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arxiv: 2511.13664 · v2 · submitted 2025-11-17 · 🧮 math.ST · stat.AP· stat.TH

Rate-optimal and computationally efficient nonparametric estimation on the circle and the sphere

Athanasios G. Georgiadis , Andrew P. Percival This is my paper

Pith reviewed 2026-05-17 20:36 UTC · model grok-4.3

classification 🧮 math.ST stat.APstat.TH
keywords nonparametric density estimationunit circleunit sphererate-optimal estimationcomputational efficiencydirectional dataspherical dataclosed-form probability estimates
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The pith

New density estimators on the circle and sphere attain optimal convergence rates while permitting direct computation from samples.

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

The paper develops nonparametric estimators for densities on the unit circle and unit sphere that match the best possible accuracy rates under smoothness assumptions and can be implemented directly without extra costs that grow with sample size. A reader would care because observations on these shapes arise often in directional and periodic data, such as animal headings or celestial positions. The work also supplies closed-form expressions for estimating probabilities over regions and backs the claims with simulations plus case studies in zoology, climatology, geophysics, and astronomy.

Core claim

The authors introduce nonparametric density estimators on the unit circle and unit sphere that achieve rate-optimality under standard smoothness assumptions while remaining computationally efficient for direct implementation. They derive closed-form expressions for probability estimates over regions of the circle and sphere. These claims are supported by simulation studies and illustrated through applications in zoology, climatology, geophysics, and astronomy.

What carries the argument

The new nonparametric density estimators that combine rate-optimality with computational efficiency for direct implementation on the circle and sphere.

If this is right

  • Closed-form expressions become available for estimating probabilities over arbitrary regions on the circle and sphere.
  • Simulation studies confirm that the estimators attain the claimed optimal rates.
  • Case studies demonstrate direct applicability to directional data in zoology, climatology, geophysics, and astronomy.
  • The methods extend to any analysis of directional or periodic phenomena on these manifolds.

Where Pith is reading between the lines

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

  • Similar constructions could be tested on other compact manifolds such as the torus for periodic data in higher dimensions.
  • The direct efficiency might support real-time processing of large sensor datasets in spatial statistics.
  • Integration with existing directional statistics software could accelerate use in applied fields.

Load-bearing premise

The rate-optimality and efficiency rest on the unknown density having standard smoothness properties and on the implementation having no hidden costs that grow with sample size.

What would settle it

A simulation where the mean integrated squared error of the proposed estimators fails to match the minimax rate for the given smoothness class as sample size grows, or where direct implementation requires computation time that scales worse than linearly with sample size.

Figures

Figures reproduced from arXiv: 2511.13664 by Andrew P. Percival, Athanasios G. Georgiadis.

Figure 1
Figure 1. Figure 1: (Left to right) The plot of the uniform distribution over [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left to right) The plot of the uniform distribution over [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Left to right) The plot of the vMF distribution over [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The estimated MISE for vMF simulated points with [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Left to right) The plot of a mixture density over [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Left to right) The plot of a mixture density over [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evaluation times of the integral of the finite KDEs over a region [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots of 𝑛 = 691 honeybee orientations (left) and the corresponding KDE with 𝑠 = 1 (right). Region Frequency KDE Prob. [−1.33, 0.81] 0.4906 0.4893 [2.46, 𝜋] ∪ (−𝜋, −3.03] 0.0637 0.0637 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plots of 𝑛 = 1455 days with nonzero precipitation in Los Angeles (left) and the corresponding KDE with 𝑠 = 2 (right). Dates (Inclusive) Frequency KDE Prob. 1 st Feb. - 31st Mar. 0.2838 0.2815 1 st Jun. - 31st Oct. 0.2027 0.2008 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Plots of 𝑛 = 1630 high magnitude earthquakes (left) and the corresponding KDE with 𝑠 = 0.05 (right). Country Min. Lat. Max. Lat. Min. Lon. Max. Lon. Frequency KDE Prob. Japan 31◦N 45.5 ◦N 129.4 ◦E 145.5 ◦E 0.0540 0.049808 Chile 55.6 ◦S 17.6 ◦S 75.6 ◦W 70◦W 0.0429 0.043289 Philippines 5.6 ◦N 18.5 ◦N 117.2 ◦E 126.5 ◦E 0.0270 0.025305 Ireland 51.7 ◦N 55.1 ◦N 10◦W 6 ◦W 0.0000 0.000002 [PITH_FULL_IMAGE:figure… view at source ↗
Figure 11
Figure 11. Figure 11: Plots of 𝑛 = 9096 bright stars (left) and the corresponding KDE with 𝑠 = 1 (right). This application highlight’s the importance of having kernel density estimators available on S 2 . If one were to have instead constructed an estimator over R 2 , using the latitude and longitude as coordinates, the periodicity in the longitude would be lost. For instance, there would be a discontinuity between the earthqu… view at source ↗
read the original abstract

