Estimate the dependence parameters in a conditional multivariate extreme values model
Description
Estimate the dependence parameters in a conditional multivariate extreme values model using the approach of Heffernan and Tawn, 2004.
Usage
1 2 3 
Arguments
x 
An object of class "migpd" as returned by 
which 
The name of the variable on which to condition. This is the
name of a column of the data that was passed into 
dqu 
See documentation for this argument in 
margins 
The form of margins to which the data are transformed for carrying out dependence estimation. Defaults to "laplace", with the alternative option being "gumbel". The choice of margins has an impact on the interpretation of the fitted dependence parameters. Under Gumbel margins, the estimated parameters a and b describe only positive dependence, while c and d describe negative dependence in this case. For Laplace margins, only parameters a and b are estimated as these capture both positive and negative dependence. 
constrain 
Logical value. Defaults to 
v 
Scalar. Tuning parameter used to carry out constrained estimation
of dependence structure under 
maxit 
The maximum number of iterations to be used by the optimizer.
Defaults to 
start 
Optional starting value for dependence estimation. This can
be: a vector of length two, with values corresponding to dependence
parameters a and b respectively, and in which case 
marTransform 
Optional form of transformation to be used for
probability integral transform of data from original to Gumbel or Laplace
margins. Takes values 
nOptim 
Number of times to run optimiser when estimating dependence
model parameters. Defaults to 1. In the case of 
PlotLikDo 
Logical value: whether or not to plot the profile likelihood surface for dependence model parameters under constrained estimation. 
PlotLikRange 
This is used to specify a region of the parameter space
over which to plot the profile loglikelihood surface. List of length 2;
each item being a vector of length two corresponding to the plotting ranges
for dependence parameters a and b respectively. If this argument is not
missing, then 
PlotLikTitle 
Used only if 
... 
Further arguments to be passed to methods. 
Details
Estimates the extremal dependence structure of the data in x
. The
precise nature of the estimation depends on the value of margins
. If
margins="laplace"
(the default) then dependence parameters a and b
are estimated after transformation of the data to Laplace marginal
distributions. These parameters can describe both positive and negative
dependence. If margins="gumbel"
then the parameters a, b, c and d in
the dependence structure described by Heffernan and Tawn (2004) are
estimated in the following two steps: first, a and b are estimated; then, if
a=0 and b is negative, parameters c and d are estimated (this is the case of
negative dependence). Otherwise c and d will be fixed at zero (this is the
case of positive dependence).
If margins="laplace"
then the option of constrained parameter
estimation is available by setting argument constrain=TRUE
. The
default is to constrain the values of the parameters
(constrain=TRUE
). This constrained estimation ensures validity of
the estimated model, and enforces the consistency of the fitted dependence
model with the strength of extremal dependence exhibited by the data. More
details are given in Keef et al. (2013). The effect of this constraint is
to limit the shape of the dependence parameter space so that its boundary is
curved rather than following the original box constraints suggested by
Heffernan and Tawn (2004). The constraint brings with it some performance
issues for the optimiser used to estimate the dependence parameters, in
particular sensitivity to choice of starting value which we describe now.
The dependence parameter estimates returned by this function can be
particularly sensitive to the choice of starting value used for the
optimisation. This is especially true when margins="laplace"
and
constrain=TRUE
, in which case the maximum of the objective function
can lie on the edge of the (possibly curved) constrained parameter space.
It is therefore up to the user to check that the reported parameter
estimates really do correspond to the maximum of the profile lilkelihood
surface. This is easily carried out by using the visual diagnostics invoked
by setting PlotLikDo=TRUE
and adjusting the plotting area by using
the argument PlotLikRange
to focus on the region containing the
surface maximum. See an example below which illustrates the use of this
diagnostic.
Value
An object of class mex
which is a list containing the
following three objects:
margins 
An object of class

dependence 
An object of class

call 
This matches the original function call. 
Author(s)
Harry Southworth, Janet E. Heffernan
References
J. E. Heffernan and J. A. Tawn, A conditional approach for multivariate extreme values, Journal of the Royal Statistical society B, 66, 497 – 546, 2004.
C. Keef, I. Papastathopoulos and J. A. Tawn. Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model, Journal of Multivariate Analysis, 115, 396 – 404, 2013
See Also
migpd
, bootmex
,
predict.mex
, plot.mex
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  data(winter)
mygpd < migpd(winter , mqu=.7, penalty="none")
mexDependence(mygpd , which = "NO", dqu=.7)
# focus on 2d example with parameter estimates on boundary of constrained parameter space:
NO.NO2 < migpd(winter[,2:3] , mqu=.7, penalty="none")
# starting value gives estimate far from true max:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.01,0.01),
PlotLikDo=TRUE,PlotLikTitle=c("NO2  NO"))
# zoom in on plotting region containing maximum:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.01,0.01),
PlotLikDo=TRUE,PlotLikTitle=c("NO2  NO"),
PlotLikRange = list(a=c(0,0.8),b=c(0.2,0.6)))
# try different starting value:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.1,0.1),
PlotLikDo=TRUE,PlotLikTitle=c("NO2  NO"),
PlotLikRange = list(a=c(0,0.8),b=c(0.2,0.6)))
