On the asymptotic normality of kernel estimators of the long run covariance of functional time series

Berkes, István; Horváth, Lajos; Rice, Gregory

doi:10.1016/j.jmva.2015.11.005

Cited by 21 publications

(18 citation statements)

References 41 publications

(60 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof We begin by establishing . According to the triangle inequality,

(||, ∥ Ĉ_{h, q}^{(p)} ∥ - ∥ C ∥) ⩽ ∥ Ĉ_{h, q}^{(p)} - C ∥ .

Under Assumptions and , we obtain from equation (2.15) of Berkes et al () that, in the case when p = 0,

E ∥ Ĉ_{h, q} - C ∥^{2} = O (\frac{h}{T} + h^{- 2 α}) .

Hence, Chebyshev's inequality implies that

∥ Ĉ_{h, q} - C ∥ = O_{P} ({(\frac{h}{T})}^{1 / 2} + h^{- α}),

which along with implies in this case. The approximation in follows similarly, which we now aim to show.…”

Section: Proofsmentioning

confidence: 69%

“…In general, later, we let ∥·∥ denote the standard norm on L 2 [0,1] d , with the dimension d being clear based on the input function. It is established by Berkes et al () that under mild conditions, which are implied by Assumptions and , that

\begin{matrix} right E ∥ Ĉ_{h, q} - C ∥^{2} = \frac{h}{T} (∥ C ∥^{2} + {(\int_{0}^{1} C (u, u) d u)}^{2}) \int_{- \infty}^{\infty} W_{q}^{2} (x) d x & left + h^{- 2 q} ∥ w C^{(q)} ∥^{2} & right \\ right & left + o (\frac{h}{T} + h^{- 2 q}), \end{matrix}

where

C^{(q)} (u, s) = \sum_{ℓ = - \infty}^{\infty} | ℓ |^{q} γ_{ℓ} (u, s),

and the constant w is defined in equation . In particular, the first two terms on the right‐hand side of represent the asymptotically leading terms in the mean‐squared norm of the error of

Ĉ_{h, q}

, which we refer to as the AMSNE.…”

Section: Statement Of Methods and Main Resultsmentioning

confidence: 99%

“…Theorem 2.1 of Politis () and equation (10) of Politis and Romano ()), which holds under a number of more primitive weak dependence conditions. The primary utility of Assumption , which is a slightly stronger version of the main assumption of Berkes et al (), is to guarantee that the integral of var(

Ĉ (t, s)

) is on the order of h / T . This could be replaced by other functional weak dependence conditions that guarantee this.…”

Section: Statement Of Methods and Main Resultsmentioning

confidence: 99%

“…The approximation in follows similarly, which we now aim to show. According to Jensen's and Lyapunov's inequalities,

\begin{matrix} (||, \int Ĉ_{h, q} (t, t) normald t - \int C (t, t) normald t) & ⩽ \int | Ĉ_{h, q} (t, t) - C (t, t) | normald t \\ ⩽ {((), \int | Ĉ_{h, q} (t, t) - C (t, t) |^{2} normald t)}^{1 / 2} . \end{matrix}

Moreover,

\begin{matrix} E \int | Ĉ_{h, q} (t, t) - C (t, t) |^{2} normald t & = \int var (Ĉ_{h, q} (t, t)) normald t + \int {(E Ĉ_{h, q} (t, t) - C (t, t))}^{2} normald t \\ = O ((), \frac{h}{T} + h^{- 2 α}), \end{matrix}

by Lemma 4.3 and Theorem 2.3 of Berkes et al , which implies . Towards establishing the rest of for 0 < p ⩽ q , one has that the integrated mean‐squared error of

Ĉ_{h, q}^{(p)}

may be broken into a variance and bias term, as follows:

E ∥ Ĉ_{h, q}^{(p)} - C^{(p)} ∥^{2} = \int \int var (

…”

Section: Proofsmentioning

confidence: 99%

“…Under Assumptions 2.1 and 2.2, we obtain from equation (2.15) of Berkes et al (2016) that, in the case when p D 0,…”

Section: Proofmentioning

confidence: 99%

See 4 more Smart Citations

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

Rice

Shang

2016

Journal Time Series Analysis

Self Cite

View full text Add to dashboard Cite

Abstract. In arenas of application including environmental science, economics, and medicine, it is increasingly common to consider time series of curves or functions.Many inferential procedures employed in the analysis of such data involve the long run covariance function or operator, which is analogous to the long run covariance matrix familiar to finite dimensional time series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies crucially on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite dimensional setting, we propose a bandwidth selection method that aims to minimize the estimator's asymptotic meanAs the AMSNE depends on unknown population quantities including the long run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.

show abstract

“…Proof We begin by establishing . According to the triangle inequality,

(||, ∥ Ĉ_{h, q}^{(p)} ∥ - ∥ C ∥) ⩽ ∥ Ĉ_{h, q}^{(p)} - C ∥ .