We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for direct implementation. After establishing these estimators, we derive closed-form expressions for probability estimates over regions of the circle and the sphere. Then, the proposed theories are supported by extensive simulation studies. The considered settings naturally arise when analyzing phenomena on the Earth's surface or in the sky (sphere), as well as directional or periodic phenomena (circle). The proposed approaches are broadly applicable, and we illustrate their usefulness through case studies in zoology, climatology, geophysics, and astronomy, which may be of independent interest. The methodologies developed here can be readily applied across a wide range of scientific domains.

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

2 major / 3 minor

Summary. The manuscript develops new nonparametric density estimators for data distributed on the unit circle and the unit sphere. The primary contributions are estimators that achieve rate-optimality under standard smoothness assumptions while remaining computationally efficient for direct implementation without hidden costs scaling with sample size. After defining the estimators, the authors derive closed-form expressions for probability estimates over arbitrary regions on the circle and sphere. Theoretical results are supported by simulation studies, and the methods are demonstrated on case studies from zoology, climatology, geophysics, and astronomy.

Significance. If the rate-optimality and direct-implementability claims hold, the work would supply a practically useful addition to directional statistics, particularly for applications involving periodic or spherical data where both statistical efficiency and computational simplicity matter. The closed-form region probabilities and the breadth of the case studies are concrete strengths that increase the potential impact beyond pure theory.

major comments (2)
  1. [§3, Theorem 2] §3, Theorem 2: the upper bound on the MISE is derived under Hölder smoothness of order β, but the matching minimax lower bound is only referenced rather than proved or sketched; without an explicit lower-bound argument or citation to a result that applies directly to the circle/sphere geometry, the rate-optimality claim remains incomplete.
  2. [§4.1, Algorithm 1] §4.1, Algorithm 1: the computational complexity is stated as O(n) for the estimator itself, yet the bandwidth selection step (cross-validation or plug-in) is not analyzed; if this step requires O(n²) operations or iterative optimization whose cost grows with n, the central claim of 'direct implementation' without hidden n-dependent costs would be undermined.
minor comments (3)
  1. [§2.2] The notation for the spherical harmonics basis in §2.2 is introduced without an explicit reference to the normalization convention used; adding a short sentence or citation would remove ambiguity for readers.
  2. [Figure 3] Figure 3 caption does not indicate the number of Monte Carlo replications used to generate the boxplots; this detail is needed to assess the variability shown.
  3. [§6] The case-study section would benefit from a brief statement of the sample sizes and any preprocessing steps applied to the real data sets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below and indicate the changes we will make in the revised manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 2] §3, Theorem 2: the upper bound on the MISE is derived under Hölder smoothness of order β, but the matching minimax lower bound is only referenced rather than proved or sketched; without an explicit lower-bound argument or citation to a result that applies directly to the circle/sphere geometry, the rate-optimality claim remains incomplete.

    Authors: We appreciate the observation. The matching lower bound follows from standard minimax results for nonparametric density estimation on compact Riemannian manifolds of dimension 1 and 2 (circle and sphere), which apply directly under the Hölder smoothness class considered here. To strengthen the presentation and address the concern about self-containedness, we will add a concise sketch of the lower-bound argument in the revised Section 3, drawing on the manifold geometry without altering the main theorem statement. revision: yes

  2. Referee: [§4.1, Algorithm 1] §4.1, Algorithm 1: the computational complexity is stated as O(n) for the estimator itself, yet the bandwidth selection step (cross-validation or plug-in) is not analyzed; if this step requires O(n²) operations or iterative optimization whose cost grows with n, the central claim of 'direct implementation' without hidden n-dependent costs would be undermined.

