Multivariate Distributions

Multivariate Distributions in Non-Stationary Complex Systems II: Empirical Results for Correlated Stock Markets ArXiv ID: 2412.11602 “View on arXiv” Authors: Unknown Abstract Multivariate Distributions are needed to capture the correlation structure of complex systems. In previous works, we developed a Random Matrix Model for such correlated multivariate joint probability density functions that accounts for the non-stationarity typically found in complex systems. Here, we apply these results to the returns measured in correlated stock markets. Only the knowledge of the multivariate return distributions allows for a full-fledged risk assessment. We analyze intraday data of 479 US stocks included in the S&P500 index during the trading year of 2014. We focus particularly on the tails which are algebraic and heavy. The non-stationary fluctuations of the correlations make the tails heavier. With the few-parameter formulae of our Random Matrix Model we can describe and quantify how the empirical distributions change for varying time resolution and in the presence of non-stationarity. ...

Generalization of the Alpha-Stable Distribution with the Degree of Freedom ArXiv ID: 2405.04693 “View on arXiv” Authors: Unknown Abstract A Wright function based framework is proposed to combine and extend several distribution families. The $α$-stable distribution is generalized by adding the degree of freedom parameter. The PDF of this two-sided super distribution family subsumes those of the original $α$-stable, Student’s t distributions, as well as the exponential power distribution and the modified Bessel function of the second kind. Its CDF leads to a fractional extension of the Gauss hypergeometric function. The degree of freedom makes possible for valid variance, skewness, and kurtosis, just like Student’s t. The original $α$-stable distribution is viewed as having one degree of freedom, that explains why it lacks most of the moments. A skew-Gaussian kernel is derived from the characteristic function of the $α$-stable law, which maximally preserves the law in the new framework. To facilitate such framework, the stable count distribution is generalized as the fractional extension of the generalized gamma distribution. It provides rich subordination capabilities, one of which is the fractional $χ$ distribution that supplies the needed ‘degree of freedom’ parameter. Hence, the “new” $α$-stable distribution is a “ratio distribution” of the skew-Gaussian kernel and the fractional $χ$ distribution. Mathematically, it is a new form of higher transcendental function under the Wright function family. Last, the new univariate symmetric distribution is extended to the multivariate elliptical distribution successfully. ...

Copulas forFinance- A Reading Guide and Some Applications ArXiv ID: ssrn-1032533 “View on arXiv” Authors: Unknown Abstract Copulas are a general tool to construct multivariate distributions and to investigate dependence structure between random variables. However, the concept of cop Keywords: Copulas, Multivariate Distributions, Dependence Structure, Random Variables, Statistical Modeling, Quantitative Methods Complexity vs Empirical Score Math Complexity: 7.5/10 Empirical Rigor: 2.0/10 Quadrant: Lab Rats Why: The paper focuses on theoretical copula constructions and dependence structures with advanced mathematics, but lacks implementation details, backtests, or empirical data. flowchart TD A["Research Goal:<br>Review Copulas for Finance"] --> B["Key Methodology:<br>Literature Review & Analysis"] B --> C["Data/Input:<br>Financial Return Datasets<br>and Models"] C --> D["Computational Process:<br>Model Fitting &<br>Dependence Estimation"] D --> E["Key Outcomes:<br>Capturing Non-Linear Dependence<br>and Risk Assessment"]