Local asymptotic normality property for fractional Gaussian noise under high-frequency observations

Brouste, Alexandre; Fukasawa, Masaaki

doi:10.1214/17-aos1611

Cited by 33 publications

(46 citation statements)

References 22 publications

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“…This matrix will play the role of the proper rate matrix of the MLE. As with [2], concerning the upper-left part of ϕ n we assume the following conditions:…”

Section: Likelihood Asymptoticsmentioning

confidence: 99%

“…The asymptotic degeneracy is an intrinsic feature coming from the self-similarity of the underlying model. Recently, in the context of the observation of fractional Brownian motion observed at high-frequency, the singularity issue has been untied in [2] by using a non-diagonal norming matrix, and a classical local asymptotic normality (LAN) property of the likelihoods has been obtained with a non-degenerate Fisher information. The associated Hájek-Le Cam asymptotic minimax theorem is derived as a corollary; we refer to [12] for a detailed account of the LAN property and its consequences.…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, this study enlarges the domain of application of the LAN theory already proved for i.i.d. classical setting [10], [15], ergodic Markov chains [22], ergodic diffusions [9], diffusions under high-frequency observations [8], diffusions with observational noise [6], [7], [20], several Lévy process or Lévy driven models [3], [14], [18] and fractional Gaussian noise [2], [4].…”

Section: Introductionmentioning

confidence: 99%

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Efficient estimation of stable Lévy process with symmetric jumps

Brouste

Masuda

2018

Stat Inference Stoch Process

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Efficient estimation of a non-Gaussian stable Lévy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.

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“…This matrix will play the role of the proper rate matrix of the MLE. As with [2], concerning the upper-left part of ϕ n we assume the following conditions:…”

Section: Likelihood Asymptoticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient estimation of stable Lévy process with symmetric jumps

Brouste

Masuda

2018

Stat Inference Stoch Process

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“…The estimation of the Hurst parameter H and/or the scaling parameter σ has been investigated in numerous papers both in low and high frequency framework. We refer to [13] for efficient estimation of the Hurst parameter H in the low frequency setting and to [9,12,16] for the estimation of (σ, H) in the high frequency setting, among many others. In the low frequency framework the spectral density methods are usually applied and the optimal convergence rate for the estimation of (σ, H) is known to be √ n. In the high frequency setting the estimation of the pair (σ, H) typically relies upon power variations and related statistics, and the optimal convergence rate is known to be ( √ n/ log(n), √ n).…”

Section: Introductionmentioning

confidence: 99%

Estimation of the linear fractional stable motion

Mazur¹,

Otryakhin²,

Podolskij³

2020

Bernoulli

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In this paper we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter σ > 0, the self-similarity parameter H ∈ (0, 1) and the stability index α ∈ (0, 2) of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in [5]. More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of (σ, α, H). We present the law of large numbers and some fully feasible weak limit theorems.

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“…When the underlying data are pure fractional Brownian motion (i.e. a Gaussian process with mean zero and covariance Cov(B σ,H (t), B σ,H (s)) = σ 2 2 (|t| 2H + |s| 2H − |t − s| 2H )) the increments have the form of a Gaussian vector with correlation matrix determined by the parameters σ and H. The maximum likelihood (ML) estimator has been shown to be optimal for high-frequency point observations of fractional Brownian motion [52]. We show below that the scale spectral estimator has essentially the same accuracy as the ML when the observations come from pure fractional Brownian motion.…”

Section: B6 Remarks On Optimality Precision and Robustnessmentioning

confidence: 99%

Emergence of turbulent epochs in oil prices

Garnier

Sølna

2019

Chaos, Solitons & Fractals

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Oil price data have a complicated multi-scale structure that may vary with time. We use time-frequency analysis to identify the main features of these variations and, in particular, the regime shifts. The analysis is based on a wavelet-based decomposition and analysis of the associated scale spectrum. The joint estimation of the local Hurst exponent and volatility is the key to detect and identify regime shifting and switching of the oil price. The framework involves in particular modeling in terms of a process of "multi-fractional" type so that both the roughness and the volatility of the price process may vary with time. Special epochs then emerge as a result of these degrees of freedom, moreover, as a result of the special type of spectral estimator used. These special epochs are discussed and related to historical events. Some of them are not detected by standard analysis based on maximum likelihood estimation. The paper presents a novel algorithm for robust detection of such special epochs and multifractional behavior in financial or other types of data. In the financial context insight about such behavior of the asset price is important to evaluate financial contracts involving the asset.

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Local asymptotic normality property for fractional Gaussian noise under high-frequency observations

Abstract: Local Asymptotic Normality (LAN) property for fractional Gaussian noise under high-frequency observations is proved with a non-diagonal rate matrix depending on the parameter to be estimated. In contrast to the LAN families in the literature, non-diagonal rate matrices are inevitable.

Cited by 33 publications

References 22 publications

Efficient estimation of stable Lévy process with symmetric jumps

Efficient estimation of stable Lévy process with symmetric jumps

Estimation of the linear fractional stable motion

Emergence of turbulent epochs in oil prices

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