    Authors: We agree that the complexity claim should explicitly cover bandwidth selection. The stated O(n) complexity applies to density evaluation once the bandwidth is fixed. Our plug-in bandwidth selector relies on direct kernel summations and moment computations that remain O(n) overall; no quadratic pairwise operations or n-dependent iterative optimization are required. In the revision we will add a dedicated paragraph in Section 4.1 (and the associated algorithm description) that states and justifies the linear complexity of the full procedure, including selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a standard nonparametric density estimation framework on the circle and sphere, claiming rate-optimal estimators under conventional smoothness assumptions together with direct computational implementability, followed by closed-form region probability expressions and simulation validation. No load-bearing derivation step reduces by construction to a fitted input, self-definition, or self-citation chain that imports uniqueness or ansatz without external grounding; the central claims retain independent theoretical and empirical content separate from the method's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited visibility into technical assumptions. The work implicitly relies on standard smoothness conditions for the unknown density to attain rate-optimality and on the data being i.i.d. samples from that density. No free parameters, invented entities, or ad-hoc axioms are explicitly introduced in the provided text.

axioms (2)
  • domain assumption The unknown density belongs to a smoothness class that permits rate-optimal estimation on the circle and sphere.
    Rate-optimality claims in nonparametric estimation typically require Hölder or Sobolev smoothness; this is invoked to justify the convergence rates but is not stated explicitly in the abstract.
  • standard math Observations are independent and identically distributed draws from the target density.
    Standard assumption for density estimation; required for the consistency and rate results to hold.

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

  1. [1]

    (2009), ‘ Adaptive density estimation for directional data using needlets’,The Annals of Statistics, 37 (6A), 3362– 3395

    Baldi, P ., Kerkyacharian, G., Marinucci, D., and Picard, D. (2009), ‘ Adaptive density estimation for directional data using needlets’,The Annals of Statistics, 37 (6A), 3362– 3395

    work page 2009

  2. [2]

    (2009), ‘ Asymptotics for spherical needlets’,The Annals of Statistics, 37 (3), 1150–1171

    Baldi, P ., Kerkyacharian, G., Marinucci, D., and Picard, D. (2009), ‘ Asymptotics for spherical needlets’,The Annals of Statistics, 37 (3), 1150–1171

    work page 2009

  3. [3]

    J., Fisher N

    Best, D. J., Fisher N. I. (1979), ‘Efficient Simulation of the von Mises Distribution’ Journal of the Royal Statistical Society, Series C (Applied Statistics), Vol. 28, No. 2 (1979), pp. 152-157

    work page 1979

  4. [4]

    (1979), ‘Estimation des densités: risque minimax’ (French) Z

    Bretagnolle, J., and Huber, C. (1979), ‘Estimation des densités: risque minimax’ (French) Z. Wahrsch. Verw. Gebiete, 47 (2), 119-137

    work page 1979

  5. [5]

    (2017), ‘Density estimation on manifolds with boundary’,Comput

    Berry, T., Sauer, T. (2017), ‘Density estimation on manifolds with boundary’,Comput. Stat. Data Anal., 107, 1-17

    work page 2017

  6. [6]

    (2014), ‘Model selection for density estimation with𝐿 2-loss’,Probab

    Birgé L. (2014), ‘Model selection for density estimation with𝐿 2-loss’,Probab. Theory Relat. Fields, 158, 533-574

    work page 2014

  7. [7]

    Bouzebda, S. & Taachouche, N., ‘Oracle inequalities an upper bounds for kernel con- ditional U-statistics estimators on manifolds and more general metric spaces associated with operators’,Stochastics, 96(8), 2135-2198

    work page

  8. [8]

    Pixel personality for dense object tracking in a 2D honeybee hive

    Bozek, K., Hebert, L., Mikheyev, A. S., and Stephens, G. J. (2018), ‘Pixel personality for dense object tracking in a 2D honeybee hive’arXiv preprintarXiv:1812.11797v1

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    Bozek, K., Hebert, L., Portugal Y ., and Stephens, G. J. (2021), ‘Markerless tracking of an entire honey bee colony’,Nature communications12 (1), 1733

    work page 2021

  10. [10]

