Calculates initial estimates and estimated standard errors (SEs) for the generalized Pareto parameters \(\sigma\) and \(\xi\) based on an assumed random sample from this distribution. Also, calculates initial estimates and estimated standard errors for \(\phi\)1 = \(\sigma\) and \(\phi\)2 = \(\xi + \sigma x\) (m), where \(x\)(m) is the sample maximum threshold exceedance.
gpd_init(gpd_data, m, xm, sum_gp = NULL, xi_eq_zero = FALSE, init_ests = NULL)A numeric vector containing positive sample values.
A numeric scalar. The sample size, i.e., the length of
gpd_data.
A numeric scalar. The sample maximum.
A numeric scalar. The sum of the sample values.
A logical scalar. If TRUE assume that the shape parameter \(\xi = 0\).
A numeric vector. Initial estimate of
\(\theta = (\sigma, \xi)\). If supplied gpd_init()
returns the corresponding initial estimate of
\(\phi\) = (\(\phi\)1,
\(\phi\)2).
If init_ests is not supplied by the user, a list is returned
with components
A numeric vector. Initial estimates of \(\sigma\) and \(\xi\).
A numeric vector. Estimated standard errors of \(\sigma\) and \(\xi\).
A numeric vector. Initial estimates of
\(\phi\)1 = \(\sigma\)
and
\(\phi\)2 = \(\xi + \sigma x\)
(m)
where \(x\)(m)
is the maximum of gpd_data.
A numeric vector. Estimated standard errors of \(\phi\)1 and \(\phi\)1.
If init_ests is supplied then only the numeric vector
init_phi is returned.
The main aim is to calculate an admissible estimate of
\(\theta\), i.e., one at which the log-likelihood is finite (necessary
for the posterior log-density to be finite) at the estimate, and
associated estimated SEs. These are converted into estimates and SEs for
\(\phi\). The latter can be used to set values of min_phi and
max_phi for input to find_lambda.
In the default setting (xi_eq_zero = FALSE and
init_ests = NULL) the methods tried are Maximum Likelihood
Estimation (MLE) (Grimshaw, 1993), Probability-Weighted Moments (PWM)
(Hosking and Wallis, 1987) and Linear Combinations of Ratios of Spacings
(LRS) (Reiss and Thomas, 2007, page 134) in that order.
For \(\xi < -1\) the likelihood is unbounded, MLE may fail when
\(\xi\) is not greater than \(-0.5\) and the observed Fisher
information for \((\sigma, \xi)\) has finite variance only if
\(\xi > -0.25\). We use the ML estimate provided that
the estimate of \(\xi\) returned from gpd_mle is greater than
\(-1\). We only use the SE if the MLE of \(\xi\) is greater than
\(-0.25\).
If either the MLE or the SE are not OK then we try PWM. We use the PWM
estimate only if is admissible, and the MLE was not OK. We use the PWM SE,
but this will be c(NA, NA) if the PWM estimate of \(\xi\) is
\(> 1/2\). If the estimate is still not OK then we try LRS. As a last
resort, which will tend to occur only when \(\xi\) is strongly negative,
we set \(\xi = -1\) and estimate sigma conditional on this.
Grimshaw, S. D. (1993) Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution. Technometrics, 35(2), 185-191. and Computing (1991) 1, 129-133. doi:10.1007/BF01889987 .
Hosking, J. R. M. and Wallis, J. R. (1987) Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics, 29(3), 339-349. doi:10.2307/1269343 .
Reiss, R.-D., Thomas, M. (2007) Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields.Birkhauser. doi:10.1007/978-3-7643-7399-3 .
gpd_sum_stats to calculate summary statistics for
use in gpd_loglik.
rgpd for simulation from a generalized Pareto
find_lambda to produce (somewhat) automatically
a list for the argument lambda of ru.
# \donttest{
# Sample data from a GP(sigma, xi) distribution
gpd_data <- rgpd(m = 100, xi = 0, sigma = 1)
# Calculate summary statistics for use in the log-likelihood
ss <- gpd_sum_stats(gpd_data)
# Calculate initial estimates
do.call(gpd_init, ss)
#> $init
#> [1] 1.04553019 -0.03455597
#>
#> $se
#> sigma[u] xi
#> 0.14393678 0.09468163
#>
#> $init_phi
#> [1] 1.0455302 0.1476632
#>
#> $se_phi
#> [1] 0.14393678 0.07877361
#>
# }