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

References

[Shi2016]

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