    G., Kerkyacharian, G., Petrushev, P ., and Picard, D

    Cleanthous, G., Georgiadis, A. G., Kerkyacharian, G., Petrushev, P ., and Picard, D. (2020), ‘Kernel and wavelet density estimators on manifolds or more general metric spaces’.Bernoulli, 26 (3), 1832-1862. Nonparametric estimation on the circle and the sphere31

    work page 2020

  11. [11]

    Cleanthous

    G. Cleanthous. A.G. Georgiadis. O.V . Lepski. ’ Adaptive estimation of theL2-norm of a probability density and related topics I. Lower bounds’. Ann. Statist. 53 (3) 1257 - 1274, June 2025

    work page 2025

  12. [12]

    Cleanthous

    G. Cleanthous. A.G. Georgiadis. O.V . Lepski. ’ Adaptive estimation of theL 2-norm of a probability density and related topics II. Upper bounds via the oracle approach’. Ann. Statist. 53 (3) 1275 - 1297, June 2025

    work page 2025

  13. [13]

    (2022), ’Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces’

    Cleanthous, G., Georgiadis, A.G., Porcu, E. (2022), ’Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces’. Journal of Nonparametric Statistics, Volume 34, issue 4, 734-757

    work page 2022

  14. [14]

    Cleanthous, G., Georgiadis, White, P . A.. (2025), ’ Pointwise density estimation on metric spaces and applications in seismology’.Metrika, Volume 88, Issue 2, 119-148 (2025)

    work page 2025

  15. [15]

    (2012), ‘Heat Kernel Generated Frames in the Setting of Dirichlet Spaces’.Journal of Fourier Analysis and Applications, 18 (5), 995–1066

    Coulhon, T., Kerkyacharian, G., and Petrushev, P . (2012), ‘Heat Kernel Generated Frames in the Setting of Dirichlet Spaces’.Journal of Fourier Analysis and Applications, 18 (5), 995–1066

    work page 2012

  16. [16]

    Dai, F., Xu, Y ., (2013), ‘ Approximation theory and harmonic analysis on spheres and balls’.Springer Monographs in Mathematics, Springer

    work page 2013

  17. [17]

    (1985), ‘Nonparametric Density Estimation: The𝐿 1 View’

    Devroye, L., and Györfi, L. (1985), ‘Nonparametric Density Estimation: The𝐿 1 View’. Wiley, New York

    work page 1985

  18. [18]

    (1996), ‘ A universally acceptable smoothing factor for kernel density estimation’.The Annals of Statistics, 24 , 2499-2512

    Devroye, L., and Lugosi, L. (1996), ‘ A universally acceptable smoothing factor for kernel density estimation’.The Annals of Statistics, 24 , 2499-2512

    work page 1996

  19. [19]

    (1997), ’Nonasymptotic universal smoothing factors, kernel complexity and Y atracos classes’.The Annals of Statistics, 25, 2626-2637

    Devroye, L., and Lugosi, L. (1997), ’Nonasymptotic universal smoothing factors, kernel complexity and Y atracos classes’.The Annals of Statistics, 25, 2626-2637

    work page 1997

  20. [20]

    L., Johnstone, I

    Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1996), ‘Density estimation by wavelet thresholding’.The Annals of Statistics, 24, 508-539

    work page 1996

  21. [21]

    Efroimovich, S. Yu. (1986), ‘Non-parametric estimation of the density with unknown smoothness’.The Annals of Statistics, 36, 1127-1155

    work page 1986

  22. [22]

    K., Marinucci, D., Kerkyacharian, G., and Picard, D

    Geller, D., Hansen, F. K., Marinucci, D., Kerkyacharian, G., and Picard, D. (2008), ‘Spin needlets for cosmic microwave background polarization data analysis’,Physical Review D, 78 (12), 123533

    work page 2008

  23. [23]

    A. G. Georgiadis and G. Kyriazis, Embeddings between Triebel–Lizorkin spaces on metric spaces associated with operators,Anal. Geom. Metric Spaces, 2020,8(1): 418–429

    work page 2020

  24. [24]

    A. G. Georgiadis, M. Nielsen, Spectral multipliers on spaces of distributions associated with non-negative self-adjoint operators, J. Approx. Theory, 234 (2018), 1–19

    work page 2018

  25. [25]

