qrmtools (version 0.0-15)

fit_GPD: Parameter Estimators of the Generalized Pareto Distribution

Description

Method-of-moments estimator, probability weighted moments estimator, log-likelihood and maximum-likelihood estimator for the parameters of the generalized Pareto distribution (GPD).

Usage

fit_GPD_MOM(x)
fit_GPD_PWM(x)
logLik_GPD(param, x)
fit_GPD_MLE(x, init = c("PWM", "MOM", "shape0"),
            estimate.cov = TRUE, control = list(), ...)

Value

fit_GEV_MOM() and fit_GEV_PWM() return a

numeric(3) giving the parameter estimates for the GPD.

logLik_GPD() computes the log-likelihood of the GPD (-Inf if not admissible).

fit_GPD_MLE() returns the return object of optim()

and, appended, the estimated asymptotic covariance matrix and standard errors of the parameter estimators, if estimate.cov.

Arguments

x

numeric vector of data. In the peaks-over-threshold method, these are the excesses (exceedances minus threshold).

param

numeric(2) containing the value of the shape \(\xi\) (a real) and scale \(\beta\) (positive real) parameters of the GPD in this order.

init

character string specifying the method for computing initial values. Can also be numeric(2) for directly providing \(\xi\) and \(\beta\).

estimate.cov

logical indicating whether the asymptotic covariance matrix of the parameter estimators is to be estimated (inverse of observed Fisher information (negative Hessian of log-likelihood evaluated at MLE)) and standard errors for the estimators of \(\xi\) and \(\beta\) returned, too.

control

list; passed to the underlying optim().

...

additional arguments passed to the underlying optim().

Author

Marius Hofert

Details

fit_GPD_MOM() computes the method-of-moments (MOM) estimator.

fit_GPD_PWM() computes the probability weighted moments (PWM) estimator of Hosking and Wallis (1987); see also Landwehr et al. (1979).

fit_GPD_MLE() uses, as default, fit_GPD_PWM() for computing initial values. The former requires the data x to be non-negative and adjusts \(\beta\) if \(\xi\) is negative, so that the log-likelihood at the initial value should be finite.

References

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Hosking, J. R. M. and Wallis, J. R. (1987). Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics 29(3), 339--349.

Landwehr, J. M., Matalas, N. C. and Wallis, J. R. (1979). Estimation of Parameters and Quantiles of Wakeby Distributions. Water Resourches Research 15(6), 1361--1379.

Examples

## Simulate some data
xi <- 0.5
beta <- 3
n <- 1000
set.seed(271)
X <- rGPD(n, shape = xi, scale = beta)

## Fitting via matching moments
(fit.MOM <- fit_GPD_MOM(X))
stopifnot(all.equal(fit.MOM[["shape"]], xi,   tol = 0.52),
          all.equal(fit.MOM[["scale"]], beta, tol = 0.24))

## Fitting via PWMs
(fit.PWM <- fit_GPD_PWM(X))
stopifnot(all.equal(fit.PWM[["shape"]], xi,   tol = 0.2),
          all.equal(fit.PWM[["scale"]], beta, tol = 0.12))

## Fitting via MLE
(fit.MLE <- fit_GPD_MLE(X))
(est <- fit.MLE$par) # estimates of xi, mu, sigma
stopifnot(all.equal(est[["shape"]], xi,   tol = 0.12),
          all.equal(est[["scale"]], beta, tol = 0.11))
fit.MLE$SE # estimated asymp. variances of MLEs = std. errors of MLEs

## Plot the log-likelihood in the shape parameter xi for fixed
## scale beta (fixed as generated)
xi. <- seq(-0.1, 0.8, length.out = 65)
logLik <- sapply(xi., function(xi..) logLik_GPD(c(xi.., beta), x = X))
plot(xi., logLik, type = "l", xlab = expression(xi),
     ylab = expression("GPD log-likelihood for fixed"~beta))

## Plot the profile likelihood for these xi's
## (with an initial interval for the nuisance parameter beta such that
##  logLik_GPD() is finite)
pLL <- sapply(xi., function(xi..) {
    ## Choose beta interval for optimize()
    int <- if(xi.. >= 0) {
               ## Method-of-Moment estimator
               mu.hat <- mean(X)
               sig2.hat <- var(X)
               shape.hat <- (1-mu.hat^2/sig2.hat)/2
               scale.hat <- mu.hat*(1-shape.hat)
               ## log-likelihood always fine for xi.. >= 0 for all beta
               c(1e-8, 2 * scale.hat)
           } else { # xi.. < 0
               ## Make sure logLik_GPD() is finite at endpoints of int
               mx <- max(X)
               -xi.. * mx * c(1.01, 100) # -xi * max(X) * scaling
               ## Note: for shapes xi.. closer to 0, the upper scaling factor
               ##       needs to be chosen sufficiently large in order
               ##       for optimize() to find an optimum (not just the
               ##       upper end point). Try it with '2' instead of '100'.
           }
    ## Optimization
    optimize(function(nuis) logLik_GPD(c(xi.., nuis), x = X),
             interval = int, maximum = TRUE)$maximum
})
plot(xi., pLL, type = "l", xlab = expression(xi),
     ylab = "GPD profile log-likelihood")