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REVIEW 4 major objections 5 minor 49 references

Geometrical Smeariness -- A new Phenomenon of Fr\'echet Means

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Pith's one-line read On spheres of dimension five and higher, a Fréchet mean can converge at $n^{-1/6}$ purely because of curvature, even when data avoids the cut locus.

desk verdict Useful conceptual distinction and a clever construction, but Theorem 3.1 overclaims the n^{-1/6} rate the proof never establishes. read the letter →

arxiv 1908.04233 v3 pith:T2GCDKJM submitted 2019-08-12 math.ST stat.TH

Benjamin Eltzner This is my paper · ORCID
classification math.STstat.TH MSC 62R3060F05
keywords Fréchetmeangeometricalsmearinesscutlocuscentrallimittheoremonmanifoldsspherescurvaturen^{-1/6}ratefinitesample
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

The paper establishes a new kind of failure of the central limit theorem on curved data spaces. On the circle, the anomalously slow $n^{-1/6}$ fluctuation of a Fréchet mean (2-smeariness) requires a special value of the probability density at the antipodal point, the cut locus. The paper proves that on spheres of dimension $m\ge 5$, the same slow rate can occur for purely geometric reasons: the mean sits at the north pole, while the data is uniform on the southern hemisphere with a hole of radius $\beta$ around the south pole plus a point mass at the north pole. For any sufficiently small $\beta$ the north pole is the unique Fréchet mean, and its asymptotic fluctuation scale is $n^{-1/6}$. This shows smeariness is not an artifact of mass sitting at the cut locus, and it can even be transplanted to a manifold diffeomorphic to Euclidean space.

What carries the argument

The machinery is the rotationally symmetric Fréchet function $F(\alpha,\beta,\psi)$, where $\psi$ is the polar angle to the north pole $\mu$. Because the measure is invariant under rotations about the polar axis, proving smeariness reduces to showing $\partial^2 F/\partial\psi^2$ vanishes at $\psi=0$ while $\partial^4 F/\partial\psi^4>0$, so the Fréchet function grows like $\psi^4$ and the sample mean fluctuates at $n^{-1/6}$. The spherical annulus $L_{m,\beta}$ is the key object: its hole around the south pole removes the cut locus from the support, isolating the curvature contribution. Geometrical smeariness is smeariness in which the negative Hessian contribution from the cut locus is absent: either $F=G$ in a neighborhood of the mean or $\mathrm{Hess}(F-G)(0)\ge 0$. The proof that the local minimum is global uses Lipschitz bounds (Lemma A.8) on how the second, third, and fourth derivatives of $F$ change with $\beta$, yielding the threshold $\beta_0 = \min(c_m/(2L_4), (1/L_2)\,\partial^2 F/\partial\psi^2(\alpha_0,0,\pi/3))$.

What would settle it

For $m=5$, numerically evaluate the exact Fréchet function $F(\alpha_\beta,\beta,\psi)$ for the uniform annulus plus point mass at the north pole, choosing $\alpha_\beta$ so the Hessian vanishes at $\psi=0$, and check whether $\partial^2 F/\partial\psi^2 > 0$ for all $\psi\in(0,\pi]$ when $\beta\le\beta_0$. If the maximal allowed hole radius is not strictly positive, or if simulations of the sample mean do not show variance scaling like $n^{-1/3}$, the uniqueness-and-rate claim of Theorem 3.1 would be refuted.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that smeariness on spheres is geometrical, not cut-locus. Theorem 2.10 states that the circle $S^1$ only admits cut locus smeariness, while every sphere $S^m$ with $m\ge 2$ only admits geometrical smeariness: near the mean, the Fréchet function using only geodesics that avoid the cut locus coincides with the full Fréchet function, because geodesics can circumvent the antipodal point. Theorem 3.1 then constructs, for $m\ge 5$, a random variable with a unique 2-smeary Fréchet mean at the north pole whose support excludes a neighborhood of the antipodal point: uniform mass $\alpha$ on the spherical annulus $L_{m,\beta} = \{q : \arccos\langle q,\mu\rangle\in[\pi/2,\pi-\beta]\}$ plus point mass $1-\alpha$ at $\mu$, with $\alpha$ tuned so the Hessian of the Fréchet function vanishes at $\mu$ while its fourth derivative is positive. The resulting asymptotic rate is $n^{-1/6}$, and a corollary deforms the opposite hemisphere to build the same phenomenon on a manifold diffeomorphic to $\mathbb{R}^m$. A further theorem gives a curse of dimensionality: as $m$ grows, the hole radius may approach $\pi/2$, so in high dimension even support barely exceeding a hemisphere can produce smeary local means.

