LevyBernsteinFunction-class | R Documentation |
A virtual superclass for all Bernstein functions which can represented
by a Lévy density (no drift or killing rate). That means that there exists
a Lévy measure \nu
such that
\psi(x) = \int_0^\infty (1 - e^{-ux}) \nu(du) , x > 0 .
For continuous Lévy densities, the values of the Bernstein function are
calculated with stats::integrate()
by using the representation
\psi(x)
= \int_{0}^{\infty} (1 - \operatorname{e}^{-ux}) \nu(du), \quad x > 0 ,
and the values of the iterated differences are calculated by using the representation
(-1)^{j-1} \Delta^{j} \psi(x)
= \int_{0}^{\infty}
\operatorname{e}^{-ux} (1 - \operatorname{e}^{-u})^j \nu(du) ,
\quad x > 0 .
For discrete Lévy densities \nu(du) = \sum_{i} y_i \delta_{u_i}(du)
,
the values of the Bernstein function are calculated by using the
representation
\psi(x)
= \sum_{i} (1 - \operatorname{e}^{-u_i x}) y_i, \quad x > 0 ,
and the values of the iterated differences are calculated by using the representation
(-1)^{j-1} \Delta^{j} \psi(x)
= \sum_{i} \operatorname{e}^{-u_i x} (1 - \operatorname{e}^{-u_i})^j y_i,
\quad x > 0 .
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
,
LinearBernsteinFunction-class
,
ParetoBernsteinFunction-class
,
PoissonBernsteinFunction-class
,
ScaledBernsteinFunction-class
,
SumOfBernsteinFunctions-class
Other Virtual Bernstein function classes:
BernsteinFunction-class
,
CompleteBernsteinFunction-class
Other Levy Bernstein function classes:
AlphaStableBernsteinFunction-class
,
CompleteBernsteinFunction-class
,
ExponentialBernsteinFunction-class
,
GammaBernsteinFunction-class
,
InverseGaussianBernsteinFunction-class
,
ParetoBernsteinFunction-class
,
PoissonBernsteinFunction-class
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