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Sampling of a multivariate Pareto distribution

Source: R/simulation_functions.R
rmpareto.Rd

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,] 4.615955e-02 6.271463e-02 4.679932e-01   1.5407978
#>  [2,] 1.195558e+00 6.472809e-01 5.718531e-01   0.4792028
#>  [3,] 1.819392e+00 3.652545e-01 3.662150e+00   0.3761302
#>  [4,] 2.045067e-01 1.301331e+00 5.665407e-02   0.1698795
#>  [5,] 4.908828e-01 1.173293e-01 2.697708e-01   1.2007516
#>  [6,] 1.896467e+00 3.308719e-01 9.912923e-02   0.1263963
#>  [7,] 3.783434e+03 7.740369e+03 1.928350e+03 955.7516474
#>  [8,] 4.075726e-02 1.452662e-01 2.243282e-01   1.3937023
#>  [9,] 1.001874e+00 4.463799e-01 6.635540e-02   0.1160326
#> [10,] 3.485334e+00 3.876919e+00 5.173686e+00   3.9681296

## A 3-dimensional logistic distribution
n <- 10
d <- 3
theta <- .6
rmpareto(n, "logistic", d, par = theta)
#>             [,1]        [,2]       [,3]
#>  [1,] 2.49378501  0.36932494 0.33515179
#>  [2,] 0.99676085  0.83357935 2.32840596
#>  [3,] 1.46294810  0.39503110 0.16716777
#>  [4,] 0.04376721  0.07510183 1.41969112
#>  [5,] 1.36830518  0.67856567 0.65795762
#>  [6,] 1.19275582  9.60772370 0.76057218
#>  [7,] 2.09232247  6.71823647 1.09297062
#>  [8,] 1.03106621  0.14916308 0.19576513
#>  [9,] 0.09086568  1.77468972 0.07800615
#> [10,] 0.90695285 10.56331525 0.77231856

## 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,] 7.3511435 3.9328194 2.6864731 6.1845938 0.4765336
#>  [2,] 1.8865512 3.3529559 7.2595824 1.0236510 3.1863134
#>  [3,] 4.0757165 0.7668703 1.4644710 7.0899404 1.9207669
#>  [4,] 7.9898612 8.1998763 2.6566668 5.5068894 1.6494460
#>  [5,] 1.0975102 1.3297490 0.9195467 0.2603207 2.3212560
#>  [6,] 0.5160742 0.7794853 0.3982861 1.2183559 1.9610998
#>  [7,] 0.1844014 1.0008575 0.1536657 0.6036499 0.2900357
#>  [8,] 1.0448853 0.2645238 0.1226914 0.1486905 0.7434475
#>  [9,] 2.1672959 0.2158939 1.9209722 0.3735742 1.2956526
#> [10,] 1.6389600 0.7371918 1.3493068 2.1050342 0.8566426

## 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,] 0.55069296 0.1046215 3.08883131 0.3686149
#>  [2,] 0.86366602 0.2726818 0.15910693 1.1708847
#>  [3,] 0.39398663 0.1560021 3.77810676 2.2249727
#>  [4,] 1.67224085 1.7114601 0.06011546 2.4097557
#>  [5,] 0.07655213 1.2815127 0.05304225 0.3487650
#>  [6,] 9.99284057 0.7795481 0.26850632 3.3858871
#>  [7,] 2.17609618 0.4388302 0.27109554 2.3983842
#>  [8,] 0.38166196 0.1797033 1.31678931 0.1632469
#>  [9,] 0.30408976 0.3562097 1.11330902 0.6831740
#> [10,] 0.19853537 1.2816536 0.12636718 0.8424658