Load-bearing premise

The proof that the local smeary minimum is the unique global mean rests on Lipschitz bounds that are asserted to exist but never explicitly evaluated, and it also assumes the smeary CLT conditions apply to a distribution with a point mass and a hole.

Editorial extensions

If this is right

  • On $S^m$ with $m\ge 5$, smeariness can occur even when the support of the random variable has a hole around the cut locus, so the antipodal density is irrelevant.
  • A Fréchet mean with vanishing Hessian and positive quartic term has $n^{-1/6}$ fluctuations; the paper shows such means exist concretely on spheres and on manifolds diffeomorphic to $\mathbb{R}^m$.
  • Deforming the opposite hemisphere to a flat space yields smeary means on a manifold diffeomorphic to $\mathbb{R}^m$, so topological triviality does not prevent the phenomenon.
  • In high dimension, the allowed hole radius grows toward $\pi/2$: random variables whose support barely exceeds a hemisphere can already have smeary local means.
  • Finite-sample smeariness of large magnitude is possible for moderately spread spherical data, and the bootstrap analysis of geomagnetic pole-reversal data finds 17 of 151 data sets consistent with it.

Reading between the lines

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

  • Because the Lipschitz constants in the proof are not evaluated, the practical size of the hole radius $\beta_0$ in concrete dimensions is unknown; computing them numerically would show whether the $n^{-1/6}$ regime is robust or confined to a tiny parameter window.
  • The same vanishing-Hessian plus positive-quartic mechanism should produce geometrically smeary means on other positively curved, rotationally symmetric spaces, beyond the spheres treated here.
  • For practitioners, checking the curvature of the empirical Fréchet function at the estimated mean may be a more useful diagnostic for smeariness than inspecting the density near the antipodal point.
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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

4 major / 5 minor

Summary. The paper introduces a distinction between cut-locus smeariness and geometrical smeariness for Fréchet means on Riemannian manifolds. It claims that on the circle smeariness is cut-locus type, while on spheres S^m with m≥2 it is geometrical (Theorem 2.10). The main construction (Theorem 3.1) considers a random variable on S^m, m≥5, uniformly distributed on a spherical annulus L_{m,β} with total mass α plus a point mass at the north pole, and claims that for sufficiently small hole radius β the north pole is the unique Fréchet mean and is 2-smeary with asymptotic rate n^{-1/6}. The paper also proves a curse-of-dimensionality result for the admissible hole radius (Theorem 3.3), discusses finite sample smeariness, presents simulations on Kendall pre-shape space, and analyzes geomagnetic pole reversal data.

Significance. The geometrical-smeariness concept is original and, if established, would be an important contribution to non-Euclidean statistics: it shows that slow n^{-1/6} convergence can occur without probability mass near the cut locus and even on manifolds diffeomorphic to R^m. The supplement contains careful, largely self-contained integral computations for the second and fourth derivatives of the Fréchet function, and the paper is honest about the places where numerical bounds or conjectures are used. The simulations and the geomagnetic data analysis provide a useful practical illustration. At present, however, the formal definition of smeariness is not adequate as written, and the n^{-1/6} claim in Theorem 3.1 is not actually derived; these issues require substantial revision rather than minor polishing.

