Optimal rates for parameter estimation of stationary Gaussian processes

Es-Sebaiy, Khalifa; Viens, Frederi

doi:10.1016/j.spa.2018.08.010

Cited by 47 publications

(58 citation statements)

References 24 publications

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“…This contrasts the asymptotic normality in time (n → ∞) of this type of estimators, which is violated for H > 3 4 (see e.g. [10], [28] or [12]) due to the strong long-range dependence.…”

Section: Estimation In Stationary Casementioning

confidence: 89%

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A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

Kříž

2019

Stoch. Dyn.

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A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the estimator as number of coordinates increases and time horizon is fixed (the space consistency). In this respect, this modification outperforms the standard (non-weighted) minimum-contrast estimator. Compared to other drift estimators studied within spectral approach (eg. maximum likelihood, trajectory fitting), the weighted MCE is rather universal. It covers discrete time as well as continuous time observations and it is applicable to processes with any value of Hurst index H ∈ (0, 1). To the author's best knowledge, this is so far the first space-consistent estimator studied for H < 1/2. 2010 Mathematics Subject Classification. 60H15, 60G22, 62M09. 1 2 PAVEL KŘÍŽreader to the articles [28] for continuous-time setting, [10] for discrete-time setting and [12] for comparison with the LSE, to name just a few. These works benefit from the relation of Malliavin calculus and central limit theorems -a popular theory initiated in [23] and further developed by many authors (see e.g.[22] and references therein). These techniques were recently applied to the MCE in infinitedimensional setting in [18] and are also utilized in the present work. Another interesting work on minimum-contrast estimation is [3], where a different version of the MCE based on fundamental-martingale technique for a real-valued fOU with Hurst index H > 1/2 is studied in the frameworks of both continuous-time and discrete-time sampling. Regarding drift parameter estimation for singular realvalued fOU processes (H < 1/2), recall the paper [12], where the LSE and the MCE for continuous-time observation are studied and the work [16], which investigates discrete-time version of the LSE.A modification of the minimum-contrast estimator (the weighted MCE) is introduced in this paper. Its construction benefits from the self-similarity property and it turns the time-consistent MCE (consistent with increasing time horizon) into a space-consistent estimator (fixed time horizon and increasing number of Fourier coordinates), which provides the best attainable speed of convergence in discrete-time setting. We believe this approach is potentially applicable to other types of timeconsistent estimators, such as the LSE, and for different types of models (but still having the self-similarity property). To the author's best knowledge, this approach is new even in the basic case of parabolic diagonalizable equations with white additive noise (in space and time).

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Section: Estimation In Stationary Casementioning

confidence: 89%

“…(39)) and the second sample moment of a real-valued fractional Ornstien-Uhlenbeck process with H > 3/4 converges to the Rosenblatt distribution with increasing time horizon (see e.g. [10]). This suggests that one might expect limiting Rosenblatt distribution for Y N as N → ∞ in case of rapidly growing θ k and H > 3/4.…”

Section: Estimation In Stationary Casementioning

confidence: 99%

A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

Kříž

2019

Stoch. Dyn.

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“…where l 1 , l 2 , are defined in the previous section in (16), (19), and C 1 , C 2 , C 3 , C 4,r , r = 1, 2, 3 and C 5 are given in the lemmas below, respectively in (15), (18), (29), (30) and (34).…”

Section: Berry-esséen Bound For the Asymptotic Normality Of The Quadrmentioning

confidence: 99%

AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation

Douissi

Es-Sebaiy

Fatimah

et al. 2022

Stochastic Processes and their Applications

Self Cite

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In this paper, we study the asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white noise in the second Wiener chaos. Using tools from the analysis on Wiener space, we give an upper bound for the total-variation speed of convergence to the normal law, which we apply to study the estimation of the model's mean-reversion. Simulations are performed to illustrate the theoretical results.

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“…Similar approach used in more general setting of the fractional Vasicek model was presented in [26] and [27], where both ergodic and non-ergodic cases are considered. Recently, the 4 th moment theorem was successfully utilized to demonstrate not only asymptotic normality, but also to establish the speed of the convergence to the normal distribution (Berry-Esseentype of bounds) of the MC estimator of the drift parameter in one-dimensional SDEs driven by fBm -see [11] for discrete-time observations with increasing time-horizon and fixed mesh or [24] for continuous-time observations or discrete-time observations with combination of increasing time-horizon and observation frequency.…”

Section: Introductionmentioning

confidence: 99%

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

Kříž

Maslowski

2019

Stochastics

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Central limit theorems and asymptotic properties of the minimumcontrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular (H < 1 2 ) and regular (H > 1 2 ) types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter H < 3 4 ) is the so-called 4 th moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

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Optimal rates for parameter estimation of stationary Gaussian processes

Cited by 47 publications

References 24 publications

A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

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