ExponentialBernsteinFunction-class: Class for Exponential Bernstein functions

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

Class for Exponential Bernstein functions

Description

For the Exponential-jump compound Poisson subordinator with \lambda > 0, the corresponding Bernstein function is

\psi(x) = \frac{x}{x + \lambda}, x>0.

Details

For the Exponential jump CPP Bernstein function, the higher order alternating iterated forward differences are known in closed form:

{(-1)}^{k-1} \Delta^k \psi(x) = \lambda \cdot B(k+1, x+\lambda), x>0, k>0 .

This Bernstein function is no. 4 in the list of complete Bernstein functions in Chp. 16 of \insertCiteSchilling2012armo.

The Exponential Bernstein function has the Lévy density \nu:

\nu(du) = \lambda \operatorname{e}^{-\lambda u}, \quad u > 0 ,

and it has the (discrete) Stieltjes density \sigma:

\sigma(du) = \delta_{\{ \lambda \}}(du), \quad u > 0 .

Slots

lambda

The index \lambda.

References

\insertAllCited

See Also

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

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

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

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

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

Examples

# Create an object of class ExponentialBernsteinFunction
ExponentialBernsteinFunction()
ExponentialBernsteinFunction(lambda = 0.5)

# Create a Lévy density
bf <- ExponentialBernsteinFunction(lambda = 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 <- ExponentialBernsteinFunction(lambda = 0.5)
stieltjes_density <- stieltjesDensity(bf)
sum(stieltjes_density$y * 1/(1 + stieltjes_density$x))

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

# Calculate shock-arrival intensities
bf <- ExponentialBernsteinFunction(lambda = 0.8)
intensities(bf, 3)
intensities(bf, 3, method = "levy")
intensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-arrival intensities
bf <- ExponentialBernsteinFunction(lambda = 0.4)
uexIntensities(bf, 3)
uexIntensities(bf, 3, method = "levy")
uexIntensities(bf, 3, tolerance = 1e-4)

# Calculate exchangeable shock-size arrival intensities
bf <- ExponentialBernsteinFunction(lambda = 0.2)
exIntensities(bf, 3)
exIntensities(bf, 3, method = "levy")
exIntensities(bf, 3, tolerance = 1e-4)

# Calculate the Markov generator
bf <- ExponentialBernsteinFunction(lambda = 0.6)
exQMatrix(bf, 3)
exQMatrix(bf, 3, method = "levy")
exQMatrix(bf, 3, tolerance = 1e-4)

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