The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S\&P 500 and SPY ETF returns). For each index, the estimation results show that the six-parameter estimations are statistically significant except for the local parameter, $μ$. The goodness-of-fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson's chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions.
Abstract Recent studies about cryptocurrency returns show that their distribution can be highly-peaked, skewed, and heavy-tailed, with a large excess kurtosis. To accommodate all these peculiarities, we propose the asymmetric Laplace scale mixture (ALSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional asymmetric Laplace (AL) distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector $$\varvec{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> providing greater flexibility to the resulting ALSM. Advantageously concerning the AL distribution, our family members allow for a wider range of values for skewness and kurtosis. For illustrative purposes, we consider different mixing distributions; they give rise to ALSMs having a closed-form probability density function where the AL distribution is obtained as a special case under a convenient choice of $$\varvec{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> . We examine some properties of our ALSMs such as hierarchical and stochastic representations and moments of practical interest. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered ALSMs. We fit these models to the returns of two cryptocurrencies, considering several classical distributions for comparison. The analysis shows how our models represent a valid alternative to the considered competitors in terms of AIC, BIC, and likelihood-ratio tests.
Yongjing Wang, Zubair Ahmad, Faridoon Khan, Dalia Kamal Alnagar · 7 authors
This paper offers the introduction of a new updated form of the Dagum distribution. The new updated form of the Dagum model is called a novel generalized-Dagum distribution. The proposed novel generalized-Dagum distribution is a prominent updated form of the Dagum model with a single additional/extra parameter. The novel generalized-Dagum model is produced by mixing the Dagum distribution with the novel generalized-M distributions approach. The heavy-tailed properties of the novel generalized-Dagum model are obtained. The derivation of the estimators and a simulation study of the novel generalized-Dagum distribution are also provided. Finally, the novel generalized-Dagum model is illustrated by analyzing two real-life data sets related to the financial sector. The first data set represents the Bitcoin exchange rates vs the United States dollars. Whereas, the second data set represents the Ethereum exchange rates vs the United States dollars. Using the Bitcoin and Ethereum exchange rates data sets, the fitting power of the novel generalized-Dagum model is compared with the transmuted Dagum distribution and a new modified Dagum distribution.
Open access
Statistical Distribution Estimation and Applications