evd (version 1.2-3)

fgev.quantile: Maximum-likelihood Fitting of the Generalized Extreme Value Distribution

Description

Maximum-likelihood fitting for the generalized extreme value distribution, including linear modelling of the location parameter, and allowing any of the parameters to be held fixed if desired.

Usage

fgev.quantile(x, start, ..., prob, nsloc = NULL, std.err = TRUE,
    corr = FALSE, method = "Nelder-Mead", warn.inf = TRUE)

Arguments

x

A numeric vector, which may contain missing values.

start

A named list giving the parameters to be optimized with initial values (see Details). If start is omitted the routine attempts to find good starting values using moment estimators.

...

Additional parameters, either for the distribution or for optim. If parameters of the distribution are included they will be held fixed (see Details).

prob

The probability in the upper tail that corresponds to the quantile parameter (see Details).

nsloc

A data frame with the same number of rows as the length of x, for linear modelling of the location parameter (see Details). The data frame is treated as a covariate matrix (excluding the intercept). A numeric vector

std.err

Logical; if TRUE (the default), the ``standard errors'' are returned.

corr

Logical; if TRUE, the ``correlation matrix'' is returned.

method

The optimization method (see optim for details).

warn.inf

Logical; if TRUE (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.

Value

• Returns an object of class "evd".

  The generic accessor functions fitted (or fitted.values), std.errors and deviance extract various features of the returned object.

  The functions profile and profile2d are used to obtain deviance profiles. In particular, profiles of the quantile $z_p$ can be calculated and plotted. The function anova compares nested models. The function plot produces diagnostic plots. An object of class "evd" is a list containing at most the following components

• estimate
  A vector containing the maximum likelihood estimates.
• std.err
  A vector containing the ``standard errors''.
• fixed
  A vector containing the parameters that have been set to fixed values within the optimization.
• param
  A vector containing all parameters (optimized and fixed).
• deviance
  The deviance at the maximum likelihood estimates.
• corr
  The ``correlation matrix''.
• convergence,counts,message
  Components taken from the list returned by optim.
• call
  The call of the current function.
• data
  The data passed to the argument x.
• tdata
  The data, transformed to stationarity (for non-stationary models).
• nsloc
  The argument nsloc.
• n
  The length of x.
• model
  A character string describing the fitted model.

code

profile2d.evd

Details

Maximization of the log-likelihood is performed. The ``standard errors'' and the ``correlation matrix'' in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution because the usual asymptotic properties of maximum likelihood estimators may not hold (Smith, 1985).

fgev.quant performs the same fit as fgev, but under a different parameterization. For stationary fitting using fgev, the likelihood is optimized over $(a,b,s)$, where $a$ is the location parameter, $b$ is the scale and $s$ is the shape. For stationary fitting using fgev.quant, the likelihood is optimized over $(z_p,b,s)$, where $\code{prob} = p$ and $$z_p = a - b/s (1 - (-\log(1 - p))^s)$$ is such that $G(z_p) = 1 - p$, where $G$ is the generalized extreme value distribution function. For non-stationary fitting the parameter $z_p$ is again given by the equation above but $a$ becomes the intercept of the linear model for the location parameter. For stationary fitting, the parameters can be passed (either as named components of start or as fixed values) using quantile, scale and shape.

For non-stationary fitting, the non-stationary parameters can be passed using the column names of the data frame nsloc with the prefix ``loc''. If nsloc1 is a vector it is converted into a one column data frame with column name ``trend'', so that the associated parameter can be passed as loctrend.

For non-stationary fitting it is recommended that the covariates within the linear model for the location parameter are (at least approximately) centered and scaled, particularly if automatic starting values are used, since the starting values for all the associated parameters are taken to be zero.

References

Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. Biometrika, 72, 67--90.

See Also

anova.evd, optim, plot.evd

Examples

uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2)
M1 <- fgev.quantile(uvdata, prob = 0.1)
M2 <- fgev.quantile(uvdata, prob = 0.01)
M1P <- profile(M1, which = "quantile")
M2P <- profile(M2, which = "quantile")
plot(M1P)
plot(M2P)