Identifiability of asymmetric circular and cylindrical distributions

Takayuki Shiohama; Toshihiro Abe; Yoichi Miyata

REVIEW 2 major objections 4 minor 27 references

Identifiability of asymmetric circular and cylindrical distributions

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Sine-skewed circular and cylindrical distributions are identifiable; the proof runs through trigonometric moment ratios and Diophantine approximation.

desk verdict A genuinely new identifiability method for skew circular/cylindrical models that is probably correct, but the proofs need fixing: the zeros of trigonometric moments are not handled in several ratio limits. read the letter →

arxiv 1908.09114 v5 pith:TKQGIRD3 submitted 2019-08-24 math.ST stat.TH

Yoichi Miyata , Takayuki Shiohama , Toshihiro Abe This is my paper · ORCID

classification math.STstat.TH MSC 62H1162E1062F12

keywords identifiabilitycirculardistributionssine-skewedcylindricaltrigonometricmomentsDiophantineapproximationvonMisesdistributionwrappedCauchy

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

Identifiability — the guarantee that different parameter vectors always produce different distributions — is a necessary condition for maximum likelihood estimators to be consistent, and for skewed families on the circle and cylinder it had not been established. This paper gives a general criterion (Theorem 1) and a constructive method: use trigonometric moments as the transforms, and invoke simultaneous Diophantine approximation to find a subsequence $p_n$ along which the ratio of moments converges to a value different from 1. Under three conditions on the cosine moments of the symmetric base density, the whole sine-skewed family $f(\theta|\gamma)=f_0(\theta-\mu|\psi)(1+\lambda\sin(\theta-\mu))$ is identifiable (Theorem 2). The authors then verify the conditions for the sine-skewed von Mises and wrapped Cauchy distributions, the Möbius-transformed cardioid, and two cylindrical models with Weibull and generalized Pareto-type conditionals. The result matters because it clears the identifiability hurdle for parameter estimation in these asymmetric directional models.

What carries the argument

The machinery is the classical ratio-of-transforms identifiability test carried over to circular statistics, with a crucial modification: the ratios are taken along a subsequence supplied by simultaneous Diophantine approximation. For two candidate parameter vectors with circular locations $\mu_1,\mu_2$, Lemma 8 guarantees infinitely many integers $p$ for which $p\mu_1$ and $p\mu_2$ are both within $\epsilon$ of a multiple of $2\pi$; along such a subsequence the oscillating sine and cosine factors stabilize, so the ratio of the chosen trigonometric moments (or of the squared mean resultant length) converges to a constant $c\neq 1$. The choice of moment varies by case: cosine moments for concentration parameters, sine moments for skewness, the first trigonometric moment for location, and conditional distribution ratios for the cylindrical linear parameters.

What would settle it

Evaluate the cosine moments in equation (13) for two distinct Möbius-transformed cardioid parameter vectors with $\mu=\pi/2$, $\xi=0$, and check whether the ratio $\alpha_p(\gamma_1)/\alpha_p(\gamma_2)$ has a subsequence $p_n\to\infty$ with limit different from 1. If no such subsequence exists, Proposition 5's proof fails; if the ratio is identically 1 for some distinct pair, the family is not identifiable.

