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M\"obius-Invariant Goodness-of-Fit Tests for the Spherical Cauchy Model

T0 review · 2 major / 8 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Goodness-of-fit tests for spherical Cauchy laws are exactly distribution-free after a Möbius-mean transform to uniformity.

desk verdict Solid, complete GOF paper that turns Möbius equivariance into exact composite distribution-freeness and cleanly links estimation to the H1 tangent space. read the letter →

arxiv 2607.24376 v1 pith:HARZYYHB submitted 2026-07-27 math.ST stat.TH

Diego Bolón , Davy Paindaveine This is my paper · ORCID
classification math.STstat.TH MSC 62H1162G10
keywords directionalstatisticsexactdistribution-freenessgoodness-of-fittestinggroupinvarianceMöbiustransformationssphericalCauchyuniformitytests
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Directional data on a sphere often need a parametric model; the spherical Cauchy family is one natural choice, closed under Möbius maps and recovering uniformity after a simple transform. This paper builds goodness-of-fit tests that first estimate the unknown parameter by the sample Möbius mean, push the data toward uniformity by the inverse map, and then apply a rotation-invariant projection uniformity statistic. Because the estimator is Möbius-equivariant and the uniformity statistic is rotation-invariant, the composite null distribution does not depend on the true parameter, so critical values can be simulated once from uniform samples. The same geometry shows that parameter estimation simply removes the degree-one spherical-harmonic piece of the uniformity statistic—the tangent space of the model at uniformity—yielding an explicit asymptotic null law, consistency against fixed alternatives, and local power only against directions orthogonal to that tangent space.

What carries the argument

The Möbius mean: the unique point φ in the open ball that maximises the expected log spherical-Cauchy density (equivalently, that centres the Möbius-transformed sample at the origin). Its sample version is equivariant, coincides with the MLE, and is asymptotically linear, so the transformed sample yields a pivotal statistic.

What would settle it

Simulate large samples from a known non-spherical-Cauchy density with an L2 density (for example a Watson law with moderate concentration) and check whether the Cramér–von Mises version of Dn rejects at rate approaching one while remaining level-exact under spherical Cauchy draws of the same size.

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Extended reading notes

Core claim

Under mild non-atomicity conditions, the goodness-of-fit statistic obtained by transforming observations with the sample Möbius mean and feeding them to a rotation-invariant projection uniformity test is exactly distribution-free under the composite spherical Cauchy null, and its asymptotic null distribution is that of the original uniformity statistic with the degree-one spherical-harmonic component removed.

Load-bearing premise

No single point on the sphere may carry probability one-half or more; otherwise the Möbius mean need not exist or be unique, and the whole construction fails.

Editorial extensions

If this is right

  • Critical values for any sample size and dimension can be tabulated once from uniform Monte Carlo, without parametric bootstrap.
  • Local alternatives that move only along degree-one harmonics are asymptotically undetectable by any Möbius-invariant test.
  • The same transform-plus-uniformity template extends immediately to other rotation-invariant uniformity statistics.
  • Omnibus consistency holds for every absolutely continuous weighting measure with positive mass when alternatives have L2 densities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Möbius-mean reduction could supply exact or asymptotically pivotal residual diagnostics inside circular–circular regression and circular time-series models that already use Möbius building blocks.
  • Power loss with dimension observed in the simulations suggests a high-d regime in which the retained spherical-harmonic eigenvalues of the projection kernel decay too fast to retain non-trivial local power.
  • Because estimation kills precisely the score space of the model, the construction is a concrete instance of how group equivariance automatically orthogonalises a test to a parametric nuisance tangent space.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 8 minor

