trHill function

Hill estimator for upper truncated data

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", ...)

Arguments

• data: Vector of $n$ observations.
• r: Trimming parameter, default is 1 (no trimming).
• tol: Numerical tolerance for stopping criterion used in Newton-Raphson iterations, default is 1e-08.
• maxiter: Maximum number of Newton-Raphson iterations, default is 100.
• logk: Logical indicating if the estimates are plotted as a function of $\log(k)$ (logk=TRUE) or as a function of $k$. Default is FALSE.
• plot: Logical indicating if the estimates of $\gamma$ should be plotted as a function of $k$, default is FALSE.
• add: Logical indicating if the estimates of $\gamma$ should be added to an existing plot, default is FALSE.
• main: Title for the plot, default is "Estimates of the EVI".
• ...: Additional arguments for the plot function, see plot for more details.

Details

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.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• gamma: Vector of the corresponding estimates for $\gamma$.

• H: Vector of corresponding trimmed Hill estimates.

References

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.

Author(s)

Tom Reynkens based on R code of Dries Cornilly.

See Also

Hill, trDT, trEndpoint, trProb, trQuant, trMLE

Examples

# 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))