PreviousCitations

UpNextFrom References: 0

From Reviews: 0MR4508191 62G07 62G20
Jeon, Jeong Min (B-KUL-ORS) ; Van Keilegom, Ingrid (B-KUL-ORS)
Density estimation for mixed Euclidean and non-Euclidean data in the presenceof measurement error. (English summary)J. Multivariate Anal. 193 (2023), Paper No. 105125, 24 pp.

Recently non-Euclidean data (e.g., circular data, spherical data) have gathered interestin statistics. As with Euclidean data, non-Euclidean data sometimes contain measurement errors. This article is concerned with the problem of estimating a joint density forcontaminated Euclidean and non-Euclidean data.
Suppose that we are interested in estimating the unknown joint density ofX = (X1 , X2 ) nonparametrically, where X1 ∈ Rd and X2 ∈ GC are Euclidean and nonEuclidean components, respectively, and where GC is a finite-dimensional compact andconnected Lie group equipped with a group operation ◦. Also suppose that we can only
observe contaminated data Z = (Z1 , Z2 ) = (U1 + X1 , U2 ◦ X2 ), where U = (U1 , U2 ) isan unobservable measurement error that is assumed to satisfy X ⊥ U and U1 ⊥ U2 .
For identification purposes of the density of X, it is also assumed that distributions ofboth U1 and U2 are known. Under this setup, a new deconvolution density estimatoris proposed. The estimator consists of a deconvolution kernel density estimator for theEuclidean component and a Fourier series estimator for the non-Euclidean component.
The weight function that constructs the deconvolution density estimator is shown tohave the unbiased scoring property, which eliminates the effect of the measurement
error U in the bias part.
Asymptotic properties of the deconvolution density estimator are explored extensively. These include rates of convergence for the cases of L2 , pointwise and uniform
convergences and asymptotic normality depending on a few different smoothness scenarios. New proof techniques based on Fourier analysis on topological groups that areused for derivations of such convergence results are also of independent interest. Usefulness of the estimator is illustrated in the simulation study and a real data analysis on
the joint density estimation for bivariate mismeasured data of the level of sulfur dioxide(SO2 ) and wind direction.
Masayuki Hirukawa
References

