Title:	Bayesian Estimation of Extreme Value Mixture Models
Version:	0.0.1
Description:	Fits extreme value mixture models, which are models for tails not requiring selection of a threshold, for continuous data. It includes functions for model comparison, estimation of quantity of interest in extreme value analysis and plotting. Reference: CN Behrens, HF Lopes, D Gamerman (2004) <doi:10.1191/1471082X04st075oa>. FF do Nascimento, D. Gamerman, HF Lopes <doi:10.1007/s11222-011-9270-z>.
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.2
URL:	https://github.com/manueleleonelli/extrememix
BugReports:	https://github.com/manueleleonelli/extrememix/issues
LinkingTo:	Rcpp, RcppProgress
Imports:	evd, ggplot2, gridExtra, mixtools, Rcpp, RcppProgress, stats, threshr
Depends:	R (≥ 2.10)
LazyData:	true
Suggests:	knitr, rmarkdown
VignetteBuilder:	knitr
NeedsCompilation:	yes
Packaged:	2024-10-03 22:53:22 UTC; manueleleonelli
Author:	Manuele Leonelli [aut, cre, cph]
Maintainer:	Manuele Leonelli <manuele.leonelli@ie.edu>
Repository:	CRAN
Date/Publication:	2024-10-04 10:10:03 UTC

Deviance Information Criterion

Description

Computation of the DIC for an extreme value mixture model

Usage

DIC(x, ...)

## S3 method for class 'evmm'
DIC(x, ...)

Arguments

Details

Let y denote a dataset and p(y|\theta) the likelihood of a parametric model with parameter \theta. The deviance is defined as D(\theta)= -2\log p(y|\theta). The deviance information criterion (DIC) is defined as

DIC = D(\hat\theta) + 2p_D,

where \hat\theta is the posterior estimate of \theta and p_D is referred to as the effective number of parameters and defined as

E_{\theta|y}(D(\theta)) - D(\hat\theta).

Models with a smaller DIC are favored.

Value

The DIC of a model estimated with extrememix

References

Spiegelhalter, David J., et al. "Bayesian measures of model complexity and fit." Journal of the Royal Statistical Society: Series B 64.4 (2002): 583-639.

See Also

WAIC

Examples

DIC(rainfall_ggpd)

Expected Shortfall

Description

Computation of the expected shortfall for an extreme value mixture model

Usage

ES(x, ...)

## S3 method for class 'evmm'
ES(x, values = NULL, cred = 0.95, ...)

Arguments

Details

The expected shortfall is the expectation of a random variable conditional of being larger of a specific Value-at-Risk (quantile). For an extreme value mixture model this is equal to:

ES_p = \frac{VaR_p}{1-\xi} +\frac{\sigma-\xi u }{1-\xi}

Value

A list with the following entries:

• quantiles: a matrix containing the estimated shortfall, the posterior credibility intervals and the empirical estimate.

• data: the dataset used to estimate the expected shortfall.

• complete: a matrix with the expected shortfall for each value in the posterior sample.

References

Lattanzi, Chiara, and Manuele Leonelli. "A changepoint approach for the identification of financial extreme regimes." Brazilian Journal of Probability and Statistics.

See Also

quant, return_level, VaR

Examples

ES(rainfall_ggpd)

Value-at-Risk

Description

Computation of the Value-at-Risk for an extreme value mixture model.

Usage

VaR(x, ...)

## S3 method for class 'evmm'
VaR(x, values = NULL, cred = 0.95, ...)

Arguments

Details

The Value-at-Risk for level q\

Value

A list with the following entries:

• quantiles: a matrix containing the estimated value at risk, the posterior credibility intervals and the empirical estimate.

• data: the dataset used to estimate the value at risk.

• complete: a matrix with the value at risk for each value in the posterior sample.

References

Lattanzi, Chiara, and Manuele Leonelli. "A changepoint approach for the identification of financial extreme regimes." Brazilian Journal of Probability and Statistics.

See Also

ES, quant, return_level

Examples

VaR(rainfall_ggpd)

Widely Applicable Information Criteria

Description

Computation of the WAIC for an extreme value mixture model.

Usage

WAIC(x, ...)

