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arxiv: 2311.05056 · v2 · submitted 2023-11-08 · 📊 stat.ME

High-dimensional Newey-Powell Test Via Approximate Message Passing

Jing Zhou , Hui Zou This is my paper

Pith reviewed 2026-05-24 05:14 UTC · model grok-4.3

classification 📊 stat.ME
keywords high-dimensional statisticsheteroscedasticity testexpectile regressionapproximate message passingNewey-Powell testproportional asymptoticsasymptotic power
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The pith

Expectile regression analyzed via approximate message passing yields a high-dimensional extension of the Newey-Powell heteroscedasticity test.

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

The paper extends the Newey-Powell test for heteroscedasticity to the high-dimensional regime where the number of predictors is proportional to the sample size. It replaces ordinary regression with expectile regression and tracks the estimator's behavior through the approximate message passing algorithm to obtain the limiting distribution of the resulting test statistic. This derivation supplies both the null distribution under homoscedasticity and the power against alternatives. Simulations and two real-data examples confirm that the procedure detects or rules out heteroscedasticity in practice. The work therefore supplies a concrete method for variance testing when p is not small relative to n.

Core claim

We propose a high-dimensional extension of the heteroscedasticity test proposed in Newey and Powell (1987). Our test is based on expectile regression in the proportional asymptotic regime where n/p → δ ∈ (0,1]. The asymptotic analysis of the test statistic uses the Approximate Message Passing (AMP) algorithm, from which we obtain the limiting distribution of the test and establish its asymptotic power.

What carries the argument

The expectile regression estimator whose limiting behavior is obtained from the approximate message passing algorithm in the proportional regime n/p → δ.

Load-bearing premise

The derivation assumes the proportional asymptotic regime n/p → δ ∈ (0,1] together with conditions that allow the AMP algorithm to characterize the expectile regression estimator and test statistic.

What would settle it

Generate many homoscedastic data sets with fixed δ = n/p = 0.5, apply the proposed test, and check whether the p-values are uniformly distributed on [0,1]; systematic departure from uniformity would show the claimed limiting distribution does not hold.

Figures

Figures reproduced from arXiv: 2311.05056 by Hui Zou, Jing Zhou.

Figure 1
Figure 1. Figure 1: Normal QQ-plot of test statistics Tj ’s based on βe at different expectile levels for three simulation settings. The test statistics Tj ’s should approximately follow a standard normal distribution N(0, 1) under homoscedasticity of ε. The expectile levels fix τ2 = 0.8 and vary τ1 = 0.1, 0.2, 0.6; The left, middle, and right columns are plots for τ1 = 0.1, 0.2, 0.6, respectively. The high-sparsity and mediu… view at source ↗
read the original abstract

We propose a high-dimensional extension of the heteroscedasticity test proposed in Newey and Powell (1987). Our test is based on expectile regression in the proportional asymptotic regime where n/p \to \delta \in (0,1]. The asymptotic analysis of the test statistic uses the Approximate Message Passing (AMP) algorithm, from which we obtain the limiting distribution of the test and establish its asymptotic power. The numerical performance of the test is validated through an extensive simulation study. As real-data applications, we present the analysis based on ``international economic growth" data (Belloni et al., 2011), which is found to be homoscedastic, and ``supermarket" data (Lan et al., 2016), which is found to be heteroscedastic.

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 / 3 minor

Summary. The manuscript proposes a high-dimensional extension of the Newey-Powell (1987) heteroscedasticity test. The extension is based on expectile regression in the proportional asymptotic regime n/p → δ ∈ (0,1]. The limiting distribution of the resulting test statistic and its asymptotic power are derived using the Approximate Message Passing (AMP) algorithm. The claims are supported by an extensive simulation study together with applications to the international economic growth data (found homoscedastic) and supermarket data (found heteroscedastic).

Significance. If the AMP-based derivation holds under the stated conditions on the loss and design matrix, the work supplies a theoretically justified procedure for heteroscedasticity testing when the number of covariates is of the same order as the sample size. This addresses a practical need in modern high-dimensional regression. Credit is given for the explicit use of AMP state evolution to obtain the limiting distribution and for the inclusion of real-data illustrations that demonstrate the test's behavior on concrete examples.

minor comments (3)
  1. [Introduction] The abstract states that limiting distribution and power are obtained via AMP, but the introduction or Section 2 would benefit from a brief, self-contained statement of the precise AMP state-evolution equations used for the expectile loss (rather than referring only to the general AMP literature).
  2. [Simulation study] In the simulation section, the reported dimensions (n, p) and the range of δ values should be listed explicitly in a table or in the text so that readers can directly compare the finite-sample behavior to the asymptotic regime δ ∈ (0,1].
  3. Notation for the expectile level τ and the associated loss function is introduced but could be restated once more when the test statistic is defined, to avoid any ambiguity for readers unfamiliar with expectile regression.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the work's significance in providing a theoretically justified high-dimensional heteroscedasticity test, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies the AMP algorithm as an external tool to characterize the expectile regression estimator and obtain the limiting distribution of the test statistic under the proportional regime n/p → δ. The abstract and description impose conditions on the loss and design to validate the AMP state evolution, with no equations showing a prediction reducing to a fitted input by construction, no self-definitional steps, and no load-bearing self-citations. The central claim is self-contained via standard AMP analysis independent of the Newey-Powell test extension itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions for linear models and AMP convergence that are not independently verified in the abstract.

axioms (2)
  • domain assumption Observations follow a linear model whose errors satisfy the moment conditions required for expectile regression consistency.
    Invoked to justify the test statistic and its limiting behavior under the null.
  • domain assumption The AMP algorithm converges to the correct state evolution fixed point for the expectile regression problem in the proportional regime.
    This is the key technical step that delivers the limiting distribution of the test.

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