Bias correction for GP distribution

Bias corrected estimates for the generalized Pareto distribution using Firth's modified score function or implicit bias subtraction.

Usage

gpd.bcor(par, dat, corr = c("subtract", "firth"), method = c("obs", "exp"))

Arguments

par: parameter vector (scale, shape)
dat: sample of observations
corr: string indicating which correction to employ either subtract or firth
method: string indicating whether to use the expected ('exp') or the observed ('obs' — the default) information matrix. Used only if corr='firth'

Value

vector of bias-corrected parameters

Details

Method subtract solves $$\tilde{\boldsymbol{\theta}} = \hat{\boldsymbol{\theta}} + b(\tilde{\boldsymbol{\theta}}$$ for $\tilde{\boldsymbol{\theta}}$, using the first order term in the bias expansion as given by gpd.bias.

The alternative is to use Firth's modified score and find the root of $$U(\tilde{\boldsymbol{\theta}})-i(\tilde{\boldsymbol{\theta}})b(\tilde{\boldsymbol{\theta}}),$$ where $U$ is the score vector, $b$ is the first order bias and $i$ is either the observed or Fisher information.

The routine uses the MLE as starting value and proceeds to find the solution using a root finding algorithm. Since the bias-correction is not valid for $\xi < -1/3$, any solution that is unbounded will return a vector of NA as the bias correction does not exist then.

Examples

set.seed(1)
dat <- rgp(n=40, scale=1, shape=-0.2)
par <- gp.fit(dat, threshold=0, show=FALSE)$estimate
gpd.bcor(par,dat, 'subtract')
#>     scale     shape 
#>  0.960597 -0.141139 
gpd.bcor(par,dat, 'firth') #observed information
#>      scale      shape 
#>  0.9625726 -0.1613215 
gpd.bcor(par,dat, 'firth','exp')
#>      scale      shape 
#>  0.9375194 -0.1123656