Abstract
We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive functional limit theorems for estimated eigenvectors and eigenvalues of the Hessian matrix. We use these results to obtain the weak convergence for the estimated ridge and asymptotic simultaneous confidence regions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, CRC Press, New York, 1994.
J. Beran and Y. Feng, SEMIFARmodels – A semiparametric framework formodelling trends, long-range dependence and nonstationarity, Comput. Stat. Data Anal., 40(2):393–419, 2002.
J. Beran and Y. Feng, Data driven bandwidth choice for SEMIFAR models, J. Comput. Graph. Stat., 11(2):690–713, 2002b.
J. Beran, Y. Feng, S. Ghosh, and R. Kulik, Long-Memory Processes, Springer, New York, 2013.
J. Beran and N. Schumm, On non parametric statistical inference for densities under long-range dependence, Commun. Stat., Theory Methods, 46(22):11296–11314, 2017.
J. Beran and K. Telkmann, On nonparametric density estimation for multivariate linear long-memory processes., Commun. Stat., Theory Methods, 47(22):5460–5473, 2018.
J.E. Chacón and T. Duong, Multivariate Kernel Smoothing and Its Applications, Chapman & Hall, CRC Press, New York, 2018.
J.E. Chacón, T. Duong, and M.P. Wand, Asymptotics for general multivariate kernel density derivative estimators, Stat. Sin., 21(2):807–840, 2011.
F. Chazal, D. Cohen-Steiner, and Q. Mérigot, Geometric inference for probability measures, Found. Comput. Math., 11(6):733–751, 2011.
Y.-C. Chen, C.R. Genovese, and L. Wasserman, Asymptotic theory for density ridges, Ann. Stat., 43(5):1896–1928, 10 2015.
Y.C. Chen, C.R. Genovese, S. Ho, and L. Wasserman, Optimal ridge detection using coverage risk, in Advances in Neural Information Processing Systems 28 (NIPS 2015), Neural Information Processing Systems Foundation, 2015, pp. 316–324.
Y.C. Chen, C.R. Genovese, and L.A. Wasserman, Generalized mode and ridge estimation, 2014, arXiv:abs/1406.1803.
C.-F. Chung, Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes, Econom. Theory, 18(1):51–78, 2002.
S. Csorgo and J. Mielniczuk, Density estimation under long-range dependence, Ann. Stat., 23(3):990–999, 1995.
P. Doukhan, G. Oppenheim, and M.S. Taqqu, Ridges in Image and Data Analysis, Springer, Dordrecht, 1996.
P. Doukhan, G. Oppenheim, and M.S. Taqqu (Eds.), Theory and Application of Long-RangeDependence, Birkhäuser, Basel, 2003.
D. Eberly, R. Gardner, B. Morse, S. Pizer, and C. Scharlach, Ridges for image analysis, J. Math. Imaging Vis., 4(4): 353–373, December 1994, ISSN 0924-9907.
C.R. Genovese, M. Perone-Pacifico, I. Verdinelli, and L. Wasserman, Nonparametric ridge estimation, Ann. Stat., 42(4):1511–1545, 2014.
S. Ghosh, Kernel Smoothing, Wiley, New York, 2018.
L. Giraitis, H.L. Koul, and D. Surgailis, Large Sample Inference for Long Memory Processes, Imperial College Press, London, 2012.
A. Gramacki, Nonparametric Kernel Density Estimation and Its Computational Aspects, Springer, New York, 2018.
P. Hall and J.D. Hart, Convergence rates in density estimation for data from infinite-order moving average processes, Probab. Theory Relat. Fields, 87(2):253–274, 1990.
P. Hall, B.-Y. Jing, and S.N. Lahiri, On the sampling window method for long-range dependent data, Stat. Sin., 8(4): 1189–1204, 1998.
P. Hall, W. Qian, and D.M. Titterington, Ridge finding from noisy data, J. Comput. Graph. Stat., 1(3):197–211, 1992.
R.M. Haralick, Ridges and valleys on digital images, Comput. Vis. Graph. Image Process., 22(1):28 – 38, 1983.
I. Horová, J. Kolácek, and J. Zelinka, Kernel Smoothing In Matlab, World Scientific, River Edge, NJ, 2012.
S. Kechagias and V. Pipiras, Definitions and representations of multivariate long-range dependent time series, J. Time Ser. Anal., 36(1):1–25, 2015.
T. Kollo and D. von Rosen, Advanced Multivariate Statistics with Matrices, Springer, Dordrecht, 2005.
S.N. Lahiri, On the moving block bootstrap under long range dependence, Stat. Probab. Lett., 18(5):405–413, 1993.
H. Liang and H. Wu, Parameter estimation for differential equation models using a framework of measurement error in regression models, J. Am. Stat. Assoc., 103(484):1570–1583, 2008.
T. Lindeberg, Edge detection and ridge detection with automatic scale selection, Int. J. Comput. Vis., 30(2):117–156, November 1998.
M. Lu, E. Pebesma, A. Sánchez, and J. Verbesselt, Spatio-temporal change detection from multidimensional arrays: Detecting deforestation from MODIS time series, ISPRS J. Photogramm. Remote Sens., 117:227–236, 2016.
X. Magnus and H. Neudecker, Matrix Differential Calculus, Wiley, New Jersey, 1998.
D. Marinucci and P.M. Robinson, Weak convergence of multivariate fractional processes, Stochastic Processes Appl., 86(1):103 – 120, 2000.
A.F. Militino, M.D. Ugarte, and U. Pérez-Goya, An introduction to the spatio-temporal analysis of satellite remote sensing data for geostatisticians, in B.S. Daya Sagar, Q. Cheng, and F. Agterberg (Eds.), Handbook of Mathematical Geosciences: Fifty Years of IAMG, Springer, Cham, 2018, pp. 239–253.
Y. Nakatsukasa, Perturbation behavior of a multiple eigenvalue in generalized Hermitian eigenvalue problems, BIT, 50(1):109–121, 2010.
G. Norgard and P.-T. Bremer, Ridge-valley graphs: Combinatorial ridge detection using Jacobi sets, Comput. Aided Geom. Des., 30:597–608, 2013.
W. Qiao and W. Polonik, Theoretical analysis of nonparametric filament estimation, Ann. Stat., 44:1269–1297, 2016.
D.W. Scott, Multivariate Density Estimation, Wiley, New Jersey, 2015.
G.W. Stewart and J.G. Sun, Matrix Perturbation Theory, Academic Press, Cambridge,MA, 1990.
J.-G. Sun, Multiple eigenvalue sensitivity analysis, Linear Algebra Appl., 137–138:183–211, 1990.
M.P. Wand and M.C. Jones, Kernel Smoothing, Chapman & Hall, CRC Press, New York, 1995.
E.J.Wegman and Q. Luo, On methods of computer graphics for visualizing densities, J. Comput. Graph. Stat., 11(1): 137–162, 2002.
W.B. Wu and J. Mielniczuk, Kernel density estimation for linear processes, Ann. Stat., 30(5):1441–1459, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Beran, J., Telkmann, K. On nonparametric ridge estimation for multivariate long-memory processes. Lith Math J 60, 291–314 (2020). https://doi.org/10.1007/s10986-020-09480-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-020-09480-y