R: Maximum-likelihood Fitting of Bivariate Extreme Value...

fbvpot {evd} | 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.

### Usage

    fbvpot(x, threshold, model = c("log", "bilog", "alog", "neglog",
        "negbilog", "aneglog", "ct", "hr", "amix"), likelihood =
        c("censored", "poisson"), start, ..., sym = FALSE, cshape =
        cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method =
        "BFGS", warn.inf = TRUE)

### Arguments

`x` | 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)

[Package evd version 2.3-5 Index]