Abstract
Using a wavelets-based estimator of the bivariate density, we introduce an estimation method for nonlinear canonical analysis. Consistency of the resulting estimators of the canonical coefficients and the canonical functions is established. Under some conditions, asymptotic normality results for these estimators are obtained. Then it is shown how to compute in practice these estimators by usingmatrix computations, and the finite-sample performance of the proposed method is evaluated through simulations.
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References
P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
A. Buja, “Remarks on Functional Canonical Variates, Alternating Least Squares Methods and ACE”, Ann. Statist. 18(3), 1032–1069 (1990).
S. Darolles, J.-P. Florens, and C. Gouriéroux, “Kernel Based Nonlinear Canonical Analysis and Time Reversibility”, J. Econom. 119(2), 323–353 (2004).
J. Dauxois and G. M. Nkiet, “Nonlinear Canonical Analysis and Independence Tests”, Ann. Statist. 26(4), 1254–1278 (1998).
J. Dauxois, and A. Pousse, “Une extension de l’analyse canonique. Quelques Applications”, Ann. Inst. H. Poincaré 11, 355–379 (1975).
J. Dauxois and A. Pousse, Some Convergence Problems in Factor Analysis, Ed. by Barra et al. in Recent Developments in Statist. (North-Holland, 1977).
Z. Ding and C. Granger, “Modelling Volatility Persistence of Speculative Returns: A New Approach” J. Econom. 73, 185–216 (1996).
L. T. Fernholz, Von Mises Calculus for Statistical Functionals in Lecture Notes in Statist. (Springer, 1983), Vol. 19.
C. Gouriéroux and J. Jasiak, “Nonlinear Autocorrelograms with Application to Intertrade Durations”, J. Time Ser. Anal. 23, 127–154 (2002).
C. Gouriéroux and J. Jasiak, “Multivariate Smooth Transitions Jacobi Process Transitions with Application”, J. Econom. 131, 475–505 (2005).
L. Hansen and J. Scheinkman, “Back to the Future: Generating Moment Implications for Continuous Time Markov Processes”, Econometrica 73, 767–807 (1995).
S. Y. Huang, “Density Estimation by Wavelet-Based Reproducing Kernels”, Statist. Sinica 9, 137–151 (1999).
M. Kessler and M. Sörensen, “Estimating Equations Based on Eigenfunctions for Discretely Observed Diffusion Process”, Bernoulli 5, 299–314 (1999).
K. Larsen and M. Sörensen, “Diffusion Models for Exchange Rates in a Target Zone”, Math. Finance 17, 285–306 (1999).
A. Massiani, “Etude asymptotique locale de l’estimateur par ondelettes”, C. R. Math. Acad. Sci. Paris 335, 553–556 (2002).
Y. Meyer, Ondelettes et opérateurs (Hermann, Paris, 1990), Vol. 1.
B. L. S. Prakasa Rao, “Nonparametric Estimation of Partial Derivatives of a Multivariate Probability Density by the Method of Wavelets”, in Asymptotics in Statistics and Probability. Festschrift for G. G. Roussas, Ed. by M. L. Puri (VSP, The Netherlands, 2000), pp. 321–330.
M. Vannucci, Nonparametric Density Estimation Using Wavelets, Discussion Paper 95-26 (ISDS, Duke University, USA, 1995).
B. Vidakovic, Statistical Modeling by Wavelets (Wiley, New York, 1999).
D.W. Wu, “Asymptotic Normality of the Multiscale Wavelet Density Estimator”, Commun. Statist. — Theory Meth. 25(9), 1957–1970 (1996).
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Niang, M.A., Nkiet, G.M. & Diop, A. Wavelets-based estimation of nonlinear canonical analysis. Math. Meth. Stat. 21, 215–237 (2012). https://doi.org/10.3103/S1066530712030039
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DOI: https://doi.org/10.3103/S1066530712030039