On the likelihood function of small time variance Gamma Lévy processes

Kawai, Reiichiro

doi:10.1080/02331888.2014.918980

Cited by 9 publications

(6 citation statements)

References 21 publications

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“…The methodology presented in Section 4 can also be used for estimating parameters from other cusped, unbounded, or even distributions with extreme leptokurtosis such as stable distri-bution with small stable index, or leptokurtic financial models for high frequency data (Kawai, 2015).…”

Section: Discussionmentioning

confidence: 99%

Maximum leave-one-out likelihood estimation for location parameter of unbounded densities

Nitithumbundit,

Chan

2016

Preprint

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Maximum likelihood estimation of a location parameter fails when the density have unbounded mode. An alternative approach is considered by leaving out a data point to avoid the unbounded density in the full likelihood. This modification give rise to the leave-oneout likelihood. We propose an ECM algorithm which maximises the leave-one-out likelihood. It was shown that the estimator which maximises the leave-one-out likelihood is consistent and super-efficient. However, other asymptotic properties such as the optimal rate of convergence and asymptotic distribution is still under question. We use simulations to investigate these asymptotic properties of the location estimator using our proposed algorithm.

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Section: Discussionmentioning

confidence: 99%

Maximum leave-one-out likelihood estimation for location parameter of unbounded densities

Nitithumbundit,

Chan

2016

Preprint

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“…It is worth mentioning [49, section 4.5] that theorem 3.3 (ii) holds true even with an uninverted subordinator (that is, {U t : t 0} in lieu of {S t : t 0}). For instance, given that all the uninverted stable, tempered stable and gamma (with a large enough t [51]) subordinators admit densities with zero at the origin, one may argue that time-changing by those uninverted subordinators is not strong enough to detain particles around the initial state, and thus no singularity.…”

Section: Theorem 33 (Regularity)mentioning

confidence: 99%

Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

He¹,

Kawai²

2022

Phys. Scr.

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We investigate the fractional Fokker-Planck equation subject to a damping force with an emphasis on its dimension dependent properties. We reveal a variety of surprising properties of its solution through the lens of the probability density function of the corresponding stochastic process with nonlinear mean square displacements, such as existence, singularity, regularity, modality, stationarity and second-order structure, which are largely dependent on the dimension and the random clock. Taking into account that the trajectory information is most often collected from multidimensional systems, the discovered facts have the potential to play important roles as key foundations and alerts for inference, model identification and prediction, when departing from the well-understood univariate framework.

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“…For the univariate case with constant mean μ , Kawai (2015) showed that the Fisher information matrix with respect to μ is not well‐defined when ν <1/2 since

{normalE}_{bold-italicθ} ({[\frac{\partial}{\partial μ} \log f_{VG} (Y ; θ)]}^{2}) = \infty

where the expectation is taken with respect to Y which is a univariate VG random variable with density function f VG in (1) ( d =1 and γ =0). The likelihood function needs to be modified so that the maximum is well‐defined even with unbounded densities.…”

Section: Model Implementationmentioning

confidence: 99%

“…One of the simulated data set is plotted in Figure 2 to demonstrate the presence of cusp lines. For the univariate case with constant mean , Kawai (2015) showed that the Fisher information matrix with respect to is not well-defined when < 1=2 since…”

Section: Leave-one-out Likelihoodmentioning

confidence: 99%

ECM algorithm for estimating vector ARMA model with variance gamma distribution and possible unbounded density

Nitithumbundit

Chan

2021

Aus NZ J of Statistics

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The simultaneous analysis of several financial time series is salient in portfolio setting and risk management. This paper proposes a novel alternating expectation conditional maximisation (AECM) algorithm to estimate the vector autoregressive moving average (VARMA) model with variance gamma (VG) error distribution in the multivariate skewed setting. We explain why the VARMA-VG model is suitable for high-frequency returns (HFRs) because VG distribution provides thick tails to capture the high kurtosis in the data and unbounded central density further captures the majority of near-zero HFRs. The distribution can also be expressed in normal-mean-variance mixtures to facilitate model implementation using the Bayesian or expectation maximisation (EM) approach. We adopt the EM approach to avoid the time-consuming Markov chain Monto Carlo sampling and solve the unbounded density problem in the classical maximum likelihood estimation. We conduct extensive simulation studies to evaluate the accuracy of the proposed AECM estimator and apply the models to analyse the dependency between two HFR series from the time zones that only differ by one hour.

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On the likelihood function of small time variance Gamma Lévy processes

Cited by 9 publications

References 21 publications

Maximum leave-one-out likelihood estimation for location parameter of unbounded densities

Maximum leave-one-out likelihood estimation for location parameter of unbounded densities

Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

ECM algorithm for estimating vector ARMA model with variance gamma distribution and possible unbounded density

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