A large deviation inequality for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>β</mml:mi></mml:math>-mixing time series and its applications to the functional kernel regression model

Krebs, Johannes

doi:10.1016/j.spl.2017.09.013

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…random variables [52], the exponent of the tail bound has a sublinear dependence on sample size due to the presence of n. n is close to n when γ is large corresponding to a faster decaying β-mixing coefficient (Definition 4). This sublinear relation with respect to n is also reported by [49] and [48] as well under exponential α and β-mixing conditions with γ = 1. They provided an O(exp(−nϵ/(log n log log n))) tail bound, which is a faster rate in n compared to Proposition 5 with γ = 1.…”

Section: Resultssupporting

confidence: 79%

“…The α-mixing concentration inequality in [47,Theorem 2] gives the tail bound on the relative deviation (scaled by variance) instead of the absolute deviation. The β-mixing results in [48] and the α-mixing results in [49] provide a subexponential bound of O(exp(−ϵ)) which is a weaker dependency on ϵ than we desired. The detailed discussion of the concentration inequalities we derived is postponed until the main results are introduced.…”

Section: Resultsmentioning

confidence: 72%

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Model-Free Change Point Detection for Mixing Processes

Chen,

Gupta,

Sun

et al. 2024

IEEE Open J. Control. Syst.

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This paper considers the change point detection problem under dependent samples. In particular, we provide performance guarantees for the MMD-CUSUM test under exponentially α, β, and fast ϕ-mixing processes, which significantly expands its utility beyond the i.i.d. and Markovian cases used in previous studies. We obtain lower bounds for average-run-length (ARL) and upper bounds for average-detection-delay (ADD) in terms of the threshold parameter. We show that the MMD-CUSUM test enjoys the same level of performance as the i.i.d. case under fast ϕ-mixing processes. The MMD-CUSUM test also achieves strong performance under exponentially α/β-mixing processes, which are significantly more relaxed than existing results. The MMD-CUSUM test statistic adapts to different settings without modifications, rendering it a completely data-driven, dependence-agnostic change point detection scheme. Numerical simulations are provided at the end to evaluate our findings.

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Section: Resultssupporting

confidence: 79%

Section: Resultsmentioning

confidence: 72%

Model-Free Change Point Detection for Mixing Processes

Chen,

Gupta,

Sun

et al. 2024

IEEE Open J. Control. Syst.

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“…Indeed, this follows with some calculations (see the proof of Theorem 11.2 [22]). Furthermore, we need a result, which follows using the β-mixing property and a lemma in [38], s,t∈I(l,u)…”

Section: Proofs Of the Results In Section 2 And Sectionmentioning

confidence: 99%

Non-parametric regression for spatially dependent data with wavelets

Krebs

2018

Statistics

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We study non-parametric regression estimates for random fields. The data satisfies certain strong mixing conditions and is defined on the regular N -dimensional lattice structure. We show consistency and obtain rates of convergence. The rates are optimal modulo a logarithmic factor in some cases. As an application, we estimate the regression function with multidimensional wavelets which are not necessarily isotropic. We simulate random fields on planar graphs with the concept of concliques (cf. [30]) in numerical examples of the estimation procedure.

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“…For (4.23), use that A n,t = Ψ(X t−1 ) − Ψ(X t−1 ) − ε. We apply Proposition 2 in Krebs (2018b) which allows us to exchange the first probability in (4.23) by the expectation w.r.t. the product measure, i.e.,…”

Section: Technical Resultsmentioning

confidence: 99%

The autoregression bootstrap for kernel estimates of smooth nonlinear functional time series

Krebs,

Franke

2018

Preprint

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Functional times series have become an integral part of both functional data and time series analysis. This paper deals with the functional autoregressive model of order 1 and the autoregression bootstrap for smooth functions. The regression operator is estimated in the framework developed by Vieu (2004) andFerraty et al. (2007) which is here extended to the double functional case under an assumption of stationary ergodic data which dates back to Laib and Louani (2010). The main result of this article is the characterization of the asymptotic consistency of the bootstrapped regression operator.

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A large deviation inequality forβ-mixing time series and its applications to the functional kernel regression model

Cited by 5 publications

References 22 publications

Model-Free Change Point Detection for Mixing Processes

Model-Free Change Point Detection for Mixing Processes

Non-parametric regression for spatially dependent data with wavelets

The autoregression bootstrap for kernel estimates of smooth nonlinear functional time series

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