On nonparametric ridge estimation for multivariate long-memory processes
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Lithuanian Mathematical Journal
On nonparametric ridge estimation for multivariate long-memory processes Jan Beran a and Klaus Telkmann a,b a
Department of Mathematics and Statistics, University of Konstanz, Universitaetsstrasse 10, 78457 Konstanz, Germany b Department of Statistics, Donald Bren School of Information and Computer Sciences, University of California, Irvine, 6210 Donald Bren Hall Irvine, CA 92697-3425, USA (e-mail: [email protected])
Received November 1, 2018; revised November 11, 2019
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. MSC: 62M10, 62G05, 62G15, 62G07, 60G18, 60F17 Keywords: kernel density estimation, linear process, long-range dependence, multivariate time series, ridge
1 Introduction Let μ1 , . . . , μn be an i.i.d. sample generated by a probability distribution with density pμ that has a compact support M ⊂ Rm . Furthermore, let X1 , . . . , Xn be generated by an m-dimensional linear process Xt =
∞
Aj εt−j
(t ∈ Z)
(1.1)
j=0
with probability density function pX , where εt = (εt,1 , . . . , εt,m )T ∈ Rm denote i.i.d. zero mean random vectors, and Aj are suitable m × m-matrices. Define the process Yt = (Yt,1 , . . . , Yt,m )T = μt + Xt
(t ∈ Z)
(1.2)
with corresponding probability density function pY = pμ pX , where denotes convolution. For k < m, the k-dimensional ridge of pY is the set of points that are local maxima of pY in at least m−k directions. In this paper, we consider kernel estimation of the ridge under long-memory assumptions. We use the i.i.d. assumption on μt for simplicity of presentation. Analogous results can be derived under more general conditions, including correlated or deterministic locations μt ∈ M . In contrast to standard smoothing methods in time series c 2020 Springer Science+Business Media, LLC 0363-1672/20/6002-0001
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J. Beran and K. Telkmann
analysis, the method developed in this paper is very general in the sense that the order in which μt traverses M does not have to be known. Processes defined by (1.2) occur, for instance, in spatio-temporal remote sensing where temporal correlations can be observed even at the level of individual pixels (see, e.g., [35] and references therein). A much discussed issue is, for example, the statical analysis of time series of the so-called Synthetic Aperture Radar (SAR) satellite data (see, e.g., [32]). Other applications, possibly with modified conditions on μt , include, for instance, dynamic systems with random perturbations Xt . For example, in [30] parameter estimation in m-dimensional ordinary differential equation (ODE) models is studied, where μt follows an ODE with unknown parameters, and observations are of the form Yt = μt + Xt with i.i.d. errors Xt
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