rexmo: Simulate from exchangeable Marshall–Olkin distributions

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

Simulate from exchangeable Marshall–Olkin distributions

Description

Draws n iid samples from a d-variate exchangeable Marshall–Olkin distribution parametrized by a vector of exchangeable shock-size arrival intensities.

Usage

rexmo(n, d, ex_intensities, method = c("MDCM", "AM", "ESM"))

Arguments

n	An integer for the number of samples.
d	An integer for the dimension.
ex_intensities	A numeric vector with the exchangeable shock-size arrival intensities.
method	A string indicating which sampling algorithm should be used. Use "MDCM" for the Markovian death-counting model, "AM" for the Arnold model, and "ESM" for the exogenous shock model. We recommend using the ESM only for small dimensions; the AM can be used up until dimension 30.

Details

The exchangeable Marshall–Olkin distribution has the survival function

\bar{F}{(t)} = \exp{\left\{ -\sum_{i=1}^{d}{ {\left[ \sum_{j=0}^{d-i}{ \binom{d-i}{j} \lambda_{j+1} } \right]} \tau_{[i]} } \right\}} , \quad t = {(t_{1}, \ldots, t_{d})} > 0 ,

for exchangeable shock arrival intensities \lambda_{i} \geq 0, 1 \leq i \leq d and t_{[1]} \geq \cdots \geq t_{[d]}, see \insertCiteMai2017armo.

The relationship of exchangeable shock-size arrival intensities to the shock-arrival intensities of the Marshall–Olkin distribution is given by:

\eta_{i} = \binom{d}{i} \lambda_{i} , \quad i \in {\{1, \ldots, n\}} .

The exchangeable shock-size arrival intensities correspond to the initial transition rates of independent exponential random variables in the Markovian death-counting model (MDCM).

Simulation algorithms

• The Markovian death-counting model (MDCM) is a simulation algorithm used to generate samples from an exchangeable Marshall-Olkin distribution. It simulates the death-counting process of the random vector, which is a Markov process, until all components are "dead". This process defines an order statistic that is then used to obtain a sample through a random permutation. For more details on this algorithm, refer to \insertCiteSloot2022armo.

• The exogenous shock model (ESM) and Arnold model (AM) simulation algorithms can be used to generate samples from the general Marshall–Olkin distribution. In these algorithms, the exchangeable shock-size arrival intensities are converted to the corresponding shock-arrival intensities and passed to the rmo() function.

Value

rexmo returns a numeric matrix of size n x d. Each row corresponds to an independently and identically (iid) distributed sample of a d-variate exchangeable Marshall–Olkin distribution with specified parameters.

References

\insertAllCited

See Also

Other sampling-algorithms: rextmo(), rmo(), rpextmo()

Examples

rexmo(
  10, 3,
  c(1.2, 0.3, 0.4)
)
## independence
rexmo(
  10, 3,
  c(3, 0, 0)
)
## comonotone
rexmo(
  10, 3,
  c(0, 0, 1)
)

rexmo(
  10, 3,
  c(1.2, 0.3, 0.4),
  method = "MDCM"
)
## independence
rexmo(
  10, 3,
  c(3, 0, 0),
  method = "MDCM"
)
## comonotone
rexmo(
  10, 3,
  c(0, 0, 1),
  method = "MDCM"
)

rexmo(
  10, 3,
  c(1.2, 0.3, 0.4),
  method = "AM"
)
## independence
rexmo(
  10, 3,
  c(3, 0, 0),
  method = "AM"
)
## comonotone
rexmo(
  10, 3,
  c(0, 0, 1),
  method = "AM"
)

rexmo(
  10, 3,
  c(1.2, 0.3, 0.4),
  method = "ESM"
)
## independence
rexmo(
  10, 3,
  c(3, 0, 0),
  method = "ESM"
)
## comonotone
rexmo(
  10, 3,
  c(0, 0, 1),
  method = "ESM"
)

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