major comments (4)
  1. [Definition 2.8] The definition of smeariness only imposes a lower bound: sup_{||x||<δ} |F(x)-F(0)| ≥ C_X δ^κ with κ>2. This condition is automatically satisfied by every Fréchet mean with positive-definite Hessian, because such an F satisfies F(x)-F(0) ≥ c||x||² in a neighborhood, so the supremum over a δ-ball is at least cδ², and for small δ<1 one has cδ² ≥ Cδ^κ for a suitable choice of C. Thus every nondegenerate mean would be classified as smeary, and the definition does not isolate the intended slower-than-n^{-1/2} phenomenon. The definition needs a two-sided growth condition, or an explicit requirement that all derivatives below order κ vanish at 0, to be mathematically meaningful.
  2. [Theorem 3.1, statement] Theorem 3.1 quantifies over arbitrary total mass 0<α<1 of the uniform annulus component, but the proof uses only the special value α=α_β defined by the condition ∂²F/∂ψ²(α_β,β,0)=0. For generic α the Hessian at the north pole need not vanish, and the claimed n^{-1/6} rate cannot hold; indeed for α<α_β one would expect standard n^{-1/2} asymptotics. The theorem should be restated with α=α_β, or it should be made explicit that the assertion applies only to the tuned value of α.
  3. [Theorem 3.1, proof / Definition 2.8 note] The asymptotic rate n^{-1/6} is asserted but never derived. Definition 2.8 concerns the population Fréchet function; the distribution of the sample Fréchet mean requires a smeary central limit theorem. The paper's own note after Definition 2.8 says that the smeary asymptotic theory relies on the stricter (Eltzner and Huckemann, 2019, Assumption 2.6), but Theorem 3.1 does not verify that assumption for a distribution with an atom at the mean and an indicator-supported annulus with a hole. In fact, Assumption 2.5(ii) requires a density in a neighborhood of Cut(μ), and the support L_{m,β} with β>0 explicitly excludes such a neighborhood. The theorem should either prove the needed CLT for this case or weaken the claim to '2-smeary in the sense of Definition 2.8' without the n^{-1/6} conclusion.
  4. [Supplement A.2.1 / Theorem 2.12] Theorem 2.12 states that for every ρ≥0 one can construct a random variable with density value ρ at the south pole and a non-smeary mean at the north pole. The proof, however, sets α=sinδ/(4π) and shows only that the south-pole density can be made arbitrarily large as δ→0. This establishes arbitrarily large density values, not that every nonnegative ρ is attained. A continuity or monotonicity argument over the full range of δ, or a modified construction covering small ρ, is needed; otherwise the theorem should be weakened to 'arbitrarily large ρ'.
minor comments (5)
  1. [Section 3, proof of Theorem 3.1] The sentence 'This is established by Lemmas A.3 and A.5 in the supplement' appears to cite the wrong lemmas; the local-minimum calculation for the annulus model with a hole is carried out in Lemmas A.6 and A.7.
  2. [Theorem 3.1 / Definition 2.8] The term '2-smeary' is used in Theorem 3.1 without being defined. Please define k-smeary, or explain the relation between the order κ in Definition 2.8 and the label '2-smeary' and the rate n^{-1/6}.
  3. [Supplement A.2] The heading 'A.2.1 Proof of Theorem 2.14' should refer to Theorem 2.12, not Theorem 2.14.
  4. [Supplement A.2.1] The sentence beginning 'No note that due to convexity...' appears to contain a typo ('No note' should likely be 'Note').
  5. [Lemma A.8 / Eq. (1)] The Lipschitz constants L2, L3, L4 in Lemma A.8 are asserted to exist but are not explicitly bounded, and the threshold β0 in the proof of Theorem 3.1 depends on them. Since the proof argues finiteness, this is not fatal, but an explicit bound or a more transparent finiteness argument would make the uniqueness window quantitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1 is an existence construction with a free parameter tuned to the degeneracy point, and the cited prior work is independent; the unresolved n^{-1/6} issue is a proof gap, not a circular reduction.