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Extended reading notes

Core claim

The paper's central discovery is that identifiability of skewed circular families can be proved by examining ratios of trigonometric moments rather than generating functions, provided one can pass to a carefully chosen subsequence. For the sine-skewed density $f(\theta|\gamma)=f_0(\theta-\mu|\psi)\{1+\lambda\sin(\theta-\mu)\}$, the cosine and sine moments have explicit formulas in terms of the base density's cosine moments (equation (4)). Theorem 2 states: if $\alpha_{0,1}(\psi)\neq 0$, if the ratio $(\alpha_{0,p-1}(\psi)-\alpha_{0,p+1}(\psi))/\alpha_{0,p}(\psi)$ is bounded away from zero uniformly in $p$, and if distinct $\psi$'s are separated by the asymptotic behaviour of $\alpha_{0,p}(\psi_1)/\alpha_{0,p}(\psi_2)$ at polynomial speed, then the family is identifiable. That theorem yields, as Propositions 3–7, identifiability of the sine-skewed wrapped Cauchy, sine-skewed von Mises, Möbius-transformed cardioid, the Abe–Ley cylindrical model, and the sine-skewed generalized Pareto-type cylindrical model. The proof of the general theorem is a short contradiction argument: equal distributions would make every moment ratio equal to 1, contradicting the constructed non-unity limit.

Load-bearing premise

For the Möbius-transformed cardioid, the proof assumes that a ratio of cosine moments can be driven to infinity along integers where the denominator moment does not vanish; the paper does not show such a subsequence exists when both cosine terms vanish, for example at $\mu=\pi/2$, $\xi=0$.

Editorial extensions

If this is right

For the sine-skewed von Mises and wrapped Cauchy models, identifiability is now a theorem, removing the identifiability obstacle to proving consistency of maximum likelihood estimators for $(\mu,\kappa,\lambda)$ or $(\mu,\rho,\lambda)$.
The Abe–Ley cylindrical model and the sine-skewed generalized Pareto-type cylindrical model are identifiable, so location, concentration, skewness, and linear-part parameters can be recovered from the joint distribution.
The Möbius-transformed cardioid family is identifiable in its four-parameter space, aside from the known degeneracy at $\bar\rho=0$ where $\xi$ is lost.
Theorem 1 provides a criterion weaker than the classical ratio test; any family whose moment ratios converge along a nontrivial subsequence inherits identifiability without a tractable moment generating function.
Identifiability is reduced to finitely many moment-ratio checks on the base density, making the theorem a reusable template for other symmetric circular families.

Reading between the lines

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

The same subsequence-based ratio argument is a natural route to generic identifiability of finite mixtures of sine-skewed circular distributions, which the paper names as an open problem; the oscillation that blocks naive moment-ratio limits is exactly what the Diophantine machinery removes.
A concrete stress test would be the inverse Batschelet distribution named in the paper: its characteristic function is known, so one could check whether its moment ratios stabilize along a Diophantine subsequence and thereby settle that open question.
A reader verifying Proposition 5 should confirm that Step 1's ratio limit can be made to hold along a subsequence that avoids integers where both cosine factors vanish, for instance $\mu=\pi/2,\xi=0$; the paper does not spell out this avoidance, and the identifiability claim depends on it.
Because the conditions of Theorem 2 are stated directly on the base density's cosine moments, every new symmetric circular family satisfying them immediately yields an identifiable sine-skewed version; the criterion acts as a reusable template rather than a one-off proof.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the paper has a genuinely new idea — using simultaneous Diophantine approximation to pick subsequences where trigonometric moment ratios converge, which extends Teicher's method to skew circular models — and the main identifiability results are very likely true. But the proofs as written have several gaps where ratios are asserted to have limits without handling parameters that make denominators vanish. These are repairable, not fatal, but they need fixing before the results are rigorous.

What's actually new: The paper proves ordinary identifiability for sine-skewed wrapped Cauchy, sine-skewed von Mises, a Möbius-transformed cardioid, and two cylindrical models (Abe-Ley and Imoto et al.). The moment formulas (4) and the subsequence trick via Lemma 8 are the core. The conditions in Theorem 2 are checkable and the verifications for SSWC and SSvM are clean, including the Bessel-function bounds. The cylindrical extensions follow from standard Weibull/GP-type conditional identifiability once the circular margin is identified. The writing is generally clear.