Summary. The paper constructs goodness-of-fit tests for the spherical Cauchy model on S^d by exploiting its structure as a transformation model under the Möbius group: observations are mapped toward uniformity via the estimated parameter and a rotation-invariant projection-based uniformity statistic (García-Portugués et al., 2023) is applied to the transformed sample. The main results are: (i) a new location functional, the Möbius mean, whose sample version coincides with the spherical Cauchy MLE, with existence/uniqueness under a max-atom-below-1/2 condition (Propositions 4.1–4.2), equivariance on tie-free samples (Theorem 4.1), and a Bahadur representation (Proposition 4.3); (ii) exact distribution-freeness of the test statistic D_n under the composite null (Theorem 3.1, Corollary 3.1), so critical values can be simulated from uniform samples; (iii) the asymptotic null law, obtained via a modernized de Wet–Randles V-statistic theorem (Theorem C.1) that replaces the problematic Lipschitz condition with an empirical-process condition verified by VC arguments — equal to that of the underlying uniformity statistic with the degree-one spherical-harmonic component removed (Corollary 5.1); (iv) consistency against fixed alternatives and omnibusness for absolutely continuous W against L² alternatives (Theorem 6.1, Corollary 6.1); (v) local power under contiguous alternatives determined by harmonic components orthogonal to H_1, with H_1 directions shown to be contiguous to spherical Cauchy参数

Significance. If the results hold — and they appear to — this is a useful and well-crafted contribution. To my knowledge it is the first goodness-of-fit procedure tailored to the spherical Cauchy family, and the first to exploit its Möbius transformation structure. The standout strength is exact distribution-freeness under the composite null, which removes the parameter-dependent bootstrap required by general-purpose competitors and yields both computational savings (a factor of ~400 in the authors' experiments) and exact validity at any n, including the n=33 real-data example. The structural result that parameter estimation removes precisely the degree-one spherical-harmonic component — identified with the model tangent space, with the constant in (5.4) correctly normalized — is clean and illuminating, and the local-power characterization via contiguity with SC(v/(2d√n)) explains elegantly why Fisher–von Mises local alternatives are undetectable at the n^{-1/2} scale. The paper ships complete proofs, verifiable Monte Carlo corroboration, and an honest statement of its scope restrictions (atom condition, L² alternatives for omnibusness).

major comments (2)
  1. [§3, Theorem 3.1 / §4, Theorem 4.1] Statement alignment for the headline exactness result: Theorem 3.1 assumes the estimator is equivariant for ALL samples x_1,...,x_n, but Theorem 4.1 establishes equivariance of the sample Möbius mean only on X_n (pairwise distinct points, n≥3). Corollary 3.1 remains valid for D_n built from ϕ̂_n because SC(ϕ) is non-atomic, but the chain of statements should be patched: define ϕ̂_n measurably and arbitrarily on the tie set, and state that Theorem 3.1 applies after a null-set modification. Since exact distribution-freeness is the paper's central selling point, the hypotheses should match precisely.
  2. [§4, Eq. (4.3)–(4.4); §7] No algorithm is given for computing ϕ̂_n (the unique solution of the gradient condition (4.4)). All practical claims — the critical values in Table 1, the power curves in Figures 3–4, the 400× speed claim in §7.3, and the real-data example — depend on solving this reliably and cheaply. Please describe the numerical scheme (e.g., Newton–Raphson on (4.4), initialization, convergence behavior given the uniqueness from Proposition 4.2), its cost per evaluation, and ideally provide reference code. This is a reproducibility gap rather than a theoretical one.
minor comments (8)
  1. [Figures 1 and 2] The axis labels contain raw R code artifacts: 'T[1, dd, ]' and 'Tgof[1, dd, ]' in Figure 1 and 'x[x >= 0 & x <= 0.4]' in Figure 2. These should be replaced with informative labels before publication.
  2. [§3, opening paragraph] First paragraph: 'We aim to test the null hypothesis that F is a distribution from the spherical Cauchy family' — F is never defined; the measure was introduced as P.
  3. [Table 1] Report Monte Carlo standard errors for the quantile estimates (with M=100,000 these are small but should be stated for reproducibility).
  4. [§7.3, computation-time comparison] The 'more than 400 times faster' claim is implementation- and hardware-dependent; please give machine details and per-replicate costs, or soften to a qualitative statement. Also note the EHM competitor was run with a single tuning grid (γ∈{0.5,1,2}, m=200, B=500); a brief caveat that EHM's power may improve with tuned γ would make the comparison more balanced.
  5. [§4, Proposition 4.3] Proposition 4.3: the placement of 'L→' inside display (4.5) is typographically confusing (it appears between the Bahadur expansion and the normal limit); split into two displays.
  6. [Appendix D, proof of Proposition 6.1] The Taylor-expansion fact for max_i |τ_n(X_i)| is sourced to Lemma S.1.5 of an unpublished supplement (Bolón et al., 2026); a standard textbook reference or a one-line statement would be preferable.
  7. [§4, Proposition 4.1] Is the atom condition P[X=x]<1/2 sharp? A short remark (e.g., an example where uniqueness fails with a half-mass atom) would help readers understand the scope of Proposition 4.1.
  8. [§3; §6, Theorem 6.1] State explicitly in §3 that the construction requires n≥3 (needed for ϕ̂_n to be well-defined via Proposition 4.2), and clarify the notation ẽ_P in Theorem 6.1 at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: exact null freeness and H1-removal asymptotics follow from equivariance and standard V-statistic theory, not from fitting the target into the null.

