calcMDCMGeneratorMatrix: Calculate the MDCM Markovian generator matrix

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

Calculate the MDCM Markovian generator matrix

Description

Calculates the infinitesimal Markov generator matrix of the corresponding (Markovian) default-counting process, used internally by rexmo().

Usage

calcMDCMGeneratorMatrix(object, d, cscale = 1, ...)

Arguments

object	An object deriving from the class BernsteinFunction.
d	A positive integer, larger than two, for the dimension.
cscale	A positive number for the composite scaling factor.
...	pass-through parameter.

Details

For a given Bernstein function, the Markov generator matrix is defined as the upper triangular matrix with elements

q_{i, j}^\ast = \binom{d-i}{j-i} \begin{cases} -\psi{(d-i)} & \text{if } i = j , \\ {(-1)}^{j-i-1} \Delta^{j-i}{ \psi{(d-i)} } & \text{if } i < j , \\ 0 & \text{otherwise} . \end{cases}

The calculation of the Markov generator matrix using this formula is usually not numerically stable. Consequently, the various alternative approaches are used dependent on the class of the Bernstein function.

The (upper triangular) infinitesimal Markov generator of the associated death-counting process is calculated recursively:

q_{0, i}^\ast = \eta_{i} , \quad i \in {\{ 1 , \ldots , d \}} ,

and

q_{i+1, j+1}^\ast = \frac{d-j}{d-i} q_{i,j}^\ast + \frac{j+1-i}{d-i} q_{i, j+1}^\ast , \quad 0 \leq i < j \leq d .

See Also

rexmo()

Other Bernstein function generics: calcExShockArrivalIntensities(), calcExShockSizeArrivalIntensities(), calcIterativeDifference(), calcShockArrivalIntensities(), calcValue(), getDefaultMethodString(), getLevyDensity(), getStieltjesDensity()

Examples

bf <- AlphaStableBernsteinFunction(alpha = 0.7)
calcMDCMGeneratorMatrix(bf, 3)

hsloot/rmo documentation built on May 1, 2024, 4:28 a.m.