Computes the Hill estimator for positive extreme value indices, adapted for upper truncation, as a function of the tail parameter \(k\) (Aban et al. 2006; Beirlant et al., 2016). Optionally, these estimates are plotted as a function of \(k\).
trHill(data, r = 1, tol = 1e-08, maxiter = 100, logk = FALSE,
plot = FALSE, add = FALSE, main = "Estimates of the EVI", ...)A list with following components:
Vector of the values of the tail parameter \(k\).
Vector of the corresponding estimates for \(\gamma\).
Vector of corresponding trimmed Hill estimates.
Vector of \(n\) observations.
Trimming parameter, default is 1 (no trimming).
Numerical tolerance for stopping criterion used in Newton-Raphson iterations, default is 1e-08.
Maximum number of Newton-Raphson iterations, default is 100.
Logical indicating if the estimates are plotted as a function of \(\log(k)\) (logk=TRUE) or as a function of \(k\). Default is FALSE.
Logical indicating if the estimates of \(\gamma\) should be plotted as a function of \(k\), default is FALSE.
Logical indicating if the estimates of \(\gamma\) should be added to an existing plot, default is FALSE.
Title for the plot, default is "Estimates of the EVI".
Additional arguments for the plot function, see plot for more details.
Tom Reynkens based on R code of Dries Cornilly.
The truncated Hill estimator is the MLE for \(\gamma\) under the truncated Pareto distribution.
To estimate the EVI using the truncated Hill estimator an equation needs to be solved. Beirlant et al. (2016) propose to use Newton-Raphson iterations to solve this equation. We take the trimmed Hill estimates as starting values for this algorithm. The trimmed Hill estimator is defined as $$H_{r,k,n} = 1/(k-r+1) \sum_{j=r}^k \log(X_{n-j+1,n})-\log(X_{n-k,n})$$ for \(1 \le r < k < n\) and is a basic extension of the Hill estimator for upper truncated data (the ordinary Hill estimator is obtained for \(r=1\)).
The equation that needs to be solved is $$H_{r,k,n} = \gamma + R_{r,k,n}^{1/\gamma} \log(R_{r,k,n}) / (1-R_{r,k,n}^{1/\gamma})$$ with \(R_{r,k,n} = X_{n-k,n} / X_{n-r+1,n}\).
See Beirlant et al. (2016) or Section 4.2.3 of Albrecher et al. (2017) for more details.
Aban, I.B., Meerschaert, M.M. and Panorska, A.K. (2006). "Parameter Estimation for the Truncated Pareto Distribution." Journal of the American Statistical Association, 101, 270--277.
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429--462.
Hill, trDT, trEndpoint, trProb, trQuant, trMLE
# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))
# Truncated Hill estimator
trh <- trHill(X, plot=TRUE, ylim=c(0,2))
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