GPD_par {AoE} | R Documentation |
Computes estimates of the parameters (gamma, σ) of the generalized Pareto distribution fitted to excesses over a high threshold.
GPD_par(data, method = "ML", k = 5:(length(data) - 1))
data |
A numeric vector. |
method |
A character string determining which method will be used: "Hill" , "ML" , or "Moment" . |
k |
Integer vector. For each element of k , the parameter estimates will be computed based on the sample of excesses over the threshold u defined as the (k+1)th largest order statistic. |
Let X_{1:n} < ... < X_{n:n} be a the increasing order statistics of the sample. Let k = 1, ..., n-1. The function fits the generalized Pareto distribution
H(z) = 1 - (1 + gamma*z/σ)^(-1/gamma)
to the sample of excesses X_{n-k+i:n} - X_{n-k:n}, i = 1, ..., k over the threshold u = X_{n-k:n}.
In case method
is "Hill"
or "Moment"
, only those elements of k
will be retained for which the corresponding order statistic is positive.
A list with the class
attribute "GPD_par"
, which is a list containing the following components:
gamma |
Numeric vector with the same length as k containing the estimates for gamma. |
sigma |
Numeric vector with the same length as k containing the estimates for σ. |
threshold |
Numeric vector of thresholds corresponding to k . |
k |
Vector of k-values that have been used effectively. |
n |
The sample size. |
Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. The Annals of Statistics 17, 1833-1855.
Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163-1174.
Smith, R.L. (1987). Estimating tails of probability distributions. The Annals of Statistics 15, 1174-1207.
# random sample of size 100 # from the unit Frechet distribution: x <- - 1/log(runif(100)) # fit GPD to sample of excesses over 21th largest observation: out <- GPD_par(x) # plot estimates of gamma as a function of k (on logarithmic scale) # together with the true gamma (= 1) plot(out$k, out$gamma, type = "l", log = "x"); abline(h = 1)