BernsteinFunction-class | R Documentation |
Bernstein functions parametrize extendible Marshall–Olkin distributions. They are closed under addition, scalar multiplication, composite scalar multiplication, and consequently also convex recombination:
Pass a Bernstein function object to rextmo()
to simulate from the
associated extendible MO distribution.
Use SumOfBernsteinFunctions for adding two Bernstein functions,
Use ScaledBernsteinFunction for scalar multiplication of a Bernstein function,
Use CompositeScaledBernsteinFunction for composite scalar multiplication of a Bernstein function, and
Use ConvexCombinationOfBernsteinFunctions for convex recombination of Bernstein functions.
A Bernstein function is a nonnegative, nondecreasing, infinitely often differentiable function whose recursive finite forward differences have alternating signs:
{(-1)}^{i-1} \Delta^{i}{ \psi{(x)} }
\geq 0 ,
\quad \forall i \in \mathbb{N}, x \geq 0 .
Bernstein functions have the so-called Lévy-Khintchine representation:
\psi{(x)}
= a x + b
+ \int_{0}^{\infty}{
{\left[ 1 - e^{-x y} \right]} {\nu{(dy)}}
} ,
\quad x \geq 0 ,
for some nonnegative constants a
and b
and a Lévy measure
\nu
. A Lévy measure is a measure on the Borel sets of the nonnegative
real line that satisfies the following properties:
\int_{0}^{\infty}{
\min{\{ 1 , y \}} {\nu{(dy)}} < \infty .
}
Bernstein functions are uniquely linked to extendible Marshall–Olkin distributions via the Lévy-frailty model.
First, each Bernstein functions is uniquely linked to a Lévy subordinator via the Lévy-Khintchine representation:
\Lambda{(t)}
= \begin{cases}
b t + \Lambda_{\nu}{(t)} , & \text{if } t < \epsilon , \\
\infty , & \text{otherwise} ,
\end{cases}
where \Lambda_{\nu}{(t)}
is the Lévy subordinator associated with the
Lévy measure \nu
and \epsilon
is an independently exponentially
distributed random variable with rate a
. If \nu
is a finite
measure, the Lévy subordinator is a compound Poisson process with intensity
\nu{(0, \infty)}
and jump sizes \nu{(dy)} / \nu{((0, \infty))}
.
Second, the Lévy subordinator is unique linked to an extendible Marshall–Olkin distribution via the stochastic representation:
\tau_{i}
= \inf{\left \{ t \geq 0 : \Lambda{(t)} > E_{i} \right \}} ,
\quad 1 \leq i \leq d ,
for independently unit exponentially distributed random variables
E_{i}
.
The Lévy-frailty model motivates the following interpretation of the parameters for ext. MO distributions:
a
is the killing rate,
b
is the drift,
gamma
is a scaling factor for the total shock-arrival intensity,
family
is the name of the pure-jump Lévy measure, and
eta
are the pure-jump family parameters.
To understand the influence of these parameters on the extendible Marshall–Olkin distribution's dependence properties, the following considerations are helpful:
A pure-killing Bernstein function (i.e., a > 0
, b = 0
, and
\nu \equiv 0
) corresponds to complete comonotonicity.
A pure-drift Bernstein function (i.e., a = 0
, b > 0
, and
\nu \equiv 0
) corresponds to independence.
A pure-jump Bernstein function (i.e., a = 0
, b = 0
, and
\nu \not\equiv 0
) can model various dependence structures. However,
larger jump intensities lead to weaker dependence and larger jump sizes
lead to stronger dependence.
Consequently, weighting these cases with the parameters a
, b
, and gamma
allows for a flexible modeling of the dependence structure.
For a given Bernstein function, the marginal rate and lower-tail dependence coefficient of the associated extendible Marshall–Olkin distribution can be calculated using as follows:
\text{Marginal rate}
= \psi{(1)}
and
\text{LTDC}
= 2 - \psi{(2)} / \psi{(1)} .
In context of extendible Marshall–Olkin distributions, the following expression is frequently evaluated:
\binom{n}{k} {(-1)}^{j-1} \Delta^{j}{ \psi{( c x )} } ,
\quad 0 \leq k \leq n , j \in \mathbb{N}, x \geq 0 .
The evaluation of Bernstein functions using this formula is usually not
numerically stable. Consequently, the various alternative approaches are used
dependent on the class of the Bernstein function. Use the method valueOf()
to evaluate or approximate this expression for a given Bernstein function.
An alternative stochastic representation of an exchangeable Marshall–Olkin distributions is given by the so-called Markovian death-counting model. It defines the components' death times as randomized order statistics simulated via the Markovian death-counting processes with infinitesimal generator matrix:
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 evaluation of the infinitesimal 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. Use the
method exQMatrix()
to evaluate or approximate this expression for a given
Bernstein function.
For the all-alive-state, the generator's first row has the interpretation of exchangeable shock-size-arrival intensities:
\eta_{i}
= \binom{d}{i} {(-1)}^{i-1} \Delta^{i}{ \psi{(d-i)} } ,
\quad 1 \leq i \leq d .
As noted above, their evaluation is usually not numerically stable, and
various alternative approaches are used dependent on the class of the
Bernstein function. Use the method exIntensities()
to evaluate or
approximate them.
Another alternative stochastic representation of Marshall–Olkin distributions is
Schilling2012armo \insertRefSloot2022armo
valueOf()
, intensities()
, uexIntensities()
, exIntensities()
,
exQMatrix()
, rextmo()
, rpextmo()
Other Bernstein function classes:
AlphaStableBernsteinFunction-class
,
CompleteBernsteinFunction-class
,
CompositeScaledBernsteinFunction-class
,
ConstantBernsteinFunction-class
,
ConvexCombinationOfBernsteinFunctions-class
,
ExponentialBernsteinFunction-class
,
GammaBernsteinFunction-class
,
InverseGaussianBernsteinFunction-class
,
LevyBernsteinFunction-class
,
LinearBernsteinFunction-class
,
ParetoBernsteinFunction-class
,
PoissonBernsteinFunction-class
,
ScaledBernsteinFunction-class
,
SumOfBernsteinFunctions-class
Other Virtual Bernstein function classes:
CompleteBernsteinFunction-class
,
LevyBernsteinFunction-class
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