Fit a two-parameters Generalised Pareto Distribution from a sample

Fit a two-parameters Generalised Pareto Distribution from a sample.

Usage

fGPD(x,
     info.observed = TRUE,
     shapeMin = -0.8,
     dCV = 1e-04,
     cov = TRUE,
     trace = 0)

Arguments

x

The sample data vector.

info.observed

Logical. The observed information matrix is used when TRUE and the expected information matrix is used when FALSE.

shapeMin

Lower bound on the shape parameter. This must be >= -1.0 since otherwise the ML estimate is obtained with the scale parameter equal to max(x).

dCV

Half-length of a small interval centered on 1.0. When the Coefficient of Variation (CV) falls in this interval, an exponential distribution is fitted, see Details.

cov

Logical. If FALSE, a minimal estimation is performed with no covariance matrix or derivative returned. This can be useful when a large number of ML estimations are required, e.g. to sample from a likelihood ratio.

trace

Integer level of verbosity. The value 0 prints nothing.

Details

This function mainly relies on the flomax and fmaxlo functions. When CV is larger than 1.0 + dCV, a Lomax distribution is fitted by Maximum Likelihood using a concentrated log-likelihood. When instead CV is smaller than 1.0 - dCV, a maxlo distribution is fitted. Finally, when CV -1.0 has absolute value <= dCV, an exponential distribution is fitted. In all cases, the result is translated into a parameter vector for the GPD.

Note that when CV is close to 1.0, fitting a Lomax or a maxlo distribution can lead to problems because the estimated values of the shape and scale parameter are large, and because the concentrated log-likelihood is a flat function of the scale parameter.

Value

A list

estimate

Vector containing the estimated values of the unknown parameters.

CV

The coefficient of variation of x computed using length(x) as denominator in the variance estimation.

logLik, dlogLik

The maximised value of the log-likelihood and the vector of its first order derivatives, which should be close to zero.

sd, cov

Vector of approximated standard deviations and covariance matrix for the estimated parameters. These are based on the inversion of expected information matrix.

sd, cov

Vector of standard deviations and covariance matrix for the estimates if the cov formal is TRUE.

cvg

Logical. Was convergence reached? This logical flag is set to TRUE and remains for compatibility reasons.

References

See the Renext Computing Details document.

Author

Yves Deville

Note

It may happen that the estimated shape parameter is < -0.5, in which case the expected information matrix can not be computed, nor does the covariance matrix and standard deviations. In this case, the cov and sd objects contain NA values. This problem can arise also when the shape parameter is greater than but close to the value -0.5. Even when info.observed is TRUE, the information matrix, covariance and standard deviations are set to NA.

When the true (unknown) value is is < -0.5, the regularity conditions required in the ML approximated inference do not hold.

The default value of info.observed was set to TRUE from version 3.0-1 because standard deviations obtained with this choice are usually better.

See also

The fmaxlo and flomax functions.

Examples

if (FALSE) {
set.seed(123456)
n <- 500
ns <- 1000
xi <- runif(ns, min = -0.5, max = 0.5)
X <- matrix(nrow = n, ncol = ns)

for (i in 1:length(xi)) {
  Xi <- rgpd(n, scale = 1, shape = xi[i])
  X[ , i] <- Xi
  res1 <- fGPD(Xi)
  res2 <- try(fpot(Xi, threshold = 0.0))
  if (inherits(res2, "try-error")) {
    cat(res2, "\n")
    break
  }
  logLik1 <- res1$loglik; logLik2 <- logLik(res2)
  if (abs(logLik1 - logLik2) > 0.001) {
    cat(sprintf("i = %d, xi = %7.4f\n", i, xi[i]))
    mat <- rbind(c(res1$estimate[1:2], logLik = logLik1),
                 c(res2$estimate[1:2], logLik = logLik2))
    rownames(mat) <- c("fGPD", "fpot")
    print(mat)
  }
}
}