R: Maximum-likelihood Fitting of Bivariate Extreme Value...
fbvpot
R Documentation
Maximum-likelihood Fitting of Bivariate Extreme Value
Distributions to Threshold Exceedances
Description
Fit models for one of nine parametric bivariate extreme-value
distributions using threshold exceedances, allowing any of the
parameters to be held fixed if desired.
A matrix or data frame with two columns. If this
contains missing values, those values are treated as if
they fell below the corresponding marginal threshold.
threshold
A vector of two thresholds.
model
The specified model; a character string. Must be
either "log" (the default), "alog", "hr",
"neglog", "aneglog", "bilog",
"negbilog", "ct" or "amix" (or any unique
partial match), for the logistic, asymmetric logistic,
Husler-Reiss, negative logistic, asymmetric negative logistic,
bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed
models respectively. The definition of each model is given in
rbvevd.
likelihood
The likelihood model; either "censored"
(the default) or "poisson". The "poisson" method
is not recommended. See Details.
start
A named list giving the initial values for all of the
parameters in the model. If start is omitted the routine
attempts to find good starting values using marginal maximum
likelihood estimators.
...
Additional parameters, either for the bivariate extreme
value model or for the optimization function optim. If
parameters of the model are included they will be held fixed at
the values given (see Examples).
sym
Logical; if TRUE, the dependence structure
of the models "alog", "aneglog" or "ct" are
constrained to be symmetric (see Details). For all other
models, the argument is ignored (and a warning is given).
cshape
Logical; if TRUE, a common shape parameter is
fitted to each margin.
cscale
Logical; if TRUE, a common scale parameter is
fitted to each margin, and the default value of cshape
is then TRUE, so that under this default common marginal
parameters are fitted.
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.
Details
For the "censored" method bivariate peaks over threshold models
are fitted by maximizing the censored likelihood as given in e.g. Section
8.3.1 of Coles(2001). For the "poisson" method models are fitted
using Equation 5.4 of Coles and Tawn (1991), see also Joe, Smith and
Weissman (1992). This method is only available for models whose spectral
measure does not contain point masses (see hbvevd). It is not
recommended as in practice it can produce poor estimates.
For either likelihood the margins are modelled using a generalized Pareto
distribution for points above the threshold and an empirical model for
those below. For the "poisson" method data lying below both thresholds
is not used. For the "censored" method the number of points lying
below both thresholds is used, but the locations of the those points are
not.
The dependence parameter names are one or more of dep,
asy1, asy2, alpha and beta, depending on
the model selected (see rbvevd).
The marginal parameter names are scale1 and shape1
for the first margin, and scale2 and shape2 for the
second margin.
If cshape is true, the models are constrained so that
shape2 = shape1. The parameter shape2 is then
taken to be specified, so that e.g. the common shape
parameter can only be fixed at zero using shape1 = 0,
since using shape2 = 0 gives an error. Similar
comments apply for cscale.
If sym is TRUE, the asymmetric logistic and
asymmetric negative logistic models are constrained so that
asy2 = asy1, and the Coles-Tawn model is constrained
so that beta = alpha. The parameter asy2 or
beta is then taken to be specified, so that e.g.
the parameters asy1 and asy2 can only
be fixed at 0.8 using asy1 = 0.8, since
using asy2 = 0.8 gives an error.
Bilogistic and negative bilogistic models constrained to
symmetry are logistic and negative logistic models
respectively. The (symmetric) mixed model (e.g. Tawn, 1998)
can be obtained as a special case of the asymmetric
logistic or asymmetric mixed models (see fbvevd).
For numerical reasons the parameters of each model are subject the
artificial constraints given in fbvevd.
Value
Returns an object of class c("bvpot","evd").
The generic accessor functions fitted (or
fitted.values), std.errors,
deviance, logLik and
AIC extract various features of the
returned object.
The functions profile and profile2d can be
used to obtain deviance profiles.
The function anova compares nested models, and the
function AIC compares non-nested models.
There is currently no plot method available.
An object of class c("bvpot","evd") is a list containing
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 fixed
at specific values within the optimization.
fixed2
A vector containing the parameters that have been set
to be equal to other model parameters.
param
A vector containing all parameters (those optimized,
those fixed to specific values, and those set to be
equal to other model parameters).
deviance
The deviance at the maximum likelihood estimates.
dep.summary
A value summarizing the strength of
dependence in the fitted model (see fbvevd).
corr
The correlation matrix.
var.cov
The variance covariance matrix.
convergence, counts, message
Components taken from the
list returned by optim.
data
The data passed to the argument x.
threshold
The argument threshold.
n
The number of rows in x.
nat
The vector of length three containing the number of
exceedances on the first, second and both margins respectively.
likelihood
The argument likelihood.
sym
The argument sym.
cmar
The vector c(cscale, cshape).
model
The argument model.
call
The call of the current function.
Warning
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 when either of the
marginal shape parameters are less than -0.5, because
the usual asymptotic properties of maximum likelihood estimators
do not then hold (Smith, 1985).
Author(s)
Chris Ferro and Alec Stephenson
References
Coles, S. G. (2001)
An Introduction to Statistical Modelling of Extreme
Values, London: Springer–Verlag.
Coles, S. G. and Tawn, J. A. (1991)
Modelling multivariate extreme events.
J. R. Statist. Soc. B, 53, 377–392.
Joe, H., Smith, R. L. and Weissman, I. (1992)
Bivariate threshold methods for extremes.
J. R. Statist. Soc. B, 54, 171–183.
Smith, R. L. (1985)
Maximum likelihood estimation in a class of non-regular cases.
Biometrika, 72, 67–90.
See Also
abvevd, anova.evd,
fbvevd, optim, rbvevd
Examples
bvdata <- rbvevd(1000, dep = 0.5, model = "log")
u <- apply(bvdata, 2, quantile, probs = 0.9)
M1 <- fbvpot(bvdata, u, model = "log")
M2 <- fbvpot(bvdata, u, "log", dep = 0.5)
anova(M1, M2)