Simulates exact samples of a multivariate Pareto distribution.
Usage
rmpareto(
n,
model = c("HR", "logistic", "neglogistic", "dirichlet"),
d = NULL,
par
)
Arguments
- n
Number of simulations.
- model
The parametric model type; one of:
HR
(default),logistic
,neglogistic
,dirichlet
.
- d
Dimension of the multivariate Pareto distribution. In some cases this can be
NULL
and will be inferred frompar
.- par
Respective parameter for the given
model
, that is,\(\Gamma\), numeric \(d \times d\) variogram matrix, if
model = HR
.\(\theta \in (0, 1)\), if
model = logistic
.\(\theta > 0\), if
model = neglogistic
.\(\alpha\), numeric vector of size
d
with positive entries, ifmodel = dirichlet
.
Details
The simulation follows the algorithm in Engelke and Hitz (2020) . For details on the parameters of the Huesler-Reiss, logistic and negative logistic distributions see Dombry et al. (2016) , and for the Dirichlet distribution see Coles and Tawn (1991) .
References
Coles S, Tawn JA (1991).
“Modelling extreme multivariate events.”
J. R. Stat. Soc. Ser. B Stat. Methodol., 53, 377--392.
Dombry C, Engelke S, Oesting M (2016).
“Exact simulation of max-stable processes.”
Biometrika, 103, 303--317.
Engelke S, Hitz AS (2020).
“Graphical models for extremes (with discussion).”
J. R. Stat. Soc. Ser. B Stat. Methodol., 82, 871--932.
See also
Other sampling functions:
rmpareto_tree()
,
rmstable()
,
rmstable_tree()
Examples
## A 4-dimensional HR distribution
n <- 10
d <- 4
G <- cbind(
c(0, 1.5, 1.5, 2),
c(1.5, 0, 2, 1.5),
c(1.5, 2, 0, 1.5),
c(2, 1.5, 1.5, 0)
)
rmpareto(n, "HR", d = d, par = G)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.6242115 1.2340786 0.28424553 0.4945530
#> [2,] 7.7306138 0.5739548 5.36225882 0.9660429
#> [3,] 6.3546256 12.9548933 4.73217785 4.2658041
#> [4,] 0.6558634 1.5353209 0.49389134 0.5448121
#> [5,] 42.8554548 8.5273448 31.33763315 55.9390571
#> [6,] 1.6246369 0.1199755 2.05703805 0.6370474
#> [7,] 1.1139422 0.2996312 0.36922986 0.8431961
#> [8,] 1.3597143 1.6552748 5.86433799 1.1239351
#> [9,] 1.6800225 0.2931093 0.08781558 0.1119706
#> [10,] 0.6128817 1.2130212 0.90993956 0.4373595
## A 3-dimensional logistic distribution
n <- 10
d <- 3
theta <- .6
rmpareto(n, "logistic", d, par = theta)
#> [,1] [,2] [,3]
#> [1,] 8.24837685 10.69857583 2.4734847
#> [2,] 1.14082522 0.58671413 1.2991381
#> [3,] 1.05199869 0.01952548 0.1518117
#> [4,] 0.09815916 2.09836473 0.9171968
#> [5,] 1.44549755 0.03732527 0.1230960
#> [6,] 0.87907805 2.96681227 1.6638072
#> [7,] 0.25061503 1.47779913 0.2151474
#> [8,] 1.05973104 0.40336846 0.1357882
#> [9,] 0.21627479 1.27101068 0.5378615
#> [10,] 0.76093562 0.58420217 1.3596907
## A 5-dimensional negative logistic distribution
n <- 10
d <- 5
theta <- 1.5
rmpareto(n, "neglogistic", d, par = theta)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.0410953 1.0368430 3.0641360 0.4899866 0.3413277
#> [2,] 0.8043985 1.4684191 0.3371993 2.4377260 2.0067296
#> [3,] 0.8398382 1.3174039 0.3297471 1.1210441 0.7363261
#> [4,] 1.6320666 0.4856634 0.2237380 0.1971754 0.6634043
#> [5,] 1.6686381 0.2726166 0.7718576 0.7949440 0.2269781
#> [6,] 0.7743841 1.7759363 4.0861415 0.3360490 3.0394581
#> [7,] 0.8424968 0.8434586 1.6880089 0.2612004 0.4319269
#> [8,] 4.2622232 5.1144089 1.3030681 5.3718415 1.0723264
#> [9,] 10.4418632 3.3544738 5.1462313 2.2192071 17.9442460
#> [10,] 1.1775948 0.5574875 1.1206045 6.1359100 1.2798100
## A 4-dimensional Dirichlet distribution
n <- 10
d <- 4
alpha <- c(.8, 1, .5, 2)
rmpareto(n, "dirichlet", d, par = alpha)
#> [,1] [,2] [,3] [,4]
#> [1,] 0.54438164 0.03495636 0.89308764 1.6750301
#> [2,] 4.66487201 0.28122718 2.04269019 1.5075597
#> [3,] 2.60696064 1.67971723 1.70213708 1.5252516
#> [4,] 0.07793149 0.38886068 1.15564558 0.1560207
#> [5,] 2.97816809 0.28430839 0.77949134 1.9339135
#> [6,] 1.44739369 1.33222612 0.01962053 0.3117005
#> [7,] 1.08362319 0.32159735 0.52307518 0.1984790
#> [8,] 0.03959297 0.01409648 0.60053325 4.2018204
#> [9,] 17.73588870 0.15360482 9.95951584 8.4454189
#> [10,] 4.03849379 0.94996230 1.76963146 1.4442792