evd (version 1.1-0)

bvmle: Maximum-likelihood Fitting of Bivariate Extreme Value Distributions

Description

Maximum-likelihood estimates for the symmetric/asymmetric logistic, symmetric/asymmetric negative logistic, bilogistic, negative bilogistic, Husler-Reiss and Coles-Tawn models, including linear modelling of the marginal location parameters, and allowing any of the parameters to be held fixed if desired.

Usage

fbvlog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE,
    method = "BFGS")
fbvalog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE,
    method = "BFGS")
fbvhr(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE,
    method = "BFGS")
fbvneglog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE,
    method = "BFGS")
fbvaneglog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE,
    method = "BFGS")
fbvbilog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE, 
    method = "BFGS") 
fbvnegbilog(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE, 
    method = "BFGS")
fbvct(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE, 
    method = "BFGS")

Arguments

x

A matrix or data frame with two columns, 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 marginal maximum likelihood estimators.

...

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

nsloc1, nsloc2

A data frame with the same number of rows as the length of x, for linear modelling of the location parameter on the first/second margin (see Details). The data frames are treated as covariate matrices, excluding the int

std.err

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

method

The optimization method (see optim for details).

Value

• A list with with components
• estimate
  A vector containing the maximum likelihood estimates.
• std.err
  A vector containing the ``standard errors'' (if $\code{std.err} = \code{TRUE}$).
• deviance
  The deviance at the maximum likelihood estimates.
• counts
  A two-element integer vector giving the number of calls to the log-likelihood function and the number gradient approximations respectively.

Details

Maximization of the log-likelihood 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).

For stationary fitting, the named components of start can be any of dep, asy (a vector of length two), alpha, beta, mar1 (a vector of length three) and mar2 (a vector of length three). Alternatively, mar1 can be passed as three separate arguments; loc1, scale1 and shape1. mar2 can be similarly passed as three separate arguments; loc2, scale2 and shape2. The asymmetry parameters for the asymmetric logistic and negative asymmetric logistic models can also be passed separately as asy1 and asy2. Associating a separate name with each parameter allows any parameter subset to be fixed at specified values. All parameters to be fixed must be passed individually (e.g. to fix the parameters on the first margin all of loc1, scale1 and shape1 must be specified; using mar1 to specify all three simultaneously results in an error).

For non-stationary fitting, the non-stationary parameters on the first margin can be passed (either in start or as fixed values) using the column names of the data frame nsloc1 with the prefix ``loc1''. The intercept of the linear model can be passed as loc1. 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 loc1trend. Parameters on the second margin can be passed similarly.

For non-stationary fitting it is recommended that the covariates within the linear models for the location parameters 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

optim

Examples

bvdata <- rbvlog(100, dep = 0.6, mar1 = c(1.2,1.4,0.4),
mar2 = c(1.2,1.4,0.4))
fbvlog(bvdata)
fbvlog(bvdata, dep = 1)
fgev(bvdata[,1])$estimate
fgev(bvdata[,2])$estimate
fbvlog(bvdata, nsloc1 = (-49:50)/100)
fbvlog(bvdata, nsloc1 = (-49:50)/100, nsloc2 = data.frame(trend =
(-49:50)/100, random = runif(100, min=-.5, max=.5)))
fbvlog(bvdata, nsloc1 = (-49:50)/100, nsloc2 = data.frame(trend =
(-49:50)/100, random = runif(100, min=-.5, max=.5)), loc2random = 0)