Description Usage Arguments Details Value Author(s) References See Also Examples
Estimate the dependence parameters in a conditional multivariate extreme values model using the approach of Heffernan and Tawn, 2004.
1 
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. 
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. (2011). 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.
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. 
Harry Southworth, Janet E. Heffernan
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.
Keef, C., Papastathopoulos, I. and Tawn, J.A., (2011) Additional Constraints for the Conditional Multivariate Extremes Model of Heffernan and Tawn, to appear.
migpd
, bootmex
, predict.mex
, plot.mex
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)))

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