Maximum-likelihood Fitting for the GPD Model

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Description

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

Usage

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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

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