qrmtools (version 0.0-15)

fit_GEV: Parameter Estimators of the Generalized Extreme Value Distribution

Description

Quantile matching estimator, probability weighted moments estimator, log-likelihood and maximum-likelihood estimator for the parameters of the generalized extreme value distribution (GEV).

Usage

fit_GEV_quantile(x, p = c(0.25, 0.5, 0.75), cutoff = 3)
fit_GEV_PWM(x)
logLik_GEV(param, x)
fit_GEV_MLE(x, init = c("shape0", "PWM", "quantile"),
            estimate.cov = TRUE, control = list(), ...)

Value

fit_GEV_quantile() and fit_GEV_PWM() return a

numeric(3) giving the parameter estimates for the GEV distribution.

logLik_GEV() computes the log-likelihood of the GEV distribution (-Inf if not admissible).

fit_GEV_MLE() returns the return object of optim()

(by default, the return value value is the log-likelihood) 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 block maxima method, these are the block maxima.

p

numeric(3) specifying the probabilities whose quantiles are matched.

cutoff

positive \(z\) after which \(\exp(-z)\) is truncated to 0.

param

numeric(3) containing the value of the shape \(\xi\) (a real), location \(\mu\) (a real) and scale \(\sigma\) (positive real) parameters of the GEV distribution in this order.

init

character string specifying the method for computing initial values. Can also be numeric(3) for directly providing \(\xi\), \(\mu\), \(\sigma\).

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 \(\xi\), \(\mu\), \(\sigma\) returned, too.

control

list; passed to the underlying optim().

...

additional arguments passed to the underlying optim().

Author

Marius Hofert

Details

fit_GEV_quantile() matches the empirical p-quantiles.

fit_GEV_PWM() computes the probability weighted moments (PWM) estimator of Hosking et al. (1985); see also Landwehr and Wallis (1979).

fit_GEV_MLE() uses, as default, the case \(\xi = 0\) for computing initial values; this is actually a small positive value since Nelder--Mead could fail otherwise. For the other available methods for computing initial values, \(\sigma\) (obtained from the case \(\xi = 0\)) is doubled in order to guarantee a finite log-likelihood at the initial values. After several experiments (see the source code), one can safely say that finding initial values for fitting GEVs via MLE is non-trivial; see also the block maxima method script about the Black Monday event on https://qrmtutorial.org.

Caution: See Coles (2001, p. 55) for how to interpret \(\xi\le -0.5\); in particular, the standard asymptotic properties of the MLE do not apply.

References

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

Hosking, J. R. M., Wallis, J. R. and Wood, E. F. (1985). Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments. Technometrics 27(3), 251--261.

Landwehr, J. M. and Wallis, J. R. (1979). Probability Weighted Moments Compared With Some Traditional Techniques in Estimating Gumbel Parameters and Quantiles. Water Resourches Research 15(5), 1055--1064.

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag.

Examples

## Simulate some data
xi <- 0.5
mu <- -2
sig <- 3
n <- 1000
set.seed(271)
X <- rGEV(n, shape = xi, loc = mu, scale = sig)

## Fitting via matching quantiles
(fit.q <- fit_GEV_quantile(X))
stopifnot(all.equal(fit.q[["shape"]], xi,  tol = 0.12),
          all.equal(fit.q[["loc"]],   mu,  tol = 0.12),
          all.equal(fit.q[["scale"]], sig, tol = 0.005))

## Fitting via PWMs
(fit.PWM <- fit_GEV_PWM(X))
stopifnot(all.equal(fit.PWM[["shape"]], xi,  tol = 0.16),
          all.equal(fit.PWM[["loc"]],   mu,  tol = 0.15),
          all.equal(fit.PWM[["scale"]], sig, tol = 0.08))

## Fitting via MLE
(fit.MLE <- fit_GEV_MLE(X))
(est <- fit.MLE$par) # estimates of xi, mu, sigma
stopifnot(all.equal(est[["shape"]], xi,  tol = 0.07),
          all.equal(est[["loc"]],   mu,  tol = 0.12),
          all.equal(est[["scale"]], sig, tol = 0.06))
fit.MLE$SE # estimated asymp. variances of MLEs = std. errors of MLEs

## Plot the log-likelihood in the shape parameter xi for fixed
## location mu and scale sigma (fixed as generated)
xi. <- seq(-0.1, 0.8, length.out = 65)
logLik <- sapply(xi., function(xi..) logLik_GEV(c(xi.., mu, sig), x = X))
plot(xi., logLik, type = "l", xlab = expression(xi),
     ylab = expression("GEV distribution log-likelihood for fixed"~mu~"and"~sigma))
## => Numerically quite challenging (for this seed!)

## Plot the profile likelihood for these xi's
## Note: As initial values for the nuisance parameters mu, sigma, we
##       use their values in the case xi = 0 (for all fixed xi = xi.,
##       in particular those xi != 0). Furthermore, for the given data X
##       and xi = xi., we make sure the initial value for sigma is so large
##       that the density is not 0 and thus the log-likelihood is finite.
pLL <- sapply(xi., function(xi..) {
    scale.init <- sqrt(6 * var(X)) / pi
    loc.init <- mean(X) - scale.init * 0.5772157
    while(!is.finite(logLik_GEV(c(xi.., loc.init, scale.init), x = X)) &&
          is.finite(scale.init)) scale.init <- scale.init * 2
    optim(c(loc.init, scale.init), fn = function(nuis)
                logLik_GEV(c(xi.., nuis), x = X),
    		        control = list(fnscale = -1))$value
})
plot(xi., pLL, type = "l", xlab = expression(xi),
     ylab = "GEV distribution profile log-likelihood")