1. D.N. Anderson, A multivariate Linnik distribution, Statist. Probab. Lett. 14 (1992)
333–336. MR1179637
2. D. Applebaum, Probability on Compact Lie Groups, Springer International Publishing Switzerland, 2014. MR3243650
3. D. Belomestny, A. Goldenshluger, Density deconvolution under general assumptions on the distribution of measurement errors, Ann. Statist. 49 (2021) 615–649.
MR4255101
4. V.N. Berestovskiı̆, V.M. Svirkin, The Laplace operator on normal homogeneous
Riemannian manifolds, Siberian Adv. Math. 20 (2010) 231–255. MR2599423
5. A. Bertrand, I. Van Keilegom, C. Legrand, Flexible parametric approach to classicalmeasurement error variance estimation without auxiliary data, Biometrics 75 (2019)297–307. MR3953730
6. A. Bhattacharya, D. Dunson, Nonparametric Bayesian density estimation on mani-folds with applications to planar shapes, Biometrika 97 (2011) 851–865. MR2746156
7. I. Dattner, A. Goldenshluger, A. Juditsky, On deconvolution of distribution functions, Ann. Statist. 39 (2011) 2477–2501. MR2906875
8. I. Dattner, M. Reiss, M. Trabs, Adaptive quantile estimation in deconvolution withunknown error distribution, Bernoulli 22 (2016) 143–192. MR3449779
9. A. Delaigle, P. Hall, On optimal kernel choice for deconvolution, Statist. Probab.
Lett. 76 (2006) 1594–1602. MR2248846
10. A. Delaigle, P. Hall, A. Meister, On deconvolution with repeated measurements,
Ann. Statist. 36 (2008) 665–685. MR2396811
11. P. Diaconis, Group Representations in Probability and Statistics, in: Volume 11 ofInstitute of Mathematical Statistics Lecture Notes -Monograph Series, Institute ofMathematical Statistics, 1988. MR0964069
12. S. Efromovich, Density estimation for the case of supersmooth measurement error,
J. Amer. Statist. Assoc. 92 (1997) 526–535. MR1467846
13. P. Etingof, O. Goldberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, Introduction to Representation Theory, American Mathematical Society, 2011.
MR2808160
14. J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Statist. 19 (1991a) 1257–1272. MR1126324
15. J. Fan, Asymptotic normality for deconvolution kernel density estimators, Sankhya
53 (1991b) 97–110. MR1177770
16. J. Fan, Y.K. Truong, Nonparametric regression with errors in variables, Ann. Statist.
21 (1993) 1900–1925. MR1245773
17. J. Faraut, Analysis on Lie Groups, Cambridge University Press, 2008. MR2426516
18. G.B. Folland, A Course in Abstract Harmonic Analysis, Chapman and Hall/CRC,
Boca Raton, 2016. MR3444405
19. E. Garcı́a-Portugués, Exact risk improvement of bandwidth selectors for kerneldensity estimation with directional data, Electron. J. Stat. 7 (2013) 1655–1685.
MR3070874
20. E. Garcı́a-Portugués, R.M. Crujeiras, W. González-Manteiga, Exploring wind direction and SO2 concentration by circular-linaer density estimation, Stoch. Environ.
Res. Risk Assess. 27 (2013a) 1055–1067.
21. E. Garcı́a-Portugués, R.M. Crujeiras, W. González-Manteiga, Kernel density estimation for directional-linear data, J. Multivariate Anal. 121 (2013b) 152–175.
MR3090475
22. E. Garcı́a-Portugués, R.M. Crujeiras, W. González-Manteiga, Central limit theorems for directional and linear random variables with applications, Statist. Sinica
25 (2015) 1207–1229. MR3410305
23. D.M. Healy, H. Hendriks, P.T. Kim, Spherical deconvolution, J. Multivariate Anal.
67 (1998) 1–22. MR1659108
24. H. Hendriks, Nonparametric estimation of a probability density on a Riemannian
manifold using Fourier expansions, Ann. Statist. 18 (1990) 832–849. MR1056339
25. R. Hielscher, Kernel density estimation on the rotation group and its application
to crystallographic texture analysis, J. Multivariate Anal. 119 (2013) 119–143.
MR3061419
26. S. Huckemann, P.T. Kim, J.-Y. Koo, A. Munk, Möbius deconvolution on the hyperbolic plane with application to impedance density estimation, Ann. Statist. 38(2010) 2465–2498. MR2676895
27. J.M. Jeon, B.U. Park, I. Ingrid, Additive regression for non-Euclidean responses
and predictors, Ann. Statist. 49 (2021) 2611–2641. MR4338377
28. J.M. Jeon, B.U. Park, I. Ingrid, Nonparametric regression on Lie groups with