Soft spots: The biggest is in the proof of Theorem 2, Step 2, where the paper says that if λ2=0, the ratio β_p(γ1)/β_p(γ2) tends to infinity. That's false as stated when μ2=0, because then β_p(γ2)=0 for every p. The fix is trivial — relabel the pair so the nonzero λ is in the denominator — but the text doesn't say that, and the missing clause leaves a hole in the main theorem that propagates to Propositions 3, 4, 6, and 7. Second, Proposition 5, Step 1, claims a ratio tends to infinity as p→∞ without restricting to a subsequence where the denominator is nonzero; for parameters like μ2=π/2, ξ2=0, the denominator vanishes on infinitely many p. Again, a subsequence argument repairs it. Third, Lemma 8's definition of A mentions rationals but doesn't use them; the statement is true for arbitrary real c, so it's a typo, not a math error. Fourth, in Proposition 7 the A3 factor has an index slip (τ1/δ and missing δ in the exponent), but the asymptotic conclusion still holds.

The self-citations are appropriate and not padding; the paper builds on the same authors' earlier estimation paper and clearly distinguishes what's new.

Who it's for: anyone working in directional statistics or on identifiability of parametric models with trigonometric moments. It deserves a serious referee; the proofs need a careful revision, not rejection.

Referee Report

2 major / 4 minor

Summary. This paper develops a general identifiability criterion (Theorem 1) based on ratios of transforms along subsequences, generalizing Teicher's approach. The main application is to sine-skewed circular densities f(θ|γ)=f0(θ−μ|ψ)(1+λ sin(θ−μ)). Under three conditions on the cosine moments of the symmetric base density, Theorem 2 asserts identifiability of this family. The authors verify the conditions for the sine-skewed wrapped Cauchy and sine-skewed von Mises families (Propositions 3 and 4), prove identifiability of the Möbius-transformed cardioid (Proposition 5), and of the Abe–Ley and sine-skewed generalized Pareto-type cylindrical models (Propositions 6 and 7). The proofs use trigonometric moments together with simultaneous Diophantine approximation (Lemma 8).

Significance. If established, the results fill a real gap: ordinary identifiability of these asymmetric circular and cylindrical models has not been systematically treated, and identifiability is a prerequisite for consistency of maximum likelihood estimators. The proposed moment-ratio method is a useful and reasonably general device, and the conditions in Theorem 2 are checkable for concrete base densities. The paper is careful to rely on standard moment formulas and known Diophantine approximation results, with no circular reasoning. The main caveats are localized proof gaps in Theorem 2, Step 2 and Proposition 5, Step 1, which appear repairable; the conceptual contribution remains valuable.

major comments (2)

[Section 4, proof of Theorem 2, Step 2] Equation (4) gives β_p(γ)=sin(pμ)α_{0,p}(ψ)+cos(pμ)λ{α_{0,p−1}(ψ)−α_{0,p+1}(ψ)}/2. In Step 2 the proof takes φ_2(p|γ)=β_p(γ) and, for λ_2=0, asserts that |β_p(γ_1)/β_p(γ_2)|→∞. This ratio is undefined when μ_2=0 or μ_2=π, because then sin(pμ_2)=0 for every integer p and hence β_p(γ_2)≡0. These parameters are admissible in Γ and include the symmetric base case, so the gap is not a boundary artifact. Since Theorem 2 is used for Propositions 3, 4, 6, and 7, this is load-bearing. The proof can be repaired by handling the λ_2=0 case separately (for example, exchanging the roles of γ_1 and γ_2, or using α_p when μ∈{0,π} and β_p otherwise), but the repair must be stated explicitly.
[Section 4, proof of Proposition 5, Step 1, equation (31)] The displayed ratio is said to tend to infinity as p→∞ when ρ_{α1}>ρ_{α2}. After normalization, the denominator contains ρ̄_2(1−ρ_{α2}^2)cos(pμ_2+ξ_2)+ρ_{α2}cos(pμ_2)/p. For parameters such as cos(pμ_2+ξ_2)=0 on an infinite subsequence, the denominator is only of order 1/p on that subsequence, and it can even vanish unless the second term is nonzero. The limit over all integers is therefore not established, and the ratio may be undefined for infinitely many p. Because Theorem 1 only needs a suitable subsequence, a simultaneous-Diophantine subsequence argument or a uniform lower bound for the denominator would repair the proof, but neither is supplied. This step is the only argument for the ρ_{α1}≠ρ_{α2} case of Proposition 5.

