Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data

Grahovac, Danijel; Jia, Mofei; Leonenko, Nikolai; Taufer, Emanuele

doi:10.1080/02331888.2014.969267

Cited by 20 publications

(20 citation statements)

References 22 publications

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“…See and Grahovac et al (2013) for the proof and more details. This shows that estimating the scaling function under infinite moments is influenced by the value of the tail index α and will yield a concave shape of the scaling function.…”

Section: Lévy Processesmentioning

confidence: 99%

Bounds on the Support of the Multifractal Spectrum of Stochastic Processes

Grahovac

Leonenko

2018

Fractals

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Multifractal analysis of stochastic processes deals with the fine scale properties of the sample paths and seeks for some global scaling property that would enable extracting the so-called spectrum of singularities. In this paper we establish bounds on the support of the spectrum of singularities. To do this, we prove a theorem that complements the famous Kolmogorov's continuity criterion. The nature of these bounds helps us identify the quantities truly responsible for the support of the spectrum. We then make several conclusions from this. First, specifying global scaling in terms of moments is incomplete due to possible infinite moments, both of positive and negative order. For the case of ergodic self-similar processes we show that negative order moments and their divergence do not affect the spectrum. On the other hand, infinite positive order moments make the spectrum nontrivial. In particular, we show that the self-similar stationary increments process with the nontrivial spectrum must be heavy-tailed. This shows that for determining the spectrum it is crucial to capture the divergence of moments. We show that the partition function is capable of doing this and also propose a robust variant of this method for negative order moments. * dgrahova@mathos.hr † LeonenkoN@cardiff.ac.uk

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Section: Lévy Processesmentioning

confidence: 99%

Bounds on the Support of the Multifractal Spectrum of Stochastic Processes

Grahovac

Leonenko

2018

Fractals

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“…More details on the method can be found in Grahovac et al (2015). It is important to note that the estimation does not depend on the particular form of the underlying distribution and the only assumption is that the sample comes from the class of heavy-tailed distributions, which in particular also includes Student's distribution.…”

Section: Estimation Of Parameter νmentioning

confidence: 99%

“…where υ is supposed to be the known value of the tail index of marginal distribution of Student's diffusion and δ̑, μ̑ and θ̑ are consistent estimators of parameters δ, μ and θ given in (12), (13) and (15), respectively. The covariance matrix ∑(υ, δ̑, μ̑, θ ̑) could be represented as D∑D τ , where and the elements of the matrix ∑ are as follows: For more details on the methodology of analysis of asymptotic properties of some of these estimators and calculation of explicit form of the covariance matrix ∑(υ, δ̑, μ̑, θ ̑) we refer to Leonenko and Šuvak (2010).…”

Section: Asymptotic Normalitymentioning

confidence: 99%

Heavy-tailed modeling of CROBEX

Grahovac

Šuvak

2015

fintp

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Classical continuous-time models for log-returns usually assume their independence and normality of distribution. However, nowadays it is widely accepted that the empirical properties of log-returns often show a specific correlation structure and deviation from normality, in most cases suggesting that their distribution is heavy-tailed. Therefore we suggest an alternative continuous-time model for logreturns, a diffusion process with Student's marginal distributions and exponentially decaying autocorrelation structure. This model depends on several unknown parameters that need to be estimated. The tail index is estimated by the method based on the empirical scaling function, while the parameters describing mean, variance and correlation structure are estimated by the method of moments. The model is applied to the CROBEX stock market index, meaning that the estimation of parameters is based on the CROBEX log-returns. The quality of the model is assessed by means of simulations, by comparing CROBEX log-returns with the simulated trajectories of Student's diffusion depending on estimated parameter values.

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“…Asymptotic properties of S q (n, t) have been considered in the context of multifractality detection ( [5,6,8]), and in particular for LFSM in [7]. We go over the methodology from [5] to establish the asymptotic properties. Instead of keeping t fixed,wetakeittobeoftheform t = n s for some s ∈ (0, 1), which allows the blocks to grow as the sample size increases…”

Section: Empirical Structure Function and The Scaling Functionmentioning

confidence: 99%

Scaling Properties of the Empirical Structure Function of Linear Fractional Stable Motion and Estimation of Its Parameters

2014

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Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data

Cited by 20 publications

References 22 publications

Bounds on the Support of the Multifractal Spectrum of Stochastic Processes

Bounds on the Support of the Multifractal Spectrum of Stochastic Processes

Heavy-tailed modeling of CROBEX

Scaling Properties of the Empirical Structure Function of Linear Fractional Stable Motion and Estimation of Its Parameters

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