Mathematical Background

This section provides the mathematical foundation for the distributions implemented in normix, based on [Shi2016].

Contents:

• The Generalized Inverse Gaussian Distribution
  • Definition
  • Alternative Parameterization
  • Moment Generating Function
  • Moments
  • Exponential Family Form
  • Maximum Likelihood Estimation
  • Numerical Challenges
  • Hellinger Distance
  • Special Cases
  • References
• The Generalized Hyperbolic Distribution
  • Definition as Normal Mixture
  • Joint GH Distribution
  • Marginal GH Density
  • Alternative Parameterization
  • Model Identifiability
  • Exponential Family Form
  • Recovering Parameters from Expectations
  • Hellinger Distance
  • Numerical Stability
  • Special Cases
  • References
• EM Algorithm for Generalized Hyperbolic Distributions
  • Overview
  • Conditional Distribution of Y given X
  • Conditional Expectations
  • The EM Algorithm
  • Parameter Regularization
  • MCECM Algorithm
  • Special Cases
  • Numerical Considerations
  • Implementation in normix
  • References
• Shrinkage with Penalized Likelihood
  • Penalized Likelihood Framework
  • Penalized EM Algorithm
  • Shrunk Sufficient Statistics
• Factor Analysis for Generalized Hyperbolic Distributions
  • Joint Distribution
  • Curved Exponential Family
  • Log-Likelihood Function
  • M-Step: Closed-Form Solutions
  • Conditional Expectations for the E-Step
  • E-Step
  • References
• Online EM Algorithm
  • Online EM for Exponential Families
  • Application to the GH Distribution
  • Limitations for Curved Exponential Families
  • References
• Mean-Risk Optimization for Normal Mixture Distributions
  • Coherent Risk Measures
  • Risk Monotonicity for Normal Mixtures
  • Portfolio Return as Normal Mixture
  • Mean-Risk Optimization
  • Dimension Reduction via the Efficient Surface
  • Worst-Case Risk Measures
  • References
• CVaR Derivatives for Normal Mixture Distributions
  • General CVaR Derivatives
  • Application to Univariate Normal Mixtures
  • Portfolio CVaR Gradient and Hessian
  • References
• Portfolio Optimization with Transaction Costs
  • Problem Formulation
  • Quadratic Approximation
  • Reduction to Quadratic Programming
• Effective Number of Bets and Minimum Torsion
  • Variance-Based Risk Decomposition
  • Uncorrelated Factor Decomposition
  • Effective Number of Bets
  • Characterizing Diagonalizations
  • Minimum Torsion
  • References
• Generalized Effective Number of Bets
  • Homogeneous Functions and Risk Decomposition
  • Local Diagonalization via the Hessian
  • Generalized ENB
  • Application to Coherent Risk Measures
  • References

References

[Shi2016]

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