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1

GARCH(1,1) models with stable residuals

Trusz M., Tserakh U.

Studia Informatica : systems and information technology

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2018

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Vol. 1-2(22)

47--57

EN

The focus of this paper is the use of stable distributions for GARCH models. Such models are applied for the analysis of financial and economic time series, which have several special properties: volatility clustering, heavy tails and asymmetry of residuals distributions. Below we compare the properties of stable and tempered stable distributions and describe methodologies for constructing models and subsequent estimation of parameters using the maximum likelihood method. We also analyze an example of building models on real data in order to illustrate that tempered stable distributions could be used in financial time series models. Moreover, such distributions can show better results in comparison with traditionally used distributions.

2

Limiting spectral distributions of sums of products of non-Hermitian random matrices

Kösters H., Tikhomirov A.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 2

359--384

EN

For fixed l ≥ 0 and m ≥ 1, let X(0)n, X(1)n,…, X(l)n be independent random n × n matrices with independent entries, let F(0)n:= X(0)n (X(l)n)−1…(X(l)n)−1, and let F(1)n,…, F(m)n be independent random matrices of the same form as F(0)n. We show that as n → ∞, the matrices F(0)n and m−(l+1)/2 (F(1)n + … + F(m)n) have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov (2015) to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we pro vide a closer description of the limiting distributions in terms of free probabilisty theory.

3

Stable distributions, generalised entropy, and fractal diagnostic models of mechanical vibration signals

Puchalski A., Komorska I.

Diagnostyka

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2017

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Vol. 18, No. 4

103--110

EN

Vibrodiagnostic analysis of wearing and/or defects of complex rotating systems confirms the presence of non-linear, nonstationary and multiscale properties as well as long-term correlations of real signals. The recorded time series of vibrations are often of an impulse character. Probability distributions are different than Gaussian distributions and exhibit heavy-tails. These are important sources of multifractal dynamics, requiring advanced, data-based modelling methods. The reliable numerical algorithms, used for calculations of functions of stable distributions and multifractal properties, were applied in the approach presented in the hereby paper. Relations between parameters of stable distributions and singularity spectra indicate the possibility of applying both methods for modelling mechanical vibrations signals in diagnostics of complex systems. The performed investigations confirmed the possibility of modelling and assessing the observed states of the powertrain of vehicles with SI engines, on the bases of parameters of alpha-stable distribution (ASD) and parameterised entropy of mechanical vibrations signals.

PL

Wibrodiagnostyczna analiza zużycia i / lub wad złożonych układów wirujących potwierdza obecność nieliniowych, niestacjonarnych i wieloskalowych właściwości oraz długookresowe korelacje sygnalozywisych. Rejestrowane szeregi czasowe drgań mają często charakter impulsowy. Rozkłady prawdopodobieństwa odbiegają od rozkładów Gaussowskich i wykazują gruboogonowość. Są to ważne źródła dynamiki multifraktalnej, wymagające zaawansowanych metod modelowania bazującego na danych. W podejściu przedstawionym w pracy wykorzystano niezawodne algorytmy numeryczne służące do obliczania funkcji stabilnych dystrybucji i cech multifraktalnych. Relacje między parametrami stabilnych rozkładów i widmami osobliwości wskazują na możliwość zastosowania obu metod do modelowania sygnałów drgań mechanicznych w diagnostyce układów złożonych. Wykonane badania potwierdziły możliwość modelowania i oceny obserwowanych stanów układu napędowego pojazdu z silnikiem o zapłonie iskrowym, na podstawie parametrów rozkładów alfastabilnych (ASD) gęstości prawdopodobieństwa i entropii parametryzowanej sygnałów drgań mechanicznych.

4

Unimodality of Boolean and monotone stable distributions

Hasebe T., Sakuma N.

Demonstratio Mathematica

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2015

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Vol. 48, nr 3

424--439

EN

We give a complete list of the Lebesgue–Jordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal.

5

Computing the portfolio conditional value-at-risk in the α-stable case

Stoyanov S., Samorodnitsky G., Rachev S. T., Ortobelli S.

Probability and Mathematical Statistics

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2006

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Vol. 26, Fasc. 1

1--22

EN

The class of α-stable distributions is an attractive probabilistic model of asset returns distribution in the field of finance. When dealing with real issues, such as optimal portfolio selection, it is important that we can compute the Conditional Value-at-Risk (CVaR) accurately. The CVaR is also known as the expected tail loss (ETL) proposed in literature as a coherent risk measure. In our paper we propose an integral expression for the calculation of the CVaR of a stable law. We compare the current approach to some existing method and we demonstrate how to relate the derived result to some common multivariate distributional assumptions.

6

Distributional analysis of the stocks comprising the DAX 30

Hoechstoetter M., Rachev S., Fabozzi F.

Probability and Mathematical Statistics

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2005

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Vol. 25, Fasc. 2

363--383

EN

In this paper, we analyze the returns of stocks comprising the German stock index DAX with respect to the α-stable distribution. We apply nonparametric estimation methods such as the Hill estimator as well as parametric estimation methods conditional on the α-stable distribution. We find for both the nonparametric and parametric estimation methods that the α-stable hypothesis cannot be rejected for the return distribution. We then employ the GARCH model; the fit of innovations modeled with an underlying α-stable distribution is compared to the fit obtained from modelling the innovations with the skew-t distribution. The α-stable distribution is found to out-perform the skew-t distribution.

7

Approximation by Penultimate Stable Laws

de Haan L., Peng L., Pereira H. I.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 1

105--121

EN

In certain cases partial sums of i.i.d. random variables with finite variance are better approximated by a sequence of stable distributions with indices αn → 2 than by a normal distribution. We discuss when this happens and how much the convergencerate can be improved by using penultimate approximations. Similar results are valid for other stable distributions.