PoissonBernsteinFunction-class: Class for Poisson Bernstein functions

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

Class for Poisson Bernstein functions

Description

The Poisson process with arrival-rate \lambda and fixed jump size \eta is a Lévy subordinator corresponding to the Bernstein function

\psi(x) = 1 - e^{-x\eta}, x>0.

Details

For the Poisson Bernstein function, the higher-order alternating iterated forward differences can be calculated in closed form:

{(-1)}^{k-1} \Delta^k \psi(x) = e^{-u\eta} (1-e^{-\eta})^k, x>0, k>0.

The Poisson Bernstein function has the (discrete) Lévy density \nu:

\nu(du) = \delta_{\eta}(du), \quad u > 0 .

Slots

eta

The fixed (positive) jump size.

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, ParetoBernsteinFunction-class, ScaledBernsteinFunction-class, SumOfBernsteinFunctions-class

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

Other Bernstein function boundary classes: ConstantBernsteinFunction-class, LinearBernsteinFunction-class

Examples

# Create an object of class PoissonBernsteinFunction
PoissonBernsteinFunction()
PoissonBernsteinFunction(eta = 2)

# Create a Lévy density
bf <- PoissonBernsteinFunction(eta = 0.7)
levy_density <- levyDensity(bf)
sum(levy_density$y * pmin(1, levy_density$x))

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

# Calculate shock-arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.8)
intensities(bf, 3)
intensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.4)
uexIntensities(bf, 3)
uexIntensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-size arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.2)
exIntensities(bf, 3)
exIntensities(bf, 3, tolerance = 1e-4)

# Calculate the Markov generator
bf <- PoissonBernsteinFunction(eta = 0.6)
exQMatrix(bf, 3)
exQMatrix(bf, 3, tolerance = 1e-4)

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