## S3 method for class 'evmm'
WAIC(x, ...)

Arguments

Details

Consider a dataset y=(y_1,\dots,y_n), p(y|\theta) the likelihood of a parametric model with parameter \theta, and (\theta^{(1)},\dots,\theta^{(S)}) a sample from the posterior distribution p(\theta|y). Define

\textnormal{llpd} = \sum_{i=1}^n \log\left(\sum_{i=1}^Sp(y_i|\theta^{(s)}\right)

and

p_\textnormal{WAIC} = \sum_{i=1}^n Var_{\theta|y}(\log p(y_i|\theta)).

Then the Widely Applicable Information Criteria is defined as

WAIC = -2\textnormal{llpd} + 2p_\textnormal{WAIC}.

Models with a smaller WAIC are favored.

Value

The WAIC of a model estimated with extrememix

References

Gelman, Andrew, Jessica Hwang, and Aki Vehtari. "Understanding predictive information criteria for Bayesian models." Statistics and computing 24.6 (2014): 997-1016.

Watanabe, Sumio. "A widely applicable Bayesian information criterion." Journal of Machine Learning Research 14.Mar (2013): 867-897.

See Also

DIC

Examples

WAIC(rainfall_ggpd)

Convergence Assessment of MCMC Algorithms

Description

Plot of the traceplot and autocorrelation function for the 0.99 quantile from the posterior sample.

Usage

check_convergence(x, ...)

## S3 method for class 'evmm'
check_convergence(x, ...)

Arguments

Value

Two plots to check if the estimation with fggpd and mgpd converged: traceplot and autocorrelation plot for the 99th quantile of the posterior density.

Examples

check_convergence(rainfall_ggpd)

GGPD Estimation

Description

Fit of the GGPD model using an MCMC algorithm.

Usage

fggpd(x, it, start = NULL, var = NULL, prior = NULL, thin = 1, burn = 0)

Arguments

Details

Estimation of the GGPD is carried out using an adaptive block Metropolis-Hastings algorithm. As standard, the user needs to specify the data to use during estimation, the number of iterations of the algorithm, the burn-in period (by default equal to zero) and the thinning interval (by default equal to one). To run the algorithm it is also needed the choice of the starting values, the starting values of the proposal variances, and the parameters of the prior distribution. If not provided, these are automatically set as follows:

• starting values: u is chosen by the function ithresh of the threshr package; \xi and \sigma are chosen by the fpot function of evd for data over the threshold; \mu and \eta are chosen as the maximum likelihood estimate of the Gamma distribution over data below the threshold.

• variances: variances are chosen as the standard deviation of the normal distribution whose 0.01 quantile is equal to 0.9 times the starting value of the associated parameter.

• prior distributions: the prior distribution for \xi and \sigma is the objective prior

  p(\xi,\sigma) = \sigma^{-1}(1+\xi)^{-1}(1+2\xi)^{-1/2}.

  The prior for the threshold u is Normal with mean chosen as for the starting values and the standard deviation is chosen so that the 0.05 quantile of the prior is equal to the median of the data. The priors for the parameters \mu and \eta are Gammas with mean chosen as for the starting values and shapes equal to 0.001 so to give high prior variances.

The user can also select any of the three inputs above.

• The starting values start must be a list with entries xi, sigma, u, mu, eta.

• The variances var must be a list with entries xi, sigma, u, mu, eta.

• The prior prior must be a list with entries u, mu, eta all containing a vector of length two (for u giving the mean and the standard deviation of the Normal prior, for mu and eta giving the mean and shape of the Gamma prior).

Value

fggpd returns a list with three elements:

• chain: a matrix of size (it - burn)/thin\times5, reporting in each column the posterior sample for each parameter.

• var: a matrix of size it\times5 reporting the variance of the proposal distribution for each parameter.

• data: the dataset used for estimation.

References

Behrens, Cibele N., Hedibert F. Lopes, and Dani Gamerman. "Bayesian analysis of extreme events with threshold estimation." Statistical Modelling 4.3 (2004): 227-244.