full rationale

The central construction is not a fit or a restatement of its inputs. In Theorem 3.1 the free mass parameter αβ is defined by requiring the Hessian to vanish, ∂²F/∂ψ²(αβ,β,0)=0, and the proof then checks by direct integration that the fourth derivative is positive and uses Lipschitz bounds (Lemma A.8) to obtain a unique global minimum for small holes. That is a legitimate mathematical construction, not an estimate renamed as a prediction, and the uniqueness argument is self-contained relative to the β=0 hemisphere model. The citations to Eltzner and Huckemann (2019) supply derivative formulas and a general smeary CLT with stated assumptions; these are prior, independently published results, not assumptions that contain the present conclusion, so the self-citation is not circular. The main caveat is that Theorem 3.1 states the n^{-1/6} asymptotic rate while the proof verifies only the Definition 2.8 growth condition and does not explicitly check the stricter Assumption 2.6 of the cited smeary CLT, and the theorem's quantification over arbitrary α is looser than the proof's α=αβ construction. These are correctness or presentation gaps that do not make the derivation circular.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's central examples rely on rotation-symmetric distributions and on previously derived derivative formulas from the author's own earlier work. The main construction tunes α=αβ to make the Hessian vanish; this is a legitimate choice for an existence theorem but means the examples are designed to sit exactly at the boundary where standard n^{-1/2} asymptotics break down. The application analyses add domain assumptions about i.i.d. sampling that are not verified.

free parameters (2)
  • alpha_beta = Implicitly defined by ∂²F/∂ψ²(αβ,β,0)=0; no closed form given
    Central to Theorem 3.1: the annulus mass α is tuned so the Hessian at the north pole vanishes, which is what produces the smeary n^{-1/6} rate. The theorem statement omits that α must equal this tuned value.
  • alpha_landmark_simulation = Not reported; described only as making the Hessian slightly positive
    In Section 4.2, α is chosen by hand so that the Hessian at the mean is slightly positive. Without the value, the finite-sample smeariness simulation cannot be reproduced exactly.
assumptions (4)
  • domain assumption Assumption 2.5: each random variable has unique population Fréchet mean and a density near the cut locus
    Used throughout the framework; for the constructed examples the density near the cut locus is replaced by a hole, so only part of the assumption is inherited.
  • domain assumption Rotation symmetry of the random variables reduces the Fréchet function to a function of polar angle ψ only
    Used in Section A.1 and throughout; all examples and applications rely on this symmetry.
  • domain assumption VGP data sets are modeled as i.i.d. samples from a fixed spherical distribution
    Section 4.3 treats each paleomagnetic data set as a random sample; temporal dependence and measurement uncertainty are not modeled.
  • domain assumption Derivative formulas f2, f4, cm and the smeary CLT from Eltzner and Huckemann (2019) are correct and applicable
    The paper recalls these prior results rather than reproving them; applicability to atomic distributions with a hole is not checked.

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Pith. "Pith review of Geometrical Smeariness -- A new Phenomenon of Fr\'echet Means." pith.science (2026). https://pith.science/paper/T2GCDKJM

@misc{pith2026190804233,
  author       = {Pith},
  title        = {Pith review of: Geometrical Smeariness -- A new Phenomenon of Fr\'echet Means},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2GCDKJM}},
  note         = {Machine review of arXiv:1908.04233}
}
abstract

In the past decades, the central limit theorem (CLT) has been generalized to non-Euclidean data spaces. Some years ago, it was found that for some random variables on the circle, the sample Fr\'echet mean fluctuates around the population mean asymptotically at a scale $n^{-\tau}$ with exponent $\tau < 1/2$ with a non-normal distribution if the probability density at the antipodal point of the mean is $\frac{1}{2\pi}$. The author and his collaborator recently discovered that $\tau = 1/6$ for some random variables on higher dimensional spheres. In this article we show that, even more surprisingly, the phenomenon on spheres of higher dimension is qualitatively different from that on the circle, as it depends purely on geometrical properties of the space, namely its curvature, and not on the density at the antipodal point. This gives rise to the new concept of geometrical smeariness. In consequence, the sphere can be deformed, say, by removing a neighborhood of the antipodal point of the mean and gluing a flat space there, with a smooth transition piece. This yields smeariness on a manifold, which is diffeomorphic to Euclidean space. We give an example family of random variables with 2-smeary mean, i.e. with $\tau = 1/6$, whose range has a hole containing the cut locus of the mean. The hole size exhibits a curse of dimensionality as it can increase with dimension, converging to the whole hemisphere opposite a local Fr\'echet mean. We observe smeariness in simulated landmark shapes on Kendall pre-shape space and in real data of geomagnetic north pole positions on the two-dimensional sphere.

Figures

Figures reproduced from arXiv: 1908.04233 by the authors.

