over-threshold estimators of this predictor are proved to be universally consistent over large classes of angular measures. A general-purpose implementation is presented and shown to perform well relative to oracle estimators, with an application to extreme solar-flare prediction.

Significance. If the central claims hold, the work supplies an explicit, theoretically justified construction for optimal homogeneous extreme predictors together with consistent estimators that apply to broad classes of tail-dependence structures. The reduction of the variational problem to a conditional quantile of a tilted measure, the universal-consistency result, and the reproducible numerical comparison against oracles are concrete strengths that advance both the theory and practice of extreme-value prediction.

minor comments (2)
  1. The abstract states that the estimators are 'universally consistent over large classes of angular measures,' but the precise definition of these classes (e.g., continuity or support conditions on the angular measure) is not visible in the provided summary; a short clarifying sentence in the introduction would help readers assess the scope.
  2. The numerical section reports that the procedure 'works remarkably well against optimal oracle estimators,' yet no quantitative table or figure reference is given in the abstract; adding a brief statement of the reported error metrics would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on optimal homogeneous extreme predictors and for recommending minor revision. No specific major comments appear in the report, so there are no points requiring point-by-point rebuttal. We will incorporate any minor editorial or presentational suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation begins from the standard multivariate regular variation assumption on (Y,X), expresses the asymptotic prediction error as the tail dependence coefficient λ(Y,h(X)) which is an integral functional of the angular measure, poses the search for optimal positive homogeneous h as a well-posed variational problem, and solves it by exhibiting an explicit non-extreme conditional quantile of a tilted version of that measure. The subsequent POT estimator consistency is established directly over large classes of angular measures without any reduction of the claimed optimum to a fitted parameter, without self-referential definitions, and without load-bearing self-citations that would make the central result equivalent to its inputs by construction. The argument is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the regular-variation assumption that supplies the angular measure; no free parameters, additional axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The random vector (Y,X) is multivariate regularly varying.
    This is the explicit modeling assumption stated in the abstract that enables the angular-measure representation and the variational problem.

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Reference graph

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    Conversely, ifZ∈RV α(Rk,{a n}, c Z, τ, σ)according to Definition 3.1, thenZ∈RV α({an}, µ) according to Definition B.1, where µ◦T −1 τ ((r,∞)×B)) =c Zr−ασ(B),for allr >0, B∈ B(S τ),(B.4) and whereT τ :R k \{τ= 0} →(0,∞)×S τ is the generalized polar coordinated homeomorphism defined asT τ(x) = (τ(x), x/τ(x)). The proof of this result can be derived from man...

  65. [65]

    quantiles

    thatZ:= (Y, X)∈RV 1(R+ ×R d,{a (Z) n :=n}, c Z, τ, σ), where withc i := (bi, ai), we have cZ = pX i=1 τ(c i) = pX i=1 bi +∥a i∥. In this case, the angular measureσisdiscreteand takes the form: σ(dθ) = 1 cZ pX i=1 τ(c i)δci/τ(c i)(dθ).(C.2) We begin with a counterpart to Proposition 3.2 and for the sake of completeness, we provide a proof. Proposition C.1....

    2010