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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 from par.

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, if model = dirichlet.

Value

Numeric \(n \times d\) matrix of simulations of the multivariate Pareto distribution.

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