References

Bibliographic entries cited across the theory notes, tutorials, and user guide. Inline citations use {ref}Key ``.

Amos1986. Amos, D. E. (1986). Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Transactions on Mathematical Software, 12(3), 265–273.

AneGeman2000. Ané, T. & Geman, H. (2000). Order flow, transaction clock, and normality of asset returns. Journal of Finance, 55(5), 2259–2284.

ArevalilloNavarro2020. Arevalillo, J. M. & Navarro, H. (2020). Data projections by skewness maximization under scale mixtures of skew-normal vectors. Advances in Data Analysis and Classification, 14(2), 435–461.

ArevalilloNavarro2021. Arevalillo, J. M. & Navarro, H. (2021). Skewness-based projection pursuit as an eigenvector problem in scale mixtures of skew-normal distributions. Symmetry, 13(6), 1056.

Artzner1999. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.

AzzaliniCapitanio2014. Azzalini, A. & Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press.

BallottaBonfiglioli2016. Ballotta, L. & Bonfiglioli, E. (2016). Multivariate asset models using Lévy processes and applications. European Journal of Finance, 22(13), 1320–1350.

BarndorffNielsen1977. Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society A, 353(1674), 401–419.

BarndorffNielsen1997. Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24(1), 1–13.

BarndorffNielsenShephard2002. Barndorff-Nielsen, O. E. & Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B, 64(2), 253–280.

Blaesild1981. Blæsild, P. (1981). The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s bean data. Biometrika, 68(1), 251–263.

BreedenGibbonsLitzenberger1989. Breeden, D. T., Gibbons, M. R. & Litzenberger, R. H. (1989). Empirical tests of the consumption-oriented CAPM. Journal of Finance, 44(2), 231–262.

Cappe2009. Cappé, O. & Moulines, E. (2009). On-line expectation-maximization algorithm for latent data models. Journal of the Royal Statistical Society: Series B, 71(3), 593–613.

CarrWu2009. Carr, P. & Wu, L. (2009). Variance risk premiums. Review of Financial Studies, 22(3), 1311–1341.

Clark1973. Clark, P. K. (1973). A subordinated stochastic process model with finite variance for speculative prices. Econometrica, 41(1), 135–155.

Day1969. Day, N. E. (1969). Estimating the components of a mixture of normal distributions. Biometrika, 56(3), 463–474.

Dempster1977. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B, 39(1), 1–38.

Devroye2014. Devroye, L. (2014). Random variate generation for the generalized inverse Gaussian distribution. Statistics and Computing, 24(2), 239–246.

DLMF. Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V., Cohl, H. S., & McClain, M. A. (eds.). NIST Digital Library of Mathematical Functions. Release 1.2.x. https://dlmf.nist.gov/.

HormannLeydold2011. Hörmann, W. & Leydold, J. (2011). Generating generalized inverse Gaussian random variates by fast inversion. Computational Statistics & Data Analysis, 55(1), 213–217.

HormannLeydold2014. Hörmann, W. & Leydold, J. (2014). Generating generalized inverse Gaussian random variates. Statistics and Computing, 24(4), 547–557.

HubermanKandelStambaugh1987. Huberman, G., Kandel, S. & Stambaugh, R. F. (1987). Mimicking portfolios and exact arbitrage pricing. Journal of Finance, 42(1), 1–9.

Hu2005. Hu, W. (2005). Calibration of multivariate generalized hyperbolic distributions using the EM algorithm.

Jorgensen2012. Jørgensen, B. (2012). Statistical Properties of the Generalized Inverse Gaussian Distribution. Springer.

Lamont2001. Lamont, O. A. (2001). Economic tracking portfolios. Journal of Econometrics, 105(1), 161–184.

Loperfido2010. Loperfido, N. (2010). Canonical transformations of skew-normal variates. TEST, 19, 146–165.

LucianoMarenaSemeraro2016. Luciano, E., Marena, M. & Semeraro, P. (2016). Dependence calibration and portfolio fit with factor-based subordinators. Quantitative Finance, 16(7), 1037–1052.

LucianoSemeraro2010. Luciano, E. & Semeraro, P. (2010). Multivariate time changes for Lévy asset models: characterization and calibration. Journal of Computational and Applied Mathematics, 233.

MadanSeneta1990. Madan, D. B. & Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business, 63(4), 511–524.

Markowitz1952. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.

McNeil2010. McNeil, A. J., Frey, R., & Embrechts, P. (2010). Quantitative Risk Management. Princeton University Press.

McLachlanKrishnan2008. McLachlan, G. J. & Krishnan, T. (2008). The EM Algorithm and Extensions, 2nd ed. Wiley. Aitken acceleration of the observed-data log-likelihood sequence: remaining gap \(\ell_\infty-\ell_{t+1}=a\Delta_2/(1-a)\) with \(a=\Delta_2/\Delta_1\).

MenciaSentana2009. Mencía, J. & Sentana, E. (2009). Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation. Journal of Econometrics, 153(2), 105–121.

MengRubin1993. Meng, X.-L. & Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika, 80(2), 267–278.

Meucci2010. Meucci, A. (2010). Managing diversification. Risk Magazine, 22(5), 74–79.

Meucci2014. Meucci, A., Santangelo, A., & Deguest, R. (2014). Measuring portfolio diversification based on optimized uncorrelated factors. SSRN Electronic Journal.

MurphyIzzeldin2010. Murphy, A. & Izzeldin, M. (2010). Recovering the moments of information flow and the normality of asset returns. Applied Financial Economics, 20.

Protassov2004. Protassov, R. S. (2004). EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions.

RauHasanov2004. Rau-Bredow, H. (2004). Value-at-risk, expected shortfall, and marginal risk contribution. In Risk Measures for the 21st Century, Wiley.

Renyi1961. Rényi, A. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 547–561. University of California Press.

RichardsonSmith1994. Richardson, M. & Smith, T. (1994). A direct test of the mixture of distributions hypothesis: measuring the daily flow of information. Journal of Financial and Quantitative Analysis, 29(1), 101–116.

RobbinsMonro1951. Robbins, H. & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics, 22(3), 400–407.

RockafellarUryasev2000. Rockafellar, R. T. & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21–41.

Semeraro2008. Semeraro, P. (2008). A multivariate variance gamma model for financial applications. International Journal of Theoretical and Applied Finance, 11(1), 1–18.

Shi2016. Shi, X. (2016). Generalized Hyperbolic Distributions and Related Topics. PhD Thesis.

Stankyavichyus2026. Stankyavichyus, A. V. (2026). Varentropy: Overview, Computational Routes, and Structural Decomposition. Preprint.

Takekawa2022. Takekawa, T. (2022). Fast parallel calculation of modified Bessel function of the second kind and its derivatives. SoftwareX, 17, 100923. https://arxiv.org/abs/2108.11560.

Tasche1999. Tasche, D. (1999). Risk contributions and performance measurement. Report of the Lehrstuhl für mathematische Statistik, TU München.

Tortora2013. Tortora, C., McNicholas, P. D., & Browne, R. P. (2013). Mixtures of multivariate generalized hyperbolic distributions.

WilsonHilferty1931. Wilson, E. B. & Hilferty, M. M. (1931). The distribution of chi-square. Proceedings of the National Academy of Sciences, 17(12), 684–688.