ExtremeRisks (version 0.0.5)

estPOT: Estimation of generalized Pareto distributions

Description

Bayesian or frequentist estimation of the scale and shape parameters of the continuous or discrete generalized Pareto distribution.

Usage

estPOT(
  data,
  k = 10L,
  pn = NULL,
  type = c("continuous", "discrete"),
  method = c("bayesian", "frequentist"),
  prior = "empirical",
  start = NULL,
  sig0 = NULL,
  nsim = 5000L,
  burn = 1000L,
  p = 0.234,
  optim.method = "Nelder-Mead",
  control = NULL,
  ...
)

Value

a list with the following elements

• Bayesian estimation case

  • Q.est matrix with nsim-burn rows and length(pn) columns containing the posterior sample of the extreme quantile estimated at level given in pn

  • post_sample matrix with nsim-burn rows and 2 columns containing the posterior sample of the scale and shape parameters for the continuous or discrete generalized Pareto distribution

  • burn double indicating the number of iterations excluded in the MCMC algorithm

  • straight.reject vector of length nsim-burn+1 indicating the iterations in which the proposed parameters do not respect basic constraints

  • sig.vec vector of length nsim\(-\lfloor(5 / (p (1 - p)))\rfloor+1\) containing the values of the variance updated at each iteration of the MCMC algorithm

  • accept.prob matrix containing the values of acceptance probability (second column) corresponding to specific iterations (first column)

  • msg character string containing an output message on the result of the Bayesian estimation procedure

  • mle vector of length 2 containing the maximum likelihood estimates of the scale and shape parameters of the continuous or discrete generalized Pareto distribution

  • t double indicating the threshold for the generalized Pareto model, corresponding to the \(n-k\)th order statistic of the sample

• Frequentist estimation case

  • est vector of length 2 containing the maximum likelihood estimates of the scale and shape parameters of the continuous or discrete generalized Pareto distribution

  • t double indicating the threshold

  • Q.est vector of dimension length(pn) containing the estimates of the extreme quantile at level given in pn

  • VarCov \(2 \times 2\) variance-covariance matrix of (gamma, sigma)

  • Q.VC variance of Q.est

  • msg character string containing an output message on the result of the frequentist estimation procedure

Arguments

data

numeric vector of length n containing complete data values (under and above threshold)

k

double indicating the effective sample size. Default: 10

pn

numeric vector containing one or more percentile level at which extreme quantiles are estimated, with \(p < k/n\). Default: NULL

type

string indicating distribution types. Default: c('continuous','discrete')

method

string indicating estimation methods. Default: c('bayesian', 'frequentist')

prior

string indicating prior distribution (uniform or empirical). Default: 'empirical'

start

list of 2 containing starting values for scale and shape parameters. Default: NULL

sig0

double indicating the initial value for the update of the variance in the MCMC algorithm. Default: NULL

nsim

double indicating the total number of iterations of the MCMC algorithm in the Bayesian estimation case. Default: 5000L

burn

double indicating the number of iterations to exclude in the MCMC algorithm of the Bayesian estimation case. Default: 1000L

p

double indicating the desired overall sampler acceptance probability. Default: 0.234

optim.method

string indicating the optimization method in the frequentist estimation case. Default: 'Nelder-Mead'

control

list containing additional parameters for the minimization function optim. Default: NULL

...

other arguments passed to the function

References

Dombry, C., S. Padoan and S. Rizzelli (2025). Asymptotic theory for Bayesian inference and prediction: from the ordinary to a conditional Peaks-Over-Threshold method, arXiv:2310.06720v2.

See Also

fpot

Examples

if (FALSE) {
# generate data
set.seed(1234)
n <- 500
samp <- evd::rfrechet(n,0,3,4)
# set effective sample size and threshold
k <- 50
threshold <- sort(samp,decreasing = TRUE)[k+1]
# preliminary mle estimates of scale and shape parameters
mlest <- evd::fpot(samp, threshold)
# empirical bayes procedure
proc <- estPOT(
  samp,
  k = k,
  pn = c(0.01, 0.005),
  type = "continuous",
  method = "bayesian",
  prior = "empirical",
  start = as.list(mlest$estimate),
  sig0 = 0.1)
 }