Class for Poisson Bernstein functions
Source:R/s4-PoissonBernsteinFunction.R
PoissonBernsteinFunction-class.RdThe 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 \(\nu\): $$ \nu(du) = \delta_{\eta}(du), \quad u > 0 . $$
See also
getLevyDensity(), calcIterativeDifference(),
calcShockArrivalIntensities(), calcExShockArrivalIntensities(),
calcExShockSizeArrivalIntensities(), calcMDCMGeneratorMatrix(),
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()
#> An object of class "PoissonBernsteinFunction"
#> (invalid or not initialized)
PoissonBernsteinFunction(eta = 2)
#> An object of class "PoissonBernsteinFunction"
#> - eta: 2
# Create a Lévy density
bf <- PoissonBernsteinFunction(eta = 0.7)
levy_density <- getLevyDensity(bf)
sum(levy_density$y * pmin(1, levy_density$x))
#> [1] 0.7
# Evaluate the Bernstein function
bf <- PoissonBernsteinFunction(eta = 0.3)
calcIterativeDifference(bf, 1:5)
#> [1] 0.2591818 0.4511884 0.5934303 0.6988058 0.7768698
# Calculate shock-arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.8)
calcShockArrivalIntensities(bf, 3)
#> [1] 0.1111786 0.1111786 0.1362539 0.1111786 0.1362539 0.1362539 0.1669847
calcShockArrivalIntensities(bf, 3, tolerance = 1e-4)
#> [1] 0.1111786 0.1111786 0.1362539 0.1111786 0.1362539 0.1362539 0.1669847
# Calculate exchangeable shock-arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.4)
calcExShockArrivalIntensities(bf, 3)
#> [1] 0.14813475 0.07285633 0.03583254
calcExShockArrivalIntensities(bf, 3, tolerance = 1e-4)
#> [1] 0.14813475 0.07285633 0.03583254
# Calculate exchangeable shock-size arrival intensities
bf <- PoissonBernsteinFunction(eta = 0.2)
calcExShockSizeArrivalIntensities(bf, 3)
#> [1] 0.364525230 0.080706891 0.005956243
calcExShockSizeArrivalIntensities(bf, 3, tolerance = 1e-4)
#> [1] 0.364525230 0.080706891 0.005956243
# Calculate the Markov generator
bf <- PoissonBernsteinFunction(eta = 0.6)
calcMDCMGeneratorMatrix(bf, 3)
#> [,1] [,2] [,3] [,4]
#> [1,] -0.8347011 0.4076860 0.3351663 0.09184884
#> [2,] 0.0000000 -0.6988058 0.4952348 0.20357094
#> [3,] 0.0000000 0.0000000 -0.4511884 0.45118836
#> [4,] 0.0000000 0.0000000 0.0000000 0.00000000
calcMDCMGeneratorMatrix(bf, 3, tolerance = 1e-4)
#> [,1] [,2] [,3] [,4]
#> [1,] -0.8347011 0.4076860 0.3351663 0.09184884
#> [2,] 0.0000000 -0.6988058 0.4952348 0.20357094
#> [3,] 0.0000000 0.0000000 -0.4511884 0.45118836
#> [4,] 0.0000000 0.0000000 0.0000000 0.00000000