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

See Also

ggpd

Examples

## Small number of iterations and burn-in for quick execution
data(rainfall)
model1 <- fggpd(rainfall, it = 250, burn = 50, thin = 25)

start <- list(xi = 0.2, sigma = 2, u = 10, mu = 5, eta = 2)
var <- list(xi = 0.01, sigma = 1, u = 3, mu = 3, eta = 1)
prior <- list(u = c(22,5), mu = c(4,16), eta = c(0.001,0.001))
model2 <- fggpd(rainfall,it = 250, start = start, var =var, prior = prior)

MGPD Estimation

Description

Fit of the MGPD model using an MCMC algorithm.

Usage

fmgpd(x, it, k, start = NULL, var = NULL, prior = NULL, thin = 1, burn = 0)

Arguments

Details

Estimation of the MGPD is carried out using an adaptive block Metropolis-Hastings algorithm. As standard, the user needs to specify the data to use during estimation, the number of mixture components for the bulk, the number of iterations of the algorithm, the burn-in period (by default equal to zero) and the thinning interval (by default equal to one). To run the algorithm it is also needed the choice of the starting values, the starting values of the proposal variances, and the parameters of the prior distribution. If not provided, these are automatically set as follows:

• starting values: u is chosen by the function ithresh of the threshr package; \xi and \sigma are chosen by the fpot function of evd for data over the threshold; \mu and \eta are chosen as estimates of the gammamixEM function from the mixtools package; w is chosen as the vector with entries 1/k.

• variances: variances for \sigma and u are chosen as the standard deviation of the normal distribution whose 0.01 quantile is equal to 0.9 times the starting value of the associated parameter. The parameters \mu_i and \eta_i are sampled jointly and the proposal variance is chosen using the same method as for \sigma with respect to the parameter \mu. The proposal variance for w is 0.1 and the proposal variance for \xi is 0.1 if negative and 0.25 if positive.

• prior distributions: the prior distribution for \xi and \sigma is the objective prior

  p(\xi,\sigma) = \sigma^{-1}(1+\xi)^{-1}(1+2\xi)^{-1/2}.

  The prior for the threshold u is Normal with mean chosen as for the starting values and the standard deviation is chosen so that the 0.05 quantile of the prior is equal to the median of the data. The priors for the parameters \mu_i and \eta_i are Gammas with mean chosen as for the starting values and shapes equal to 0.001 so to give high prior variances. The prior for the weigths is the non-informative Dirichlet with parameter 1.

The user can also select any of the three inputs above.

• The starting values start must be a list with entries xi, sigma, u, mu, eta and w. The length of mu, eta and w must be k.

• The variances var must be a list with entries xi, sigma, u, mu and w. The length of mu must be k.

• The prior prior must be a list with entries u, mu_mu, mu_eta, eta_mu and eta_eta. u gives the mean and the standard deviation of the Normal prior for u. The vectors of length k, mu_mu and eta_mu give the prior means of \mu and \eta, whilst mu_eta and eta_eta give their prior shapes.

Value

fmgpd returns a list with three elements:

• chain: a matrix of size (it - burn)/thin\times5, reporting in each column the posterior sample for each parameter.

• var: a matrix of size it\times5 reporting the variance of the proposal distribution for each parameter.

• data: the dataset used for estimation.

References

Behrens, Cibele N., Hedibert F. Lopes, and Dani Gamerman. "Bayesian analysis of extreme events with threshold estimation." Statistical Modelling 4.3 (2004): 227-244.

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

See Also

fggpd, mgpd

Examples

data(rainfall)
## Small number of iterations and burn-in for quick execution
model1 <- fmgpd(rainfall, k = 2, it = 250, burn = 50, thin = 25)
start <- list(xi = 0.2, sigma = 2, u = 10, mu = c(2,5), eta = c(2,2) , w = c(0.4,0.6))
var <- list(xi = 0.01, sigma = 1, u = 3, mu = c(3,3), w = 0.01)
prior <- list(u = c(22,5), mu_mu = c(2,5), mu_eta = c(0.01,0.01),
         eta_mu = c(3,3),eta_eta = c(0.01,0.01))

model2 <- fmgpd(rainfall, k= 2, it = 250, start = start, var =var, prior = prior)

The GGPD distribution

Description

Density, distribution function, quantile function and random generation for the GGPD distribution.