full rationale

The central claims are derived rather than assumed. Exact distribution-freeness (Corollary 3.1) follows from Möbius equivariance of the sample Möbius mean (Theorem 4.1) plus rotational invariance of the projection uniformity statistic (Theorem 3.1); under the spherical Cauchy transformation model this makes Dn pivotal, which is a group-invariance argument, not a definitional identity with the quantity being tested. The population/sample Möbius means are defined as maximizers of the log-density functional and shown to solve the zero-mean estimating equation; uniqueness under a mild atom condition (Props. 4.1–4.2) and the Bahadur representation (Prop. 4.3) are proved, not imported as self-justifying uniqueness theorems. The asymptotic null law (Thm. 5.1, Cor. 5.1) is obtained by applying V-statistic theory with estimated parameters and verifying the empirical-process conditions; the explicit kernel correction removes precisely the degree-one spherical-harmonic component because that is the tangent space of the model at uniformity—an algebraic consequence of the score, not a fitted prediction. Consistency and local power (Sec. 6) are characterized against fixed and contiguous alternatives orthogonal to H1, with Monte Carlo checks on external Fisher–von Mises and Watson alternatives and an external competitor. Citations (Kato–McCullagh, García-Portugués et al., classical empirical-process results) supply independent building blocks. No step reduces the claimed output to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The work sits on standard empirical-process and directional-statistics foundations plus the Kato–McCullagh spherical Cauchy / Möbius transformation model. No physical constants or data-fitted tuning constants enter the central theorems; the only practitioner choices are the weight measure W and Monte Carlo sample size for critical values. Invented terminology (Möbius mean) names a functional already tied to the SC MLE, not a new ontological entity.

