Maximum likelihood estimators from discrete data modeled by mixed fractional Brownian motion with application to the Nordic stock markets

Pynnönen, Seppo; Sottinen, Tommi

doi:10.1080/03610918.2020.1764581

Cited by 9 publications

(7 citation statements)

References 31 publications

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“…N , κ2 N , σ 2 N ) := q(τ N ). In other words, in order to construct strongly consistent estimators, we need to solve the estimation equation f (ϑ) = τ N (7) with respect to unknown parameters. The solution to (7) can be found in explicit form, see Subsection 5.3 for details.…”

Section: Construction Of Strongly Consistent Estimatorsmentioning

confidence: 99%

“…to define the estimator of θ . By subtracting X Nh N from previously constructed statistics (4) we can obtain a system of equations similar to (7).…”

Section: Construction Of Strongly Consistent Estimatorsmentioning

confidence: 99%

“…In Refs. [2,7,14,17] the parameter estimation for model (2) with H 1 = 1/2 was studied. The simultaneous estimation of all four parameters of this model was studied in [7] by the maximum likelihood method, and in [14], where the approach of [6] was generalized and, moreover, ergodic-type estimators were proposed.…”

Section: Introductionmentioning

confidence: 99%

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Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

Ralchenko,

Yakovliev

2023

Modern Stochastics: Theory and Applications

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The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

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Section: Construction Of Strongly Consistent Estimatorsmentioning

confidence: 99%

“…to define the estimator of θ . By subtracting X Nh N from previously constructed statistics (4) we can obtain a system of equations similar to (7).…”

Section: Construction Of Strongly Consistent Estimatorsmentioning

confidence: 99%

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Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

Ralchenko,

Yakovliev

2023

Modern Stochastics: Theory and Applications

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“…The parameter of α stable distribution is directly estimated by maximizing the log-likelihood function. 29 For a given independent and same distributed observation vector x=x1,x2,…,xn, the MLE of the α stable distribution parameter θ = (α, β, γ, δ) is to maximize the following log-likelihood function which is given aswhere f(xi;θ) is the probability density function of α stable distribution.…”

Section: Protection Levels Calculation Under Multi-parameter Stable O...mentioning

confidence: 99%

A multi-parameter stable-distribution-based protection levels calculation method for ground-based augmentation systems integrity assessment

Liu

Shi

Sun

et al. 2022

Proceedings of the Institution of Mechanical Engineers, Part G:

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The Protection Levels (PL) of Ground-Based Augmentation Systems (GBAS) are bounds of the positioning errors associated with given integrity levels. However, the practical Gaussian Overbounding Method (GOM) cannot describe the non-Gaussian nature of the error model such as “thick tail,” which makes PL tend to be overestimated. To overcome this problem, an error overbounding model based on stable distribution is proposed in this paper. Four stable distribution parameters are involved to accurately describe the GBAS ranging error. The weighted summation feature of the stable distribution is combined with the maximum likelihood estimation when calculating PL. Thus, it is possible to achieve a higher accuracy of PL under similar calculation complexity. In addition, this paper puts forward the Concept of Tightness (CoT) to describe the overbounding effect intuitively and quantitatively. In order to verify the effect of this method, multiple receivers are set to simulate the GBAS system, theyn the Stable Overbounding Method (SOM) is used to calculate the PL in two directions (Vertical and Lateral) under the hypotheses of H0 and H1. The CoT convincingly proves that the SOM has a better overbounding effect and improves the continuity of GBAS. It is evident that the multi-parameter SOM makes the overbounding of the protection level tighter and reduces the probability that the protection level is close to the Alarm Limit (AL), thus improving the accuracy of PL computing and the availability of GBAS systems.

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“…where W is a Wiener process, B H is a fractional Brownian motion with Hurst index H, B H is independent of W . Parameter estimation problem for the model (1) was studied in Cai, Chigansky, and Kleptsyna (2016); Mishura, Ralchenko, and Shklyar (2017); Dufitinema, Pynnönen, and Sottinen (2022); Kukush, Lohvinenko, Mishura, and Ralchenko (2022). Cai et al (2016) considered the estimation of the drift parameter θ assuming that H and σ are known and κ = 1.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Normality of Parameter Estimators for~Mixed Fractional Brownian Motion with Trend

Ralchenko,

Yakovliev

2023

AJS

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We investigate the mixed fractional Brownian motion of the form Xt = θt+σWt +κBtH , driven by a standard Brownian motion W and a fractional Brownian motion B H with Hurst parameter H. We consider strongly consistent estimators of unknown model parameters (θ, H, σ, κ) based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for H ∈ (0, 12 ).

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Maximum likelihood estimators from discrete data modeled by mixed fractional Brownian motion with application to the Nordic stock markets

Cited by 9 publications

References 31 publications

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

A multi-parameter stable-distribution-based protection levels calculation method for ground-based augmentation systems integrity assessment

Asymptotic Normality of Parameter Estimators for~Mixed Fractional Brownian Motion with Trend

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