Figure 1
Figure 1. Example of a manifold which is diffeomorphic to R m, obtained by deforming a sphere S m (visualized for m = 2). For dimension m ≥ 5 we give an example of a random variable on such a manifold which features a smeary mean. We show • that cut locus smeariness occurs on the circle and the torus and can therefore be understood exhaustively by studying the properties of the random variable in a neighborhood of the cut loc… view at source ↗
Figure 2
Figure 2. Examples of the landmark pre-shapes used here. The shapes are quadrangles, therefore the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Variances of sample means for increasing sample size. For every sample size, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Examples of data sets which exhibit smeariness. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Bootstrap variances of the mean for samples which exhibit smeariness. (a) Finite sample [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Plots of θ 7→ f2(θ, 0) = 1 2g(θ) ∂ 2Fθ ∂ψ2 |ψ=0 for different dimension m. One can clearly see that for m → ∞ the position of the zero approaches θ = π/2 from above. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Plots of θ 7→ f4(θ, 0) = 1 2g(θ) ∂ 4Fθ ∂ψ4 |ψ=0 for different dimension m. One can clearly see that for m → ∞ the lower bound of the region where the fourth derivative is positive approaches θ = π/2 from above. The function θ 7→ f2(θ, 0) is plotted for several m in [P…
Figure 8
Figure 8. Figure 8: Numerically determined values for θm,2 and θm,4 for m ≤ 100. One can clearly see that the values approach π/2 from above. Lemma A.3. The position θm,2 of the zero of θ 7→ f2(θ, 0) is bounded by π 2 + 1 3(m − 1) ≤ θm,2 ≤ π 2 + 1 m − 1 . Proof. This is positive while 1 m…
Figure 9
Figure 9. Figure 9: Numerically determined values for βm,2 and βm,4 which bound the radius of the hole from above for m ≤ 100. One can clearly see that the values approach π/2 from below. To show that the local minimum at ψ = 0 is indeed a global minimum at least for some β > 0, we use a …

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Works this paper leans on

49 extracted references · 46 canonical work pages

  1. [1]

    Afsari, B. (2009). Means and averaging on Riemannian manifolds . University of Maryland

    work page 2009

  2. [2]

    Afsari, B. (2011). Riemannian L ^p center of mass: existence, uniqueness, and convexity. Proceedings of the American Mathematical Society\/ 139 , 655--773

    work page 2011

  3. [3]

    Le Gouic, and Q

    Ahidar-Coutrix, A., T. Le Gouic, and Q. Paris (2019). Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics. Probability Theory and Related Fields\/ 177 , 323--368

    work page 2019

  4. [4]

    Arnaudon, M. and L. Miclo (2014). Means in complete manifolds: uniqueness and approximation. ESAIM: Probability and Statistics\/ 18 , 185--206

    work page 2014

  5. [5]

    Le, and M

    Barden, D., H. Le, and M. Owen (2013). Central limit theorems for F r \'e chet means in the space of phylogenetic trees. Electronic Journal of Probability\/ 18\/ (25), 1--25

    work page 2013

  6. [6]

    Le, and M

    Barden, D., H. Le, and M. Owen (2018). Limiting behaviour of fr \'e chet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics\/ 70\/ (1), 99--129

    work page 2018

  7. [7]

    Bhattacharya, A. and R. Bhattacharya (2008). Statistics on riemannian manifolds: asymptotic distribution and curvature. Proceedings of the American Mathematical Society\/ 136 , 2959--2967

    work page 2008

  8. [8]

    Bhattacharya, A. and R. Bhattacharya (2012). Nonparametric Inference on Manifolds . Cambridge University Press

    work page 2012

Show all 49 references
  1. [9]

    Bhattacharya, R. and L. Lin (2017). Omnibus CLT for F r\'echet means and nonparametric inference on non-euclidean spaces. Proceedings of the American Mathematical Society\/ 145\/ (1), 413--428

    2017

  2. [10]

    Bhattacharya, R. N. and V. Patrangenaru (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds I . Annals of Statistics\/ 31\/ (1), 1--29

    2003

  3. [11]

    Bhattacharya, R. N. and V. Patrangenaru (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds II . Annals of Statistics\/ 33\/ (3), 1225--1259

    2005

  4. [12]

    Chen, X. and T. M. Christensen (2015). Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions . Journal of Econometrics\/ 188\/ (2), 447--465

    2015

  5. [13]

    Clement, B. and D. Kent (1985). A comparison of two sequential geomagnetic polarity transitions (upper olduvai and lower jaramillo) from the southern hemisphere. Physics of the Earth and Planetary Interiors (Special Issue Irregularities in the Secular Variation and Geodynamic ...