Usage

dggpd(x, xi, sigma, u, mu, eta, log = FALSE)

pggpd(q, xi, sigma, u, mu, eta, lower.tail = TRUE)

qggpd(p, xi, sigma, u, mu, eta, lower.tail = TRUE)

rggpd(N, xi, sigma, u, mu, eta)

Arguments

Value

The GGPD distribution is an extreme value mixture model with density

f_{GGPD}(x|\xi,\sigma,u,\mu,\eta,w)=\left\{\begin{array}{ll} f_{GA}(x|\mu,\eta), & x\leq u \\ (1-F_{GA}(u|\mu,\eta))f_{GPD}(x|\xi,\sigma,u), &\mbox{otherwise}, \end{array}\right.

where f_{GA} is the density of the Gamma parametrized by mean \mu and shape \eta, F_{GA} is the distribution function of the Gamma and f_{GPD} is the density of the Generalized Pareto Distribution, i.e.

f_{GPD}(x|\xi,\sigma,u)=\left\{\begin{array}{ll} 1- (1+\frac{\xi}{\sigma}(x-u))^{-1/\xi}, & \mbox{if } \xi\neq 0,\\ 1- \exp\left(-\frac{x-u}{\sigma}\right), & \mbox{if } \xi = 0, \end{array}\right.

where \xi is a shape parameter, \sigma > 0 is a scale parameter and u>0 is a threshold.

dggpd gives the density, pggpd gives the distribution function, qggpd gives the quantile function, and rggpd generates random deviates. The length of the result is determined by N for rggpd and by the length of x, q or p otherwise.

References

Behrens, Cibele N., Hedibert F. Lopes, and Dani Gamerman. "Bayesian analysis of extreme events with threshold estimation." Statistical Modelling 4.3 (2004): 227-244.

Examples

dggpd(3, xi = 0.5, sigma = 2, u = 5, mu = 3, eta = 3)

Log-likelihood Method

Description

Computation of the log-likelihood of an extreme value mixture model (thus also AIC and BIC are available).

Usage

## S3 method for class 'evmm'
logLik(object, ...)

Arguments

Value

The log-likelihood of a model estimated with extrememix

Examples

logLik(rainfall_ggpd)

The Gamma Mixture Distribution

Description

Density, distribution function, quantile function and random generation for the mixture of Gamma distribution.

Usage

dmgamma(x, mu, eta, w, log = FALSE)

pmgamma(q, mu, eta, w, lower.tail = TRUE)

qmgamma(p, mu, eta, w, lower.tail = TRUE)

rmgamma(N, mu, eta, w)

Arguments

Details

The Gamma distribution has density

f_{GA}(x|\mu,\eta)= \frac{(\eta/\mu)^\eta}{\Gamma(\eta)}x^{\eta-1}\exp(-(\eta/\mu)x), \hspace{1cm} x>0,

where \mu>0 is the mean of the distribution and \eta>0 is its shape. The density of a mixture of Gamma distributions with k components is defined as

f_{MG}(x|\mu,\eta,w)=\sum_{i=1}^k w_if_{GA}(x|\mu_i,\eta_i),

where w_i,\mu_i,\eta_i >0, for i=1,\dots,k, w_1+\cdots+w_k=1, \mu=(\mu_1,\dots,\mu_k), \eta = (\eta_1,\dots,\eta_k) and w=(w_1,\dots,w_k).

Value

dmgamma gives the density, pmgamma gives the distribution function, qmgamma gives the quantile function, and rmgamma generates random deviates.

The length of the result is determined by N for rmgamma and by the length of x, q or p otherwise.

References

Wiper, Michael, David Rios Insua, and Fabrizio Ruggeri. "Mixtures of gamma distributions with applications." Journal of Computational and Graphical Statistics 10.3 (2001): 440-454.

Examples

dmgamma(3, mu = c(2,3), eta = c(1,2), w = c(0.3,0.7))

The MGPD distribution

Description

Density, distribution function, quantile function and random generation for the MGPD distribution.