minor comments (4)

[Section 4, Lemma 8] The definition A:={cπ|c∈[0,2)}, followed by the sentence 'where Q denotes the set of all rational numbers,' is confusing because Q is unused and A as written is just [0,2π). The proof works for arbitrary real c_i via simultaneous Diophantine approximation, but the statement should be cleaned up.
[Section 2.2, Proposition 6] The introduction to Proposition 6 states that the result follows from Theorem 1 and Proposition 4, but the proof invokes Proposition 3 (the SSWC identifiability). The reference should be corrected; since the marginal is SSWC, Proposition 3 is the appropriate one.
[Section 4, proof of Proposition 7, τ_1≠τ_2 case] In the factor A_3, the exponent should be δ(1/τ_2−1/τ_1) rather than 1/τ_2−1/τ_1, and the base should be consistent with the preceding display. The conclusion is unaffected because the signs of the two exponents are the same.
[Section 4, equation (21)] Equation (21) has a missing closing parenthesis in the numerator (α_{0,p}(ψ_1)/α_{0,p}(ψ_2)); this typo should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identifiability proofs derive conclusions from external moment identities, Bessel-function estimates, a standard Diophantine approximation theorem, and elementary identifiability of the Weibull family.

full rationale

The paper's central claim is that coinciding sine-skewed densities imply coinciding parameter vectors by comparing transforms whose ratios have non-unity limits. The moment expressions (4) and (5) are cited from Abe and Pewsey (2011), but these are algebraic identities for the trigonometric moments of densities of the form f0(θ−μ){1+λ sin(θ−μ)}, not the identifiability conclusion. Theorem 1 is a formalization of the standard Teicher-style argument: if two distributions are equal, every transform ratio must equal 1, contradicting the constructed non-unity limit. Lemma 8 is proved from Schmidt's simultaneous Diophantine approximation theorem, an external mathematical result, and is not assumed to imply the theorem. Conditions (i)–(iii) in Theorem 2 are conditions on the symmetric base density only, not on the target identifiability claim, and Propositions 3–7 verify those conditions using explicit computations: ρ^p for wrapped Cauchy, the Bessel recurrence I_{p−1}(κ)−I_{p+1}(κ)=(2p/κ)I_p(κ) for von Mises, and direct asymptotic estimates for the Möbius-transformed cardioid. Proposition 6 identifies (μ,κ,λ) from the marginal SSWC identifiability and then identifies (α,β) from the ordinary identifiability of the Weibull family; this is a composition of previously established facts, not circular reuse of the same claim. The cylindrical proof in Proposition 7 similarly uses the marginal SSWC identifiability and then separates δ, τ, and σ by asymptotic behavior of the conditional density ratio at x→0 and x→∞ and by evaluation at x=0. The only self-citations are to Abe and Pewsey (2011) for moment formulas and to Miyata et al. (2019) in the concluding remarks as a statement of prior related work and an open direction; neither is load-bearing for the identifiability theorems. Possible mathematical gaps in individual ratio limits, such as denominators vanishing along the chosen sequence, are correctness concerns, not circularity, because the proof does not secretly assume the identifiability conclusion as an input. Overall, the derivation chain is self-contained against external mathematical facts and does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new fitted constants and no new entities. Its central results rest on standard theorems from Diophantine approximation, published moment formulas for the specific distribution families, standard Bessel identities, and elementary identifiability of the Weibull family. The main proof-specific load is carried by the moment formulas, which are quoted from prior work rather than derived in the paper.

