rmo | R Documentation |
Draws n
iid samples from a d
-variate Marshall–Olkin distribution
parametrized by shock-arrival intensities.
rmo(n, d, intensities, method = c("AM", "ESM"))
n |
an integer for the number of samples. |
d |
an integer for the dimension. |
intensities |
a numeric vector for the shock-arrival intensities. |
method |
a string indicating which sampling algorithm should be used. Use "AM" for the Arnold model and "ESM" for the exogenous shock model. We recommend using the ESM for small dimensions only; the AM can be used up until dimension 30. |
The Marshall–Olkin distribution was introduced in \insertCiteMarshall1967armo and has the survival function
\bar{F}{(t)} = \exp{≤ft\{ - ∑_{I} λ_I \max_{i \in I} t_i \right\}} , \quad t = {(t_{1}, …, t_{d})} > 0 ,
for shock-arrival intensities λ_I ≥q 0, \emptyset \neq I \subseteq {\{ 1 , …, d \}}. They are called shock-arrival intensities as they correspond to the rates of independent exponential random variables from the exogenous shock model (ESM), and a shock-arrival intensity λ_{I} of shock I equal to zero implies that the the shock I never arrives. We use the following binary representation to map a subsets of {1, …, d} to integers 0, …, 2^d-1:
I \equiv ∑_{k \in I}{ 2^{k-1} }
The exogenous shock model (ESM) simulates a Marshall–Olkin distributed random vector via independent exponentially distributed shock times for all non-empty subsets of components and defines each component as the minimum of all shock times corresponding to a subset containing this component, see \insertCite@see pp. 104 psqq. @Mai2017armo and \insertCiteMarshall1967armo.
The Arnold model (AM) simulates a Marshall–Olkin distributed random vector by simulating a marked homogeneous Poisson process with set-valued marks. The process is stopped when all components are hit by a shock, see \insertCite@see Sec. 3.1.2 @Mai2017armo and \insertCiteArnold1975armo.
rmo
returns a numeric matrix with n
rows and d
columns, with
the rows corresponding to iid distributed samples of a d
-variate
Marshall–Olkin distribution with shock-arrival intensities
intensities
.
Other sampling-algorithms:
rexmo()
,
rextmo()
,
rpextmo()
rmo( 10, 3, c(0.4, 0.4, 0.1, 0.4, 0.1, 0.1, 0.4) ) ## independence rmo( 10, 3, c(1, 1, 0, 1, 0, 0, 0) ) ## comonotone rmo( 10, 3, c(0, 0, 0, 0, 0, 0, 1) ) rmo( 10, 3, c(0.4, 0.4, 0.1, 0.4, 0.1, 0.1, 0.4), method = "ESM" ) ## independence rmo( 10, 3, c(1, 1, 0, 1, 0, 0, 0), method = "ESM" ) ## comonotone rmo( 10, 3, c(0, 0, 0, 0, 0, 0, 1), method = "ESM" ) rmo( 10, 3, c(0.4, 0.4, 0.1, 0.4, 0.1, 0.1, 0.4), method = "AM" ) ## independence rmo( 10, 3, c(1, 1, 0, 1, 0, 0, 0), method = "AM" ) ## comonotone rmo( 10, 3, c(0, 0, 0, 0, 0, 0, 1), method = "AM" )
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