A minimal contrast estimator for the linear fractional stable motion

Ljungdahl, Mathias Mørck; Podolskij, Mark

doi:10.1007/s11203-020-09216-2

Cited by 8 publications

(8 citation statements)

References 26 publications

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“…Obviously, the proposed estimation procedure assumes prior knowledge of the parameter β, since we need to choose p ∈ (0, β). In the univariate setting the papers [17,27,28,29] have suggested to use negative powers p ∈ (−1, 0) to estimate the parameter H for unknown β. A similar idea should apply in the random field setting, although negative power variations are beyond the scope of our paper.…”

Section: Resultsmentioning

confidence: 99%

Power variations for fractional type infinitely divisible random fields

Basse-O'Connor,

Pilipauskaitė,

Podolskij

2020

Preprint

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This paper presents new limit theorems for power variation of fractional type symmetric infinitely divisible random fields. More specifically, the random field X = (X(t)) t∈[0,1] d is defined as an integral of a kernel function g with respect to a symmetric infinitely divisible random measure L and is observed on a grid with mesh size n −1 . As n → ∞, the first order limits are obtained for power variation statistics constructed from rectangular increments of X. The present work is mostly related to [8,9], who studied a similar problem in the univariate context. We will see, however, that the asymptotic theory in the random field setting is much richer compared to [8,9] as it contains new limits, which depend on the precise structure of the kernel g. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional β-stable sheet, and the moving average fractional β-stable field, and discuss potential consequences for statistical inference.

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Section: Resultsmentioning

confidence: 99%

Power variations for fractional type infinitely divisible random fields

Basse-O'Connor,

Pilipauskaitė,

Podolskij

2020

Preprint

Self Cite

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show abstract

“…Obviously, the proposed estimation procedure assumes prior knowledge of the parameter β, since we need to choose p ∈ (0, β). In the case d = 1 the papers [17,27,28,29] have suggested to use negative powers p ∈ (−1, 0) to estimate the parameter H for unknown β. A similar idea should apply in the random field setting, although negative power variations are beyond the scope of our paper.…”

Section: Example 34 (Continuation Of Example 21)mentioning

confidence: 99%

“…We remark that the aforementioned probabilistic results are of immense importance for statistical applications. Indeed, they have been applied in [27,28,29] to obtain complete parametric estimation of the linear fractional stable models and related processes in low and high frequency settings. Earlier studies on similar estimation problems, which are mainly concerned with estimation of the self-similarity parameter, can be found in [3,17,34,40].…”

Section: Introductionmentioning

confidence: 99%

Power variations for fractional type infinitely divisible random fields

Basse-O'Connor

Pilipauskaitė

Podolskij

2021

Electron. J. Probab.

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This paper presents new limit theorems for power variations of fractional type symmetric infinitely divisible random fields. More specifically, the random field X = (X(t)) t∈[0,1] d is defined as an integral of a kernel function g with respect to a symmetric infinitely divisible random measure L and is observed on a grid with mesh size n −1 . As n → ∞, the first order limits are obtained for power variation statistics constructed from rectangular increments of X. The present work is mostly related to [8,9], who studied a similar problem in the case d = 1. We will see, however, that the asymptotic theory in the random field setting is much richer compared to [8,9] as it contains new limits, which depend on the precise structure of the kernel g. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional β-stable sheet, and the moving average fractional β-stable field, and discuss potential consequences for statistical inference.

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“…Previous estimation methods suggested in Ljungdahl and Podolskij (2019, 2020) and Mazur et al (2020) relied on functionals of the one‐dimensional marginal law of the process and specific properties of the process at hand. Since the marginal distribution of the considered models have been symmetric

β

‐stable, only the scale and the stability parameters can be estimated via such statistics.…”

Section: Introductionmentioning

confidence: 99%

“…Instead of relying on existing theory Basse‐O'Connor et al (2017, 2019) and Pipiras and Taqqu (2003), which only accounts for the marginal law of the underlying model, we shall use the framework from the recent paper Azmoodeh et al (2020), which is tailor‐made for the study of Gaussian fluctuations of functionals of multiple heavy‐tailed moving averages, to estimate the multidimensional parameter. While our approach is similar in spirit to the univariate framework of Ljungdahl and Podolskij (2019, 2020) from the theoretical viewpoint, there are some important differences. First of all, the parameter identifiability and nondegeneracy condition (see condition (A)(4) below) are not trivial in the multidimensional setting and we demonstrate the corresponding techniques for various examples.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional parameter estimation of heavy‐tailed moving averages

Ljungdahl

Podolskij

2021

Scandinavian J Statistics

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In this article we present a parametric estimation method for certain multiparameter heavy-tailed Lévy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

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A minimal contrast estimator for the linear fractional stable motion

Cited by 8 publications

References 26 publications

Power variations for fractional type infinitely divisible random fields

Power variations for fractional type infinitely divisible random fields

Power variations for fractional type infinitely divisible random fields

Multidimensional parameter estimation of heavy‐tailed moving averages

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