Limit theorems for Hilbert space-valued linear processes under long range dependence

Düker, Marie-Christine

doi:10.1016/j.spa.2017.07.015

Cited by 6 publications

(17 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is a positive slowly varying function (c.f., Bingham et al 1987) which depends on i. Our limit distribution theory is comparable to theorems in Characiejus & Rauckauskas (2014) and Düker (2018) which consider Hilbert space-valued long-range dependent linear processes with derivation heavily relying on the theory of multiplication operator. However, it seems challenging to directly apply the methodology in Section 3 below to their model framework, making it difficult to achieve dimension reduction via FPCA.…”

Section: Large-sample Propertiesmentioning

confidence: 98%

See 1 more Smart Citation

Long-Range Dependent Curve Time Series

Robinson

Shang

2019

Journal of the American Statistical Association

102

View full text Add to dashboard Cite

We introduce methods and theory for functional or curve time series with longrange dependence. The temporal sum of the curve process is shown to be asymptotically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run covariance function, which we use, via functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of the curves. In a semiparametric context, we propose an estimate of the memory parameter and establish its consistency. A Monte-Carlo study of finitesample performance is included, along with two empirical applications. The first of these finds a degree of stability and persistence in intra-day stock returns. The second finds similarity in the extent of long memory in incremental age-specific fertility rates across some developed nations.

show abstract

Section: Large-sample Propertiesmentioning

confidence: 98%

“…This assumption is restrictive, but, motivated by Characiejus & Rauckauskas (2014) and Düker (2018), using the multiplication operator, it may be possible to extend (4.1) to…”

Section: The Functional Farima Modelmentioning

confidence: 99%

Long-Range Dependent Curve Time Series

Robinson

Shang

2019

Journal of the American Statistical Association

102

View full text Add to dashboard Cite

show abstract

“…The bulk of the literature studies functional data which are either independent or stationary SRD (e.g., Bosq, 2000;Ramsay and Silverman, 2005;Ferraty and Vieu, 2006;Hörmann and Kokoszka, 2010;Horváth and Kokoszka, 2012;Berkes et al, 2013;Hsing and Eubank, 2015). Li et al (2020) is among the first to extend the functional framework from SRD to LRD (see also Characiejus and Rackauskas, 2014;Düker, 2018). They not only establish the central limit theorem for a temporal sum of LRD functional observations, but also develop functional principal component analysis (FPCA) and estimate the memory parameter for the projected process via semiparametric R/S.…”

Section: Introductionmentioning

confidence: 99%

Local Whittle estimation of long‐range dependence for functional time series

Robinson

Shang

2021

Journal Time Series Analysis

View full text Add to dashboard Cite

This article studies stationary functional time series with long-range dependence, and estimates the memory parameter involved. Semiparametric local Whittle estimation is used, where periodogram is constructed from the approximate first score, which is an inner product of the functional observation and estimated leading eigenfunction. The latter is obtained via classical functional principal component analysis. Under the restrictive condition of constancy of the memory parameter over the function support, and other conditions which include rather unprimitive ones on the first score, the estimate is shown to be consistent and asymptotically normal with asymptotic variance free of any unknown parameter, facilitating inference, as in the scalar time series case. Although the primary interest lies in long-range dependence, our methods and theory are relevant to short-range dependent or negative dependent functional time series. A Monte Carlo study of finite sample performance and an empirical example are included.

show abstract

“…For example, ARMA processes have been discussed in [3,24,16], a spectral theory is constructed in [20,21,25] and several estimation methods have been studied in [11,13,12,14,17,2,18,7]. However, the literature mainly focuses on short-memory processes and the study of long-memory processes valued in a separable Hilbert space is a more recent topic, see [23,4,5,9,19]. In particular, in [19,Section 4], the authors propose a generalization of the fractionally integrated autoregressive moving average (often shortened as ARFIMA but we prefer to use the abbreviation FIARMA for reasons that will be made explicit in Remark 3.1) processes to the case of curve (or functional ) time series.…”

Section: Introductionmentioning

confidence: 99%

“…A formulation not restricted to an L 2 space was proposed in [9] where the author considers long-memory processes of the form…”

Section: Introductionmentioning

confidence: 99%

Hilbert valued fractionally integrated autoregressive moving average processes with long memory operators

Durand¹,

Roueff²

2020

Preprint

View full text Add to dashboard Cite

Fractionally integrated autoregressive moving average processes have been widely and successfully used to model univariate time series exhibiting long range dependence. Vector and functional extensions of these processes have also been considered more recently. Here we rely on a spectral domain approach to extend this class of models in the form of a general Hilbert valued processes. In this framework, the usual univariate long memory parameter d is replaced by a long memory operator D acting on the Hilbert space. Our approach is compared to processes defined in the time domain that were previously introduced for modeling long range dependence in the context of functional time series.

show abstract

Limit theorems for Hilbert space-valued linear processes under long range dependence

Cited by 6 publications

References 10 publications

Long-Range Dependent Curve Time Series

Long-Range Dependent Curve Time Series

Local Whittle estimation of long‐range dependence for functional time series

Hilbert valued fractionally integrated autoregressive moving average processes with long memory operators

Contact Info

Product

Resources

About