evd (version 1.1-0)

fbvall: Simultaneous Maximum-likelihood Fitting of Bivariate Extreme Value Distributions

Description

Maximum-likelihood estimates for the bivariate extreme value distributions considered in this package, including linear modelling of the marginal location parameters.

Usage

fbvall(x, nsloc1 = NULL, nsloc2 = NULL, omit = NULL, boxcon = TRUE, 
    std.err = TRUE, orderby = c("AIC", "BIC", "SC"),
    control = list(maxit = 250))

Arguments

x

A matrix or data frame with two columns, which may contain missing values.

nsloc1, nsloc2

A data frame with the same number of rows as the length of x, to be passed to the individual fitting functions, for linear modelling of the location parameter on the first/second margin. The data frames are treated as covariat

omit

A character vector giving the models which are not to be fit, containing any of ``log'', ``alog'', ``hr'', ``neglog'', ``aneglog'', ``bilog'', ``negbilog'' or ``ct'' for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymm

boxcon

Logical; if TRUE (the default), the L-BFGS-B optimization method is used, which includes box constraints. If FALSE the BFGS method is used. The BFGS method is much faster, and sh

std.err

Logical; if TRUE (the default), the ``standard errors'' are returned. A logical vector can also be given. This should be the same length as the number of models being fitted.

orderby

The (column) order of the models in the returned list components. The models are ordered by best fit, as measured by Akaike's information criterion (the default), Bayesian information criterion or Schwarz's criterion.

control

To be passed to optim. Only options which are independent of the number of parameters within the optimization should be given. Some options are related to the optimization method used. This is def

Value

• A list with with components
• estimate
  A matrix containing the maximum likelihood estimates.
• std.err
  A matrix containing the ``standard errors'' (if $\code{std.err} = \code{TRUE}$).
• deviance
  The deviance at the maximum likelihood estimates for each fitted model.
• criteria
  A matrix containing Akaike's information criterion, Bayesian information criterion and Schwarz's criterion for each fitted model.
• dep.summary
  A matrix containing three values summarizing the dependence structure for each fitted model (see Details).

Details

Maximization of the log-likelihood for every bivariate model is performed. The std.err component of the returned list is taken from the observed information, calculated by a numerical approximation. The ``standard errors'' must be interpreted with caution because the usual asymptotic properties of maximum likelihood estimators may not hold (Smith, 1985).

Any bivariate extreme value distribution can be written as $$G(z_1,z_2) = \exp\left[-(y_1+y_2)A\left( \frac{y_1}{y_1+y_2}\right)\right]$$ for some function $A(\cdot)$ defined on $[0,1]$, where $$y_i = {1+s_i(z_i-a_i)/b_i}^{-1/s_i}$$ for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,2$, with the marginal parameters given by $(a_i,b_i,s_i)$, $b_i > 0$.

$A(\cdot)$ is called (by some authors) the dependence function. It follows that $A(0)=A(1)=1$, and that $A(\cdot)$ is a convex function with $\max(x,1-x) \leq A(x)\leq 1$ for all $0\leq x\leq1$. $A(\cdot)$ does not depend on the marginal parameters.

The component dep.summary of the returned list contains three values summarizing the dependence structure for each fitted model, based on $A(\cdot)$. There are two measures of dependence. The first is given by $2(1-A(1/2))$. The second is the integral of $4(1 - A(x))$, taken over $0\leq x\leq1$. These appear in the rows of dep.summary labelled by dep and intdep respectively. Both measures are zero at independence and one at complete dependence. The final row of dep.summary, labelled intasy, contains a measure of asymmetry given by the integral of $4(A(x) - A(1-x))/(3 - 2\sqrt2)$, taken over $0 \leq x \leq 0.5$. This integral lies in the closed interval [-1,1] (conjecture), with larger absolute values representing stronger asymmetry. As a rough guide, any value within the interval $(-0.25,0.25)$ suggests that the dependence structure is close to symmetric. For the symmetric models $A(x) = A(1-x)$ for all $0 \leq x \leq 0.5$, so the integral will be zero.

References

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

See Also

fbvlog, optim

Examples

data(sealevel)
fbvall(sealevel)
fbvall(sealevel, nsloc1 = (-40:40)/100)