AlphaStableBernsteinFunction-class: Class for the alpha-stable Bernstein function

In hsloot/rmo: A package for the Marshall-Olkin distribution

Class for the \alpha-stable Bernstein function

Description

For the \alpha-stable Lévy subordinator with 0 < \alpha < 1, the corresponding Bernstein function is the power function with exponent \alpha, i.e.

\psi(x) = x^\alpha, \quad x>0.

Details

For the \alpha-stable Bernstein function, the higher order alternating iterated forward differences are known in closed form but cannot be evaluated numerically without the danger of loss of significance. But we can use numerical integration (here: stats::integrate()) to approximate it with the following representation:

{(-1)}^{k-1} \Delta^k \psi(x) = \int_0^\infty e^{-ux} (1-e^{-u})^k \alpha \frac{1}{\Gamma(1-\alpha) u^{1+\alpha}} du, x>0, k>0 .

This Bernstein function is no. 1 in the list of complete Bernstein functions in Chp. 16 of \insertCiteSchilling2012armo.

The \alpha-stable Bernstein function has the Lévy density \nu:

\nu(du) = \frac{\alpha}{\Gamma(1-\alpha)} u^{-1 - \alpha} , \quad u > 0 ,

and it has the Stieltjes density \sigma:

\sigma(du) = \frac{\sin(\alpha \pi)}{\pi} u^{\alpha - 1}, \quad u > 0 .

Slots

alpha

The index \alpha.

References

\insertAllCited

See Also

levyDensity(), stieltjesDensity(), valueOf(), intensities(), uexIntensities(), exIntensities(), exQMatrix(), rextmo(), rpextmo()

Other Bernstein function classes: BernsteinFunction-class, CompleteBernsteinFunction-class, CompositeScaledBernsteinFunction-class, ConstantBernsteinFunction-class, ConvexCombinationOfBernsteinFunctions-class, ExponentialBernsteinFunction-class, GammaBernsteinFunction-class, InverseGaussianBernsteinFunction-class, LevyBernsteinFunction-class, LinearBernsteinFunction-class, ParetoBernsteinFunction-class, PoissonBernsteinFunction-class, ScaledBernsteinFunction-class, SumOfBernsteinFunctions-class

Other Levy Bernstein function classes: CompleteBernsteinFunction-class, ExponentialBernsteinFunction-class, GammaBernsteinFunction-class, InverseGaussianBernsteinFunction-class, LevyBernsteinFunction-class, ParetoBernsteinFunction-class, PoissonBernsteinFunction-class

Other Complete Bernstein function classes: ExponentialBernsteinFunction-class, GammaBernsteinFunction-class, InverseGaussianBernsteinFunction-class

Other Algebraic Bernstein function classes: ExponentialBernsteinFunction-class, InverseGaussianBernsteinFunction-class, ParetoBernsteinFunction-class

Examples

# Create an object of class AlphaStableBernsteinFunction
AlphaStableBernsteinFunction()
AlphaStableBernsteinFunction(alpha = 0.5)

# Create a Lévy density
bf <- AlphaStableBernsteinFunction(alpha = 0.7)
levy_density <- levyDensity(bf)
integrate(
  function(x) pmin(1, x) * levy_density(x),
  lower = attr(levy_density, "lower"),
  upper = attr(levy_density, "upper")
)

# Create a Stieltjes density
bf <- AlphaStableBernsteinFunction(alpha = 0.5)
stieltjes_density <- stieltjesDensity(bf)
integrate(
  function(x) 1/(1 + x) * stieltjes_density(x),
  lower = attr(stieltjes_density, "lower"),
  upper = attr(stieltjes_density, "upper")
)

# Evaluate the Bernstein function
bf <- AlphaStableBernsteinFunction(alpha = 0.3)
valueOf(bf, 1:5)

# Calculate shock-arrival intensities
bf <- AlphaStableBernsteinFunction(alpha = 0.8)
intensities(bf, 3)
intensities(bf, 3, method = "stieltjes")
intensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-arrival intensities
bf <- AlphaStableBernsteinFunction(alpha = 0.4)
uexIntensities(bf, 3)
uexIntensities(bf, 3, method = "stieltjes")
uexIntensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-size arrival intensities
bf <- AlphaStableBernsteinFunction(alpha = 0.2)
exIntensities(bf, 3)
exIntensities(bf, 3, method = "stieltjes")
exIntensities(bf, 3, tolerance = 1e-4)

# Calculate the Markov generator
bf <- AlphaStableBernsteinFunction(alpha = 0.6)
exQMatrix(bf, 3)
exQMatrix(bf, 3, method = "stieltjes")
exQMatrix(bf, 3, tolerance = 1e-4)

hsloot/rmo documentation built on April 25, 2024, 10:41 p.m.