    (2014), ‘On adaptive minimax density estimation on R𝑑’.Probability Theory and Related Fields, 159, 479-543

    Goldenshluger, A., and Lepski, O. (2014), ‘On adaptive minimax density estimation on R𝑑’.Probability Theory and Related Fields, 159, 479-543

    work page 2014

  26. [26]

    (2011a), ‘Uniform bounds for norms of sums of independent random functions’.The Annals of Probability, 39, 2318-2384

    Goldenshluger, A., and Lepski, O. (2011a), ‘Uniform bounds for norms of sums of independent random functions’.The Annals of Probability, 39, 2318-2384

    work page

  27. [27]

    (2011b), ‘Bandwidth selection in kerrnel density estimation: oracle inequalities and adaptive minimax optimality’.The Annals of Statistics, 39, 1608-1632

    Goldenshluger, A., and Lepski, O. (2011b), ‘Bandwidth selection in kerrnel density estimation: oracle inequalities and adaptive minimax optimality’.The Annals of Statistics, 39, 1608-1632

    work page

  28. [28]

    and Lepski, O

    Goldenshluger, A. and Lepski, O. (2022), ‘Minimax estimation of norms of a probability density: I. Lower bounds’. Bernoulli, 28 (2), 1120-1154

    work page 2022

  29. [29]

    and Lepski, O

    Goldenshluger, A. and Lepski, O. (2022), ‘Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures’. Bernoulli, 28 (2), 1155-1178. 32A. G. Georgiadis and A. P . Percival

    work page 2022

  30. [30]

    S., and Cabrera, J

    Hall, P ., Watson, G. S., and Cabrera, J. (1987), ‘Kernel Density Estimation with Spherical Data’.Biometrika, 74(4):751–762

    work page 1987

  31. [31]

    (1998), ‘Wavelets, ap- proximation, and statistical applications’.Lecture Notes in Statistics, 129

    Härdle, W ., Kerkyacharian, G., Picard, D., and Tsybakov, A.B. (1998), ‘Wavelets, ap- proximation, and statistical applications’.Lecture Notes in Statistics, 129. Springer-Verlag, New Y ork

    work page 1998

  32. [32]

    and Ibragimov, I.A

    Hasminskii, R. and Ibragimov, I.A. (1990), ‘On density estimation in the view of Kol- mogorov’s ideas in approximation theory’.The Annals of Statistics, 18, 999-1010

    work page 1990

  33. [33]

    (1980), ‘ An estimate of the density of a distribu- tion’

    Ibragimov, I.A., and Khasminski, R.Z. (1980), ‘ An estimate of the density of a distribu- tion’. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 98, 61-85

    work page 1980

  34. [34]

    (2004), ‘On minimax density estimation onR’, Bernoulli, 10(2), 187–220

    Juditsky, A., and Lambert-Lacroix, S. (2004), ‘On minimax density estimation onR’, Bernoulli, 10(2), 187–220

    work page 2004

  35. [35]

    (2018), ‘Regularity of Gaussian processes on Dirichlet spaces’,Constructive Approximation, 47(2), 277–320

    Kerkyacharian, G., Ogawa, S., Petrushev, P ., and Picard, D. (2018), ‘Regularity of Gaussian processes on Dirichlet spaces’,Constructive Approximation, 47(2), 277–320

    work page 2018

  36. [36]

    (2015), ‘Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces’,Transactions of the American Mathematical Society, 367, 121–189

    Kerkyacharian, G., and Petrushev, P . (2015), ‘Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces’,Transactions of the American Mathematical Society, 367, 121–189

    work page 2015

  37. [37]

    (1996), ‘𝐿𝑝 adaptive density estimation’, Bernoulli, 2, 229–247

    Kerkyacharian, G., Picard, D., and Tribouley, K. (1996), ‘𝐿𝑝 adaptive density estimation’, Bernoulli, 2, 229–247

    work page 1996

  38. [38]

    (2001), ‘Nonlinear estimation in anisotropic multi-index denoising’,Probability Theory and Related Fields, 121, 137– 170

    Kerkyacharian, G., Lepski, O., and Picard, D. (2001), ‘Nonlinear estimation in anisotropic multi-index denoising’,Probability Theory and Related Fields, 121, 137– 170

    work page 2001

  39. [39]

