Adaptive nonparametric drift estimation of an integrated jump diffusion process

Funke, Benedikt; Schmisser, Emeline

doi:10.1051/ps/2018005

Cited by 8 publications

(13 citation statements)

References 26 publications

(29 reference statements)

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“…The pointwise estimators considered here and in Chen and Zhang (2015), Song (2017), and Song, Lin, and Wang (2013) are based on kernel approach, whereas the adaptive estimator studied in Funke and Schmisser (2018) was done based on a model selection approach and focused on L 2 -risk. Moreover, Funke and Schmisser (2018) only focused on the regression-type estimator for the drift coefficient and could not provide the exact bias term or the central limit theorem.…”

Section: Remarkmentioning

confidence: 99%

“…Not assuming the specific form of the coefficients, Nicolau (2007) systematically studied Nadaraya-Watson estimators, Wang and Lin (2011) presented the local linear estimations using symmetric kernel for bias correction, Hanif (2015) discussed the Nadaraya-Watson estimators using Gamma asymmetric kernel which didn't coincide with the asymmetric kernel-based method introduced in Chen (2000). For model (2) with ( , ) ≠ 0, Song (2017) provided the nonparametric estimators for the infinitesimal coefficients μ(x) and 2 ( ) + ∫ ℰ 2 ( , ) ( ) in high frequency data based on Gaussian symmetric kernel, Chen and Zhang (2015) discussed the local linear estimators for them based on symmetric kernels, Song, Lin, and Wang (2013) proposed a re-weighted Nadaraya-Watson estimator for the infinitesimal conditional expectation, Funke and Schmisser (2018) constructed adaptive nonparametric estimator for the drift coefficient based on penalized least squares method.…”

Section: Introductionmentioning

confidence: 99%

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Variance reduction estimation for return models with jumps using gamma asymmetric kernels

Song

Hou

Sheng-yi

2019

Studies in Nonlinear Dynamics &Amp; Econometrics

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This paper discusses Nadaraya-Watson estimators for the unknown coefficients in second-order diffusion model with jumps constructed with Gamma asymmetric kernels. Compared with existing nonparametric estimators constructed with Gaussian symmetric kernels, local constant smoothing using Gamma asymmetric kernels possesses some extra advantages such as boundary bias correction, variance reduction and resistance to sparse design points, which is validated through theoretical details and finite sample simulation study. Under the regular conditions, the weak consistency and the asymptotic normality of these estimators are presented. Finally, the statistical advantages of the nonparametric estimators are depicted through 5-minute high-frequency data from Shenzhen Stock Exchange in China.

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Variance reduction estimation for return models with jumps using gamma asymmetric kernels

Song

Hou

Sheng-yi

2019

Studies in Nonlinear Dynamics &Amp; Econometrics

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“…For example, Gugushvili and Spereij () studied the α −mixing rate for a high frequency sampling data discretely observed from a univariate ergodic diffusion process. Funke and Schmisser () mentioned the definition of the β −mixing with a high frequency data for integrated jump‐diffusion in analogy to Comte et al (). We have clarified the definition of the ρ −mixing for

false{X_{i Δ_{n}}; i = 1, 2, ⋯ false}

in analogy to Lin and Bai () or Chen et al () as

ρ (m) = \sup_{k \in N} \sup_{X \in L_{2} {(ℱ}_{1}^{k}), Y \in L_{2} {(ℱ}_{k + n}^{\infty})} \frac{| E (X Y) - E (X) E (Y) |}{\sqrt{V a r (X) \times V a r (Y)}}

where

scriptN = false{1, 2, ⋯ false}, ℱ_{1}^{k} = σ false(X_{i Δ_{n}}, 1 \leq i \leq k false), ℱ_{k + n}^{\infty} = σ false(X_{i Δ_{n}}, i \geq k + n false)

and

L_{2} {false(ℱ}_{1}^{k} false)

L_{2} {false(ℱ}_{k + n}^{\infty} false)

denotes the sets of random variables Z , which are

ℱ_{1}^{k} -

measurable or

ℱ_{k + n}^{\infty <...>}

…”

Section: Local Linear Estimators and Large Sample Propertiesmentioning

confidence: 99%

“…Remark The BDG result is also denoted as Kunita's first inequality in Applebaum (). Schmisser () and Funke and Schmisser () stated its formulation for the detailed proof, one can also refer to them.…”

Section: Appendix Amentioning

confidence: 99%

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Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

Song

Chen

Wang

2018

Journal Time Series Analysis

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In this article, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous‐time asset return models with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps, are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality are provided through the conditional Lindeberg theorem as the time span T→∞ and the sample interval Δ n →0. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns of stock index under 5‐minute high sampling frequency for real application.

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Invariant density adaptive estimation for ergodic jump–diffusion processes over anisotropic classes

Amorino

Gloter

2021

Journal of Statistical Planning and Inference

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We consider the solution X = (Xt) t≥0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density µ. We assume that a continuous record of observations X T = (Xt) 0≤t≤T is available. In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic Hölder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density µ of the jump diffusion X.

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Adaptive nonparametric drift estimation of an integrated jump diffusion process

Cited by 8 publications

References 26 publications

Variance reduction estimation for return models with jumps using gamma asymmetric kernels

Variance reduction estimation for return models with jumps using gamma asymmetric kernels

Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

Invariant density adaptive estimation for ergodic jump–diffusion processes over anisotropic classes

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