gpd.fit: Maximum-likelihood Fitting for the GPD Model

In ismev: An Introduction to Statistical Modeling of Extreme Values

Description

Maximum-likelihood fitting for the GPD model, including generalized linear modelling of each parameter.

Usage

gpd.fit(xdat, threshold, npy = 365, ydat = NULL, sigl = NULL,
    shl = NULL, siglink = identity, shlink = identity, siginit = NULL,
    shinit = NULL, show = TRUE,
    method = "Nelder-Mead", maxit = 10000, ...)

Arguments

xdat	A numeric vector of data to be fitted.
threshold	The threshold; a single number or a numeric vector of the same length as xdat.
npy	The number of observations per year/block.
ydat	A matrix of covariates for generalized linear modelling of the parameters (or NULL (the default) for stationary fitting). The number of rows should be the same as the length of xdat.
sigl, shl	Numeric vectors of integers, giving the columns of ydat that contain covariates for generalized linear modelling of the scale and shape parameters repectively (or NULL (the default) if the corresponding parameter is stationary).
siglink, shlink	Inverse link functions for generalized linear modelling of the scale and shape parameters repectively.
siginit, shinit	numeric giving initial value(s) for parameter estimates. If NULL, default is sqrt(6 * var(xdat))/pi and 0.1 for the scale and shape parameters, resp. If using parameter covariates, then these values are used for the constant term, and zeros for all other terms.
show	Logical; if TRUE (the default), print details of the fit.
method	The optimization method (see optim for details).
maxit	The maximum number of iterations.
...	Other control parameters for the optimization. These are passed to components of the control argument of optim.

Details

For non-stationary fitting it is recommended that the covariates within the generalized linear models are (at least approximately) centered and scaled (i.e.\ the columns of ydat should be approximately centered and scaled).

The form of the GP model used follows Coles (2001) Eq (4.7). In particular, the shape parameter is defined so that positive values imply a heavy tail and negative values imply a bounded upper value.

Value

A list containing the following components. A subset of these components are printed after the fit. If show is TRUE, then assuming that successful convergence is indicated, the components nexc, nllh, mle, rate and se are always printed.

nexc	single numeric giving the number of threshold exceedances.
nllh	nsingle umeric giving the negative log-likelihood value.
mle	numeric vector giving the MLE's for the scale and shape parameters, resp.
rate	single numeric giving the estimated probability of exceeding the threshold.
se	numeric vector giving the standard error estiamtes for the scale and shape parameter estimates, resp.
trans	An logical indicator for a non-stationary fit.
model	A list with components sigl and shl.
link	A character vector giving inverse link functions.
threshold	The threshold, or vector of thresholds.
nexc	The number of data points above the threshold.
data	The data that lie above the threshold. For non-stationary models, the data is standardized.
conv	The convergence code, taken from the list returned by optim. A zero indicates successful convergence.
nllh	The negative logarithm of the likelihood evaluated at the maximum likelihood estimates.
vals	A matrix with three columns containing the maximum likelihood estimates of the scale and shape parameters, and the threshold, at each data point.
mle	A vector containing the maximum likelihood estimates.
rate	The proportion of data points that lie above the threshold.
cov	The covariance matrix.
se	A vector containing the standard errors.
n	The number of data points (i.e.\ the length of xdat).
npy	The number of observations per year/block.
xdata	The data that has been fitted.

References

Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag, London, U.K., 208pp.

See Also

gpd.diag, optim, gpd.prof, gpd.fitrange, mrl.plot, pp.fit

Examples

data(rain)
gpd.fit(rain, 10)

Example output

Loading required package: mgcv
Loading required package: nlme
This is mgcv 1.8-17. For overview type 'help("mgcv-package")'.
$threshold
[1] 10

$nexc
[1] 2003

$conv
[1] 0

$nllh
[1] 6123.465

$mle
[1] 7.43768624 0.05045225

$rate
[1] 0.1142547

$se
[1] 0.23606472 0.02256649

ismev documentation built on May 1, 2019, 9:10 p.m.