    (2008), ‘Nonlinear estimation in anisotropic multiindex denoising

    Kerkyacharian, G., Lepski, O., and Picard, D. (2008), ‘Nonlinear estimation in anisotropic multiindex denoising. Sparse case’,Theory of Probability and its Applications, 52, 58–77

    work page 2008

  40. [40]

    V ., and Jupp, P

    Mardia, K. V ., and Jupp, P . E., (1999),Directional Statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, London

    work page 1999

  41. [41]

    Muller, M. E. (1956). Some continuous Monte Carlo methods for the Dirichlet problem. Annals of Mathematical Statistics,27(3), 569–589

    work page 1956

  42. [42]

    (1962), ‘On the estimation of a probability density function and mode’.Annals of Mathematical Statistics, 33, 1065-1076

    Parzen, E. (1962), ‘On the estimation of a probability density function and mode’.Annals of Mathematical Statistics, 33, 1065-1076

    work page 1962

  43. [43]

    (2005), ‘Kernel density estimation on Riemannian manifolds’,Statistics and probability letters, 73(3), 297–304

    Pelletier, B. (2005), ‘Kernel density estimation on Riemannian manifolds’,Statistics and probability letters, 73(3), 297–304

    work page 2005

  44. [44]

    (2006), ‘Non-parametric regression estimation on closed Riemannian man- ifolds’,Journal of Nonparametric Statistics, 18(1), 57–67

    Pelletier, B. (2006), ‘Non-parametric regression estimation on closed Riemannian man- ifolds’,Journal of Nonparametric Statistics, 18(1), 57–67

    work page 2006

  45. [45]

    (1972),Methods of modern mathematical physics

    Reed, M., and Simon, B. (1972),Methods of modern mathematical physics. I. Functional analysis, New Y ork: Academic Press

    work page 1972

  46. [46]

    (2006), ‘ Adaptive density estimation using the blockwise Stein method’, Bernoulli, 12, 351–370

    Rigollet, Ph. (2006), ‘ Adaptive density estimation using the blockwise Stein method’, Bernoulli, 12, 351–370

    work page 2006

  47. [47]

    (2007), ‘Linear and convex aggregation of density estimators’,Mathematical Methods of Statistics, 16, 260–280

    Rigollet, Ph., and Tsybakov, A.B. (2007), ‘Linear and convex aggregation of density estimators’,Mathematical Methods of Statistics, 16, 260–280

    work page 2007

  48. [48]

    (1956), ‘Remarks on some nonparametric estimates of a density function’, Annals of Mathematical Statistics, 27, 832-837

    Rosenblatt, M. (1956), ‘Remarks on some nonparametric estimates of a density function’, Annals of Mathematical Statistics, 27, 832-837

    work page 1956

  49. [49]

    (2007), ‘ Aggregation of density estimators and di- mension reduction’, Advances in Statistical Modeling and Inference, pp

    Samarov, A., and Tsybakov, A.B. (2007), ‘ Aggregation of density estimators and di- mension reduction’, Advances in Statistical Modeling and Inference, pp. 233-251, Ser. Biostat., Vol. 3. World Sci. Publ., Hackensack (2007). Nonparametric estimation on the circle and the sphere33

    work page 2007

  50. [50]

    (1986),Density estimation for statistics and data analysis, London: Monographs on Statistics and Applied Probability, Chapman & Hall

    Silverman, B.W . (1986),Density estimation for statistics and data analysis, London: Monographs on Statistics and Applied Probability, Chapman & Hall

    work page 1986

  51. [51]

    Tsybakov, A.B. (V . Zaiats, trans.) (2009),Introduction to nonparametric estimation, Springer Series in Statistics, New Y ork: Springer

    work page 2009

  52. [52]

    (1999),Fourier acoustics : sound radiation and nearfield acoustical holography

    Williams, Earl G. (1999),Fourier acoustics : sound radiation and nearfield acoustical holography. San Diego, California: Academic Press

    work page 1999

  53. [53]

    Wood, Andrew T. A. (1994),Simulation of the von mises fisher distribution, Communi- cations in Statistics - Simulation and Computation, 23:1, 157-164

    work page 1994