measurement errors, Ann. Statist. 50 (2022) 2973–3008. MR4500632
29. J. Johannes, Deconvolution with unknown measurement error distribution, Ann.
Statist. 37 (2009) 2301–2323. MR2543693
30. J. Johannes, M. Schwarz, Adaptive circular deconvolution by model selection under
unknown error distribution, Bernoulli 19 (2013) 1576–1611. MR3129026
31. R.A. Johnson, T.E. Wehrly, Some angular-linear distributions and related regression
models, J. Amer. Statist. Assoc. 73 (1978) 602–606. MR0514163
32. J. Jost, Riemannian Geometry and Geometric Analysis, Springer Berlin, Heidelberg,
2011. MR2829653
33. J. Kappus, G. Mabon, Adaptive density estimation in deconvolution problems withunknown error distribution, Electron. J. Stat. 8 (2014) 2879–2904. MR3299125
34. Y. Katznelson, An Introduction to Harmonic Analysis, Cambridge University Press,
2004. MR2039503
35. P.T. Kim, Deconvolution density estimation on SO(N), Ann. Statist. 26 (1998)
1083–1102. MR1635446
36. P.T. Kim, J.-Y. Koo, Asymptotic minimax bounds for stochastic deconvolution overgroups, IEEE Trans. Inform. Theory 54 (2008) 289–298. MR2446754
37. P.T. Kim, D.St.P. Richards, Deconvolution density estimation on compact Liegroups, Contemp. Math. 287 (2001) 155–171. MR1873674
38. S. Kotz, N. Balakrishnan, N.L. Johnson, Continuous Multivariate Distributions,
Volume 1: Models and Applications, Wiley-Interscience, 2019. MR2424354
39. S. Kotz, T.J. Kozubowski, K. Podgórski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineeringand Finance, Birkhäuser, Boston, 2001. MR1935481
40. S. Kotz, S. Nadarajah, Multivariate T-Distributions and their Applications, Cambridge University Press, 2004. MR2038227
41. A.S. Krishnamoorthy, M. Parthasarathy, A multivariate gamma-type distribution,
Ann. Math. Stat. 22 (1951) 549–557. MR0044790
42. C.A. Leó, J.-C. Massé, L.-P. Rivest, A statistical model for random rotations, J.
Multivariate Anal. 97 (2006) 412–430. MR2234030
43. M. Lesosky, P.T. Kim, D.W. Kribs, Regularized deconvolution on the 2D-euclidean
motion group, Inverse Problems 24 (2008) 055017. MR2438952
44. Z.-M. Luo, P.T. Kim, T.Y. Kim, J.-Y. Koo, Deconvolution on the euclidean motion
group SE(3), Inverse Problems 27 (2011) 035014. MR2772533
45. K.V. Mardia, P.E. Jupp, Directional Statistics, John Wiley & Sons, Inc, 1999.
MR1828667
46. J.S. Marron, A.M. Alonso, Overview of object oriented data analysis, Biometical J.
5 (2014) 732–753. MR3258083
47. E. Masry, Strong consistency and rates for deconvolution of multivariate densitiesof stationary processes, Stochastic Process. Appl. 47 (1993a) 53–74. MR1232852
48. E. Masry, Asymptotic normality for deconvolution estimators of multivariate densities of stationary processes, J. Multivariate Anal. 44 (1993b) 47–68. MR1208469
49. A. Meister, Deconvolution Problems in Nonparametric Statistics, Springer-Verlag
Berlin Heidelberg, 2009. MR2768576
50. D.F. Myers, Pointwise and uniform convergence of Fourier series on SU(2), (Ph.D.
thesis), Missouri University of Science and Technology, 2016. MR3597695
51. S.J. Nadarajah, Y. Zhang, Wrapped: An R package for circular data, PLoS ONE 12(2017) e0188512.
52. M. Oliveira, R.M. Crujeiras, A. Rodrı́guez-Casal, A plug-in rule for bandwidthselection in circular density estimation, Comput. Statist. Data Anal. 56 (2012)
3898–3908. MR2957840

53. A. Pewsey, E. Garcı́a-Portugués, Recent advances in directional statistics, Test 30(2021) 1–58. MR4242171
54. Y. Qui, D.J. Nordman, S.B. Vardeman, A wrapped trivariate normal distribution
and Bayes inference for 3-D rotations, Statist. Sinica 24 (2014) 897–917. MR3235404
55. M. Schwarz, S. Van Bellegem, Consistent density deconvolution under partiallyknown error distribution, Statist. Probab. Lett. 80 (2010) 236–241. MR2575451
56. T. Sei, H. Shibata, A. Takemura, K. Ohara, N. Takayama, Properties and applications of Fisher distribution on the rotation group, J. Multivariate Anal. 116 (2013)
440–445. MR3049915
57. J. Söhl, M. Trabs, A uniform central limit theorem and efficiency for deconvolution
estimators, Electron. J. Stat. 6 (2012) 2486–2518. MR3020273
58. L.A. Stefanski, R.J. Carroll, Deconvolving kernel density estimators, Statistics 21(1990) 169–184. MR1054861
59. A. Vollrath, The nonequispaced fast SO(3) Fourier transform, generalisations andapplications, (Ph.D. thesis), Universität Lübeck, 2010.
60. P.L. Walker, Lipschitz classes on finite dimensional groups, Math. Proc. Camb. Phil.
Soc. 66 (1969) 31–38. MR0240565
61. W. Wang, T. Lee, Matrix Fisher-Gaussian distribution on SO(3) × Rn andBayesian attitude estimation, IEEE Trans. Automat. Control 67 (2022) 2175–2191.
MR4413693
Note: This list reflects references listed in the original paper as
accurately as possible with no attempt to correct errors.