    1985

  6. [14]

    Clement, B. and D. Kent (1987). Geomagnetic polarity transition records from five hydraulic piston core sites in the north atlantic. Initial Reports of the Deep Sea Drilling Project\/ 94 , 831--852

    1987

  7. [15]

    Kent, and N

    Clement, B., D. Kent, and N. D. Opdyke (1982, 08). Brunhes-matuyama polarity transition in three deep-sea sediment cores. Philosophical Transactions of the Royal Society of London\/ 306 , 113--119

    1982

  8. [16]

    Clement, B. M., L. L. Sierra, E. Smith, and P. Rodda (1996). A record of the upper cochiti polarity transition from the southern hemisphere (fiji). Surveys in Geophysics\/ 17\/ (2), 189--196

    1996

  9. [17]

    Galaz-Garcia, S

    Eltzner, B., F. Galaz-Garcia, S. F. Huckemann, and W. Tuschmann (2019). Stability of the cut locus and a central limit theorem for fr \'e chet means of riemannian manifolds. arXiv math.DG 1909.00410\/

    2019 arXiv

  10. [18]

    Eltzner, B. and S. F. Huckemann (2019). A smeary central limit theorem for manifolds with application to high dimensional spheres. Annals of Statistics\/ 47\/ (6), 3360--3381

    2019

  11. [19]

    Groisser, D. (2005). On the convergence of some P rocrustean averaging algorithms. Stochastics: An International Journal of Probability and Stochastic Processes\/ 77\/ (1), 51--60

    2005

  12. [20]

    Gurariy, G. (1988). Geomagnetic field during reversals in the late Cenozoic (Nauka, Moscow) , pp.\ pp. 207. Academy of Sciences of the USSR

    1988

  13. [21]

    Holt, J. W. and J. L. Kirschvink (1995). The upper olduvai geomagnetic field reversal from death valley, california: a fold test of transitional directions. Earth and Planetary Science Letters\/ 133\/ (3), 475--491

    1995

  14. [22]

    Hotz, T. and S. Huckemann (2015). Intrinsic means on the circle: Uniqueness, locus and asymptotics. Annals of the Institute of Statistical Mathematics\/ 67\/ (1), 177--193

    2015

  15. [23]

    Huckemann, H

    Hotz, T., S. Huckemann, H. Le, J. S. Marron, J. Mattingly, E. Miller, J. Nolen, M. Owen, V. Patrangenaru, and S. Skwerer (2013). Sticky central limit theorems on open books. Annals of Applied Probability\/ 23\/ (6), 2238--2258

    2013

  16. [24]

    Huckemann, S. (2011a). Inference on 3 D P rocrustes means: Tree boles growth, rank-deficient diffusion tensors and perturbation models. Scandinavian Journal of Statistics\/ 38\/ (3), 424--446

    2011

  17. [25]

    Huckemann, S. (2011b). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. Annals of Statistics\/ 39\/ (2), 1098--1124

    2011

  18. [26]

    Mattingly, E

    Huckemann, S., J. Mattingly, E. Miller, and J. Nolen (2015). Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability\/ 20\/ (78), 34 pp

    2015

  19. [27]

    Hundrieser, S. (2017). Exploring Smeary Limit Theorems . Bachelor Thesis, University of G\"ottingen

    2017

  20. [28]

    Eltzner, and S

    Hundrieser, S., B. Eltzner, and S. Huckemann (2020). Finite sample smeariness of fr\'echet means and application to climate. arXiv stat.ME 2005.02321\/

    2020 arXiv

  21. [29]

    Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics\/ XXX , 509--541

    1977

  22. [30]

    Karcher, H. (2014). Riemannian center of mass and so called karcher mean. arXiv math.HO 1407.2087\/

    2014 arXiv

  23. [31]

