# 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

 ```1 2 3 4``` ```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.

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

## Examples

 ```1 2``` ```data(rain) gpd.fit(rain, 10) ```

### Example output

```Loading required package: mgcv
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 11, 2018, 1:03 a.m.