Usage

dmgpd(x, xi, sigma, u, mu, eta, w, log = FALSE)

pmgpd(q, xi, sigma, u, mu, eta, w, lower.tail = TRUE)

qmgpd(p, xi, sigma, u, mu, eta, w, lower.tail = TRUE)

rmgpd(N, xi, sigma, u, mu, eta, w)

Arguments

Details

The MGPD distribution is an extreme value mixture model with density

f_{MGPD}(x|\xi,\sigma,u,\mu,\eta,w)=\left\{\begin{array}{ll} f_{MG}(x|\mu,\eta,w), & x\leq u \\ (1-F_{MG}(u|\mu,\eta,w))f_{GPD}(x|\xi,\sigma,u), &\mbox{otherwise}, \end{array}\right.

where f_{MG} is the density of the mixture of Gammas, F_{MG} is the distribution function of the mixture of Gammas and f_{GPD} is the density of the Generalized Pareto Distribution, i.e.

f_{GPD}(x|\xi,\sigma,u)=\left\{\begin{array}{ll} 1- (1+\frac{\xi}{\sigma}(x-u))^{-1/\xi}, & \mbox{if } \xi\neq 0,\\ 1- \exp\left(-\frac{x-u}{\sigma}\right), & \mbox{if } \xi = 0, \end{array}\right.

where \xi is a shape parameter, \sigma > 0 is a scale parameter and u>0 is a threshold.

Value

dmgpd gives the density, pmgpd gives the distribution function, qmgpd gives the quantile function, and rmgpd generates random deviates. The length of the result is determined by N for rmgpd and by the length of x, q or p otherwise.

References

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

Examples

dmgpd(3, xi = 0.5, sigma = 2,5, u = 5, mu = c(2,3), eta = c(1,2), w = c(0.3,0.7))

Plot of Extreme Value Mixture Models

Description

Plotting method for objects of class evmm giving an overview of an estimated model.

Usage

## S3 method for class 'evmm'
plot(x, ...)

Arguments

Details

The plot method for objects of class evmm reports four plots:

• An histogram of the posterior distribution of xi.

• An histogram of the posterior distribution of sigma.

• A line plot of the estimated quantiles, their posterior credibility interval, and the empirical ones.

• A plot of the predictive distribution together with the data histogram.

Value

Plots of a model estimated with extrememix.

Examples

plot(rainfall_ggpd)

Plot Upper Bounds

Description

Plotting method for the posterior distribution of the upper bound. No plot is reported if the posterior sample of xi has only positive values (unbounded distribution).

Usage

## S3 method for class 'upper_bound'
plot(x, xlim = c(min(x$bound), max(x$bound)), ...)

Arguments

Value

A histogram for the posterior estimated upper bound of the distribution.

Examples

plot(upper_bound(rainfall_ggpd))

Plot Methods for Summaries

Description

Plotting methods for objects created with quant, ES, return_level or VaR.

Usage

## S3 method for class 'quant'
plot(x, ylim = NULL, ...)

## S3 method for class 'return_level'
plot(x, ylim = NULL, ...)

## S3 method for class 'VaR'
plot(x, ylim = NULL, ...)

## S3 method for class 'ES'
plot(x, ylim = NULL, ...)

Arguments

Details

Two types of plot can be output: either a line plot in the case the functions quant, ES, return_level or VaR where called with more than one value for the input values, or an histogram otherwise.

Value

Appropriate plots for quantities computed with extrememix.

Examples

plot(return_level(rainfall_ggpd)) ## for line plot
plot(return_level(rainfall_ggpd, values = 100)) ## for histogram

Predictive Distribution

Description

Plot of the predictive distribution of an extreme value mixture model.

Usage

pred(x, ...)

## S3 method for class 'evmm'
pred(
  x,
  x_axis = seq(min(x$data), max(x$data), length.out = 1000),
  cred = 0.95,
  xlim = c(min(x$data), max(x$data)),
  ylim = NULL,
  ...
)

Arguments

Details

Consider an extreme value mixture model f(y|\theta) and suppose a sample (\theta^{(1)},\dots,\theta^{(S)}) from the posterior distribution is available. The predictive distribution at the point y is estimated as

\frac{1}{S}\sum_{s=1}^Sf(y|\theta^{(s)})

Value

A plot of the estimate of the predictive distribution together with the data histogram.