assumptions (6)

standard math Simultaneous Diophantine approximation (Schmidt 1996, Theorem 1A) provides, for any real c_1,...,c_s and any Q, integers q,p_i such that |c_i - p_i/q| <= 1/(qQ).
Used in Lemma 8 to build infinitely many integers p with p a_i (mod 2pi) arbitrarily small for real a_i = c_i pi.
domain assumption Trigonometric moment formulas for sine-skewed densities, alpha_p(gamma)=cos(p mu) alpha_0,p(psi)-sin(p mu) lambda(alpha_0,p-1(psi)-alpha_0,p+1(psi))/2 and beta_p analogously, from Abe and Pewsey (2011).
Quoted in equations (4); the entire verification of Theorem 2 depends on these expressions.
domain assumption Moment formulas for the Mobius-transformed cardioid, alpha_p=p rho_bar rho_alpha^(p-1)(1-rho_alpha^2)cos(p mu+xi)+rho_alpha^p cos(p mu), from Wang and Shimizu (2012).
Used as the starting point of Proposition 5.
standard math Bessel function recurrence I_{nu-1}(z)-I_{nu+1}(z)=(2 nu/z) I_nu(z) from Abramowitz and Stegun (1972).
Verifies condition (ii) for the sine-skewed von Mises distribution in Proposition 4.
standard math Identifiability of the Weibull family from its densities.
Used in Proposition 6 to identify alpha and beta from the conditional Weibull densities.
standard math Equality of all trigonometric moments determines a circular distribution.
The identifiability proofs compare distributions by comparing cosine and sine moments; this standard inversion is the formal link.

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Pith. "Pith review of Identifiability of asymmetric circular and cylindrical distributions." pith.science (2026). https://pith.science/paper/TKQGIRD3

@misc{pith2026190809114,
  author       = {Pith},
  title        = {Pith review of: Identifiability of asymmetric circular and cylindrical distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKQGIRD3}},
  note         = {Machine review of arXiv:1908.09114}
}

read the original abstract

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a M\"{o}bius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

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Works this paper leans on

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, " * write output.state after.block = add.period write newline

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Int Stat Rev 83(2):193--197

Abe T (2015) Discussion: `` O n families of distributions with shape parameters''. Int Stat Rev 83(2):193--197

work page 2015

[2] [2]

Econom Stat 4:91--104

Abe T, Ley C (2017) A tractable, parsimonious and flexible model for cylindrical data, with applications. Econom Stat 4:91--104

work page 2017

[3] [3]

Statist Papers 52(3):683--707

Abe T, Pewsey A (2011) Sine-skewed circular distributions. Statist Papers 52(3):683--707

work page 2011

[4] [4]

For sale by the Superintendent of Documents, U.S

Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C

work page 1972

[5] [5]

In: Pushing the limits of contemporary statistics: contributions in honor of J ayanta K

Bhattacharya A, Bhattacharya R (2008) Nonparametric statistics on manifolds with applications to shape spaces. In: Pushing the limits of contemporary statistics: contributions in honor of J ayanta K . G hosh, Inst. Math. Stat. (IMS) Collect., vol 3, Inst. Math. Statist., Beachwood, OH, pp 282--301

work page 2008

[6] [6]

Springer-Verlag, New York

Chikuse Y (2003) Statistics on special manifolds, Lecture Notes in Statistics, vol 174. Springer-Verlag, New York

work page 2003

[7] [7]

John Wiley & Sons, Ltd., Chichester

Dryden IL, Kent JT (eds) (2015) Geometry driven statistics. John Wiley & Sons, Ltd., Chichester

work page 2015

[8] [8]

Springer Series in Statistics, Springer, New York

Fr\" u hwirth-Schnatter S (2006) Finite mixture and M arkov switching models. Springer Series in Statistics, Springer, New York

work page 2006