Under Assumptions and , we obtain from equation (2.15) of Berkes et al () that, in the case when p = 0,

E ∥ Ĉ_{h, q} - C ∥^{2} = O (\frac{h}{T} + h^{- 2 α}) .

Hence, Chebyshev's inequality implies that

∥ Ĉ_{h, q} - C ∥ = O_{P} ({(\frac{h}{T})}^{1 / 2} + h^{- α}),

which along with implies in this case. The approximation in follows similarly, which we now aim to show.…”

Section: Proofsmentioning

confidence: 69%

\begin{matrix} right E ∥ Ĉ_{h, q} - C ∥^{2} = \frac{h}{T} (∥ C ∥^{2} + {(\int_{0}^{1} C (u, u) d u)}^{2}) \int_{- \infty}^{\infty} W_{q}^{2} (x) d x & left + h^{- 2 q} ∥ w C^{(q)} ∥^{2} & right \\ right & left + o (\frac{h}{T} + h^{- 2 q}), \end{matrix}

where

C^{(q)} (u, s) = \sum_{ℓ = - \infty}^{\infty} | ℓ |^{q} γ_{ℓ} (u, s),

and the constant w is defined in equation . In particular, the first two terms on the right‐hand side of represent the asymptotically leading terms in the mean‐squared norm of the error of

Ĉ_{h, q}

, which we refer to as the AMSNE.…”

Section: Statement Of Methods and Main Resultsmentioning

confidence: 99%

Ĉ (t, s)

) is on the order of h / T . This could be replaced by other functional weak dependence conditions that guarantee this.…”

Section: Statement Of Methods and Main Resultsmentioning

confidence: 99%

“…The approximation in follows similarly, which we now aim to show. According to Jensen's and Lyapunov's inequalities,

\begin{matrix} (||, \int Ĉ_{h, q} (t, t) normald t - \int C (t, t) normald t) & ⩽ \int | Ĉ_{h, q} (t, t) - C (t, t) | normald t \\ ⩽ {((), \int | Ĉ_{h, q} (t, t) - C (t, t) |^{2} normald t)}^{1 / 2} . \end{matrix}

Moreover,

\begin{matrix} E \int | Ĉ_{h, q} (t, t) - C (t, t) |^{2} normald t & = \int var (Ĉ_{h, q} (t, t)) normald t + \int {(E Ĉ_{h, q} (t, t) - C (t, t))}^{2} normald t \\ = O ((), \frac{h}{T} + h^{- 2 α}), \end{matrix}

by Lemma 4.3 and Theorem 2.3 of Berkes et al , which implies . Towards establishing the rest of for 0 < p ⩽ q , one has that the integrated mean‐squared error of

Ĉ_{h, q}^{(p)}

may be broken into a variance and bias term, as follows:

E ∥ Ĉ_{h, q}^{(p)} - C^{(p)} ∥^{2} = \int \int var (

…”

Section: Proofsmentioning

confidence: 99%

“…Under Assumptions 2.1 and 2.2, we obtain from equation (2.15) of Berkes et al (2016) that, in the case when p D 0,…”

Section: Proofmentioning

confidence: 99%

See 3 more Smart Citations

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

Rice

Shang

2016

Journal Time Series Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

Functional Data

Horváth,

Rice

2023

Springer Series in Statistics

View full text Add to dashboard Cite

A Darling–Erdős-type CUSUM-procedure for functional data

Torgovitski

2014

Metrika

View full text Add to dashboard Cite

This article considers testing for mean-level shifts in functional data. The class of the famous Darling-Erdős-type cumulative sums (CUSUM) procedures is extended to functional time series under short range dependence conditions which are satisfied by functional analogues of many popular time series models including the linear functional AR and the non-linear functional ARCH. We follow a data driven, projection-based approach where the lower-dimensional subspace is determined by (long run) functional principal components which are eigenfunctions of the long run covariance operator. This second-order structure is generally unknown and estimation is crucial -it plays an even more important role than in the classical univariate setup because it generates the finitedimensional subspaces. We discuss suitable estimates and demonstrate empirically that altogether this change-point procedure performs well under moderate temporal dependence.Moreover, Darling-Erdős-type change-point estimates based on (long run) functional principal components as well as the corresponding »fully-functional« counterparts are provided and the testing procedure is finally applied to publicly accessible electricity data from a German power company.

show abstract

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

Cited by 21 publications

References 41 publications

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

Functional Data

A Darling–Erdős-type CUSUM-procedure for functional data

Contact Info

Product

Resources

About