References

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

Examples

pred(rainfall_ggpd)

Printing Methods

Description

Collection of printing methods for various objects created by extrememix.

Usage

## S3 method for class 'evmm'
print(x, ...)

## S3 method for class 'summary.ggpd'
print(x, ...)

## S3 method for class 'quantile'
print(x, ...)

## S3 method for class 'return_level'
print(x, ...)

## S3 method for class 'VaR'
print(x, ...)

## S3 method for class 'ES'
print(x, ...)

## S3 method for class 'upper_bound'
print(x, ...)

Arguments

Value

A printed output of a model estimated with extrememix.

Estimated Quantiles

Description

Computation of posterior quantiles for an extreme value mixture model

Usage

quant(x, ...)

## S3 method for class 'evmm'
quant(x, values = NULL, cred = 0.95, ...)

Arguments

Details

For a random variable X the p-quantile is the value x such that P(X>x)=1-p. For an extreme value mixture model this can be computed as

x = u +\frac{\sigma}{\xi}((1-p^*)^{-\xi}-1),

where

p^* = \frac{p-F_\textnormal{bulk}(u|\theta)}{1-F_\textnormal{bulk}(u|\theta)},

and F_\textnormal{bulk} is the distribution function of the bulk, parametrized by \theta.

Value

A list with the following entries:

• quantiles: a matrix containing the quantiles, the posterior credibility intervals and the empirical estimate.

• data: the dataset used to estimate the quantiles.

• complete: a matrix with the quantiles for each value in the posterior sample.

References

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

Examples

quant(rainfall_ggpd)

Monthly Maxima Daily Rainfall in Madrid

Description

Monthly maxima of the daily rainfall (measured in mms) recorded at the Retiro station in the city centre of Madrid, Spain, between 1985 and 2020.

Usage

data(rainfall)

Format

A positive numeric vector of length 414. Observations where the monthly maxima are zero were discarded.

Source

Instituto de Estadistica, Communidad de Madrid.

Rainfall FGGPD Output

Description

Estimated ggpd model over the rainfall dataset

Usage

data(rainfall_ggpd)

Format

A list storing the output of the fggpd function over the rainfall dataset.

Rainfall FMGPD Output

Description

Estimated mgpd model over the rainfall dataset

Usage

data(rainfall_mgpd)

Format

A list storing the output of the fmgpd function over the rainfall dataset.

Return Levels

Description

Computation of the return levels for an extreme value mixture model

Usage

return_level(x, ...)

## S3 method for class 'evmm'
return_level(x, values = NULL, cred = 0.95, ...)

Arguments

Details

A return level at T units of time is defined as the 1-1/T quantile.

Value

A list with the following entries:

• quantiles: a matrix containing the estimated return levels, the posterior credibility intervals and the empirical estimate.

• data: the dataset used to estimate the return levels.

• complete: a matrix with the return levels for each value in the posterior sample.

References

do Nascimento, Fernando Ferraz, Dani Gamerman, and Hedibert Freitas Lopes. "A semiparametric Bayesian approach to extreme value estimation." Statistics and Computing 22.2 (2012): 661-675.

See Also

ES, quant, VaR

Examples

return_level(rainfall_ggpd)

Summary Method

Description

Posterior estimates and credibility intervals for the parameters of extreme value mixture models.

Usage

## S3 method for class 'evmm'
summary(object, ...)

Arguments

Value

A printed summary of a model estimated with extrememix or any quantity associated with it.

Upper Bound

Description

Computation of the upper bound of the distribution

Usage

upper_bound(x, ...)

## S3 method for class 'evmm'
upper_bound(x, cred = 0.95, ...)

Arguments

Details

For an extreme value mixture model with a shape parameter xi < 0 the distribution is right-bounded with upper limit equal to u-\sigma/\xi.

Value

upper_bound returns a list with entries:

• bound: a sample from the posterior distribution of the upper limit of the model, taken over the posterior values of xi which are negative.

• prob: the posterior probability that the distribution is unbounded.

• cred: the requested amplitude of the posterior credibility intervals.

References

Coles, Stuart, et al. An introduction to statistical modeling of extreme values. Vol. 208. London: Springer, 2001.

Examples

upper_bound(rainfall_ggpd)