    Nakajima, K

    Kawai, N., T. Nakajima, K. Hirooka, and K. Kobayashi (1973). The transition of field at the brunhes and jaramillo boundaries in the matuyama geomagnetic epoch. Proceedings of the Japan Academy\/ 49\/ (10), 820--824

    1973

  24. [32]

    Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image I : Uniqueness and fine existence. Proceedings of the London Mathematical Society\/ 61 , 371--406

    1990

  25. [33]

    Le, H. (2001). Locating F r\' e chet means with an application to shape spaces. Advances of Applied Probability (SGSA)\/ 33\/ (2), 324--338

    2001

  26. [34]

    Le, H. and D. Barden (2014). On the measure of the cut locus of a F r\'echet mean. Bulletin of the London Mathematical Society\/ 46\/ (4), 698--708

    2014

  27. [35]

    Paris, P

    Le Gouic, T., Q. Paris, P. Rigollet, and A. Stromme (2019). Fast convergence of empirical barycenters in alexandrov spaces and the wasserstein space. arXiv math.ST 1908.00828\/

    2019 arXiv

  28. [36]

    Li, Z. and C. Ma (2015). Asymptotic properties of estimators in a stable cox-ingersoll-ross model. Stochastic Processes and their Applications\/ 125\/ (8), 3196--3233

    2015

  29. [37]

    Linssen, J. (1988). Preliminary results of a study of four successive sedimentary geomagnetic reversal records from the mediterranean (upper thvera, lower and upper sidufjall, and lower nunivak). Physics of the Earth and Planetary Interiors\/ 52\/ (3), 207--231

    1988

  30. [38]

    Linssen, J. (1991). Properties of Pliocene sedimentary geomagnetic reversal records from the Mediterranean (Ph.D thesis, University of Utrecht) , Volume 80. Geologica Ultraiectina

    1991

  31. [39]

    Liu, J. and X. Yang (2012). The convergence rate and asymptotic distribution of the bootstrap quantile variance estimator for importance sampling. Advances in Applied Probability\/ 44\/ (3), 815--841

    2012

  32. [40]

    Kent, and J

    Mardia, K., J. Kent, and J. Bibby (1979). Mulitvariate Analysis . Probability and Mathematical Statistics. Academic Press

    1979

  33. [41]

    McElhinny, M. W. and J. Lock (1996). IAGA paleomagnetic databases with A ccess. Surveys in Geophysics\/ 17\/ (5), 575--591

    1996

  34. [42]

    Pennec, X. (2019). Curvature effects on the empirical mean in riemannian and affine manifolds: a non-asymptotic high concentration expansion in the small-sample regime. arXiv math.DG 1906.07418\/

    2019 arXiv

  35. [43]

    Rolph, T. C. (1993). The matuyama-jaramillo r-n transition recorded in a loess section near lanzhou, p. r. china. Journal of Geomagnetism and Geoelectricity\/ 45\/ (4), 301--318

    1993

  36. [44]

    Sch\"otz, C. (2019). Convergence rates for the generalized fréchet mean via the quadruple inequality. Electronic Journal of Statistics\/ 13\/ (2), 4280--4345

    2019

  37. [45]

    Sturm, K.-T. (2003). Probability measures on metric spaces of nonpositive curvature. In Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces: Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs , Volume 338, pp....

    2003

  38. [46]

    van der Vaart, A. (2000). Asymptotic Statistics . Cambridge University Press

    2000

  39. [47]

    Van Hoof, A. A. M. and C. G. Langereis (1992). The upper kaena sedimentary geomagnetic reversal record from southern sicily. Journal of Geophysical Research: Solid Earth\/ 97\/ (B5), 6941--6957

    1992

  40. [48]

    van Zijl, J. S. V., K. W. T. Graham, and A. L. Hales (1962). The palaeomagnetism of the stormberg lavas, ii. the behaviour of the magnetic field during a reversal. Geophysical Journal of the Royal Astronomical Society\/ 7\/ (2), 169--182

    1962

  41. [49]

    Wellman, P., M. W. McElhinny, and I. McDougall (1969). On the polar-wander path for australia during the cenozoic. Geophysical Journal International\/ 18\/ (4), 371--395

    1969

Pith tools

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