ParetoBernsteinFunction-class: Class for Pareto Bernstein functions

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

Class for Pareto Bernstein functions

Description

For the Pareto-jump compound Poisson process with index 0 < \alpha < 1 and cutoff point x0, the corresponding Bernstein function is

\psi(x) = 1 - e^{-x x_0} + (x_0 x)^\alpha \Gamma(1-\alpha, x_0 x) , x>0 .

Details

For this 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_{x_0}^\infty e^{-ux} (1-e^{-u})^k \alpha \frac{{x_0}^\alpha}{t^{1+\alpha}} du, x>0, k>0 .

The Pareto Bernstein function has the Lévy density \nu:

\nu(du) = \alpha \frac{x_0^\alpha}{u^{\alpha + 1}}, \quad u > x_0 .

The Pareto Bernstein function, in combination with a linear Bernstein function can be used to approximate the Bernstein function of an \alpha-stable subordinator, see Sec. 5.3 of \insertCiteFernandez2015armo.

Slots

alpha

The index \alpha

x0

The cutoff point x_0

References

\insertAllCited

See Also

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

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

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

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

Examples

# Create an object of class ParetoBernsteinFunction
ParetoBernsteinFunction()
ParetoBernsteinFunction(alpha = 0.2, x0 = 1e-2)

# Create a Lévy density
bf <- ParetoBernsteinFunction(alpha = 0.7, x0 = 1e-2)
levy_density <- levyDensity(bf)
integrate(
  function(x) pmin(1, x) * levy_density(x),
  lower = attr(levy_density, "lower"),
  upper = attr(levy_density, "upper")
)

# Evaluate the Bernstein function
bf <- ParetoBernsteinFunction(alpha = 0.3, x0 = 1)
valueOf(bf, 1:5)

# Calculate shock-arrival intensities
bf <- ParetoBernsteinFunction(alpha = 0.8, x0 = 1e-2)
intensities(bf, 3)
intensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-arrival intensities
bf <- ParetoBernsteinFunction(alpha = 0.4, x0 = 1e-2)
uexIntensities(bf, 3)
uexIntensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-size arrival intensities
bf <- ParetoBernsteinFunction(alpha = 0.2, x0 = 1e-2)
exIntensities(bf, 3)
exIntensities(bf, 3, tolerance = 1e-4)

# Calculate the Markov generator
bf <- ParetoBernsteinFunction(alpha = 0.6, x0 = 1e-2)
exQMatrix(bf, 3)
exQMatrix(bf, 3, tolerance = 1e-4)

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