free parameters (3)
  • Weight measure W on [0,1] = CvM W(x)=x used in simulations and data example
    Chooses the projection GOF functional (e.g. Cramér–von Mises W(x)=x). Affects eigenvalues and finite-sample power but is a user design choice, not fitted to the null theorems.
  • Monte Carlo size M for critical values = M=100000 (Table 1); M=10000 in data example
    Approximation accuracy of cn,α; Table 1 uses M=100,000. Not a model parameter.
  • EHM competitor tunings γ, m, B = γ∈{0.5,1,2}, m=200, B=500
    Only for comparison power study; not used in the proposed tests.
assumptions (5)
  • domain assumption Spherical Cauchy family is a Möbius transformation model with uniform as central member (Kato & McCullagh 2020).
    Used from §2 onward as the structural reason the inverse map sends SC(φ) to uniformity.
  • standard math Standard V-statistic / empirical-process limit theorems (Serfling; van der Vaart–Wellner; de Wet–Randles-type estimated-parameter V-statistics).
    Theorem C.1 and proofs of Thm 5.1, 6.1, 6.2 rely on these classical tools.
  • domain assumption Non-atomic P with P({x})<1/2 for all x (population); pairwise distinct sample points, n≥3 (sample).
    Propositions 4.1–4.2; needed for unique Möbius mean and well-defined equivariant estimator.
  • domain assumption Projection uniformity kernel ψ and its spherical-harmonic eigenvalues from García-Portugués et al. (2023).
    Building block of Dn; Corollary 5.1 recycles ηψ_ℓ for ℓ≥2.
  • ad hoc to paper W nonnegative bounded Borel measure on [0,1]; for omnibusness, absolutely continuous with W([0,1])>0 and alternatives in L2(νd).
    Stated in Thm 5.1 / Cor 6.1 as sufficient conditions for the asymptotic and consistency claims as written.
invented entities (1)
  • Möbius mean (population functional) independent evidence
    purpose: Canonical parameter/location that characterizes SC laws via E[MI,−φ(X)]=0 and coincides with the SC MLE target.
    Named and developed in §4; it is the argmax of expected log SC density, not a new physical object. Independent handle: equals φ under SC(φ) and is uniquely defined for non-atomic laws with no heavy atoms.

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Pith. "Pith review of M\"obius-Invariant Goodness-of-Fit Tests for the Spherical Cauchy Model." pith.science (2026). https://pith.science/paper/HARZYYHB

@misc{pith2026260724376,
  author       = {Pith},
  title        = {Pith review of: M\"obius-Invariant Goodness-of-Fit Tests for the Spherical Cauchy Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HARZYYHB}},
  note         = {Machine review of arXiv:2607.24376}
}
read the original abstract

We introduce a class of goodness-of-fit tests for the spherical Cauchy model on the unit hypersphere. The proposed procedures exploit the invariance of the spherical Cauchy family under M\"obius transformations: after estimating the parameter by the sample M\"obius mean, the observations are transformed to approximate spherical uniformity, and a projection-based uniformity statistic is applied to the resulting sample. We show that, under mild conditions, the resulting tests are exactly distribution-free under the null hypothesis, so that exact critical values can be arbitrarily well approximated by Monte Carlo simulation. We study the M\"obius mean as a population functional, establish its existence and uniqueness under mild conditions, and prove equivariance and asymptotic linearity of its empirical counterpart, which coincides with the spherical Cauchy maximum likelihood estimator. We derive the asymptotic null distribution of the test statistic and show that it coincides with that of the underlying uniformity statistic after removing the degree-one spherical-harmonic component, which corresponds to the tangent space of the spherical Cauchy model. We establish consistency against fixed alternatives and characterize local powers through the spherical-harmonic decomposition of contiguous alternatives. Monte Carlo experiments demonstrate the finite-sample accuracy of the asymptotic approximations and the empirical power of the proposed tests. A real data example is treated.

Figures

Figures reproduced from arXiv: 2607.24376 by the authors.

Figure 1
Figure 1. Monte Carlo null distributions of P W n (1st row) and Dn (2nd row) for d = 1, 2, 3. The histograms are based on M = 10,000 independent samples of size n = 200 from the uniform distribution on S d . The solid curves are kernel density estimates based on 10,000 independent draws from the corresponding asymptotic null distributions. 6 Asymptotic non-null behaviour of the proposed tests We start with consistency against… view at source ↗
Figure 2
Figure 2. Monte Carlo local-alternative distributions of the Cram´er–von Mises version [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Rejection frequencies of our test at level [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Rejection frequencies of several goodness-of-fit tests at level [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Paleomagnetic pole positions reported by [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Boxplots of the p-values of the EHM tests applied to the paleomagnetic pole positions reported by Schmidt (1976). For each value of γ, the p-values were computed using A = 1,000 independently generated artificial samples. The significance level α = 5% is shown for refe…

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