smoothtail (version 2.0.6)

smoothtail-package: Smooth Estimation of GPD Shape Parameter

Description

Given independent and identically distributed observations \(X_1 < \ldots < X_n\) from a Generalized Pareto distribution with shape parameter \(\gamma \in [-1,0]\), offers three methods to compute estimates of \(\gamma\). The estimates are based on the principle of replacing the order statistics \(X_{(1)}, \ldots, X_{(n)}\) of the sample by quantiles \(\hat X_{(1)}, \ldots, \hat X_{(n)}\) of the distribution function \(\hat F_n\) based on the log--concave density estimator \(\hat f_n\). This procedure is justified by the fact that the GPD density is log--concave for \(\gamma \in [-1,0]\).

Author

Kaspar Rufibach (maintainer), ,
http://www.kasparrufibach.ch

Samuel Mueller,

Kaspar Rufibach acknowledges support by the Swiss National Science Foundation SNF, http://www.snf.ch

Details

Package:	smoothtail
Type:	Package
Version:	2.0.5
Date:	2016-07-12
License:	GPL (>=2)

Use this package to estimate the shape parameter \(\gamma\) of a Generalized Pareto Distribution (GPD). In extreme value theory, \(\gamma\) is denoted tail index. We offer three new estimators, all based on the fact that the density function of the GPD is log--concave if \(\gamma \in [-1,0]\), see Mueller and Rufibach (2009). The functions for estimation of the tail index are:

pickands
falk
falkMVUE
generalizedPick

This package depends on the package logcondens for estimation of a log--concave density: all the above functions take as first argument a dlc object as generated by logConDens in logcondens.

Additionally, functions for density, distribution function, quantile function and random number generation for a GPD with location parameter 0, shape parameter \(\gamma\) and scale parameter \(\sigma\) are provided:

dgpd
pgpd
qgpd
rgpd.

Let us shortly clarify what we mean with log--concave density estimation. Suppose we are given an ordered sample \(Y_1 < \ldots < Y_n\) of i.i.d. random variables having density function \(f\), where \(f = \exp \varphi\) for a concave function \(\varphi : [-\infty, \infty) \to R\). Following the development in Duembgen and Rufibach (2009), it is then possible to get an estimator \(\hat f_n = \exp \hat \varphi_n\) of \(f\) via the maximizer \(\hat \varphi_n\) of

$$L(\varphi) = \sum_{i=1}^n \varphi(Y_i) - \int \exp \varphi (t) d t$$

over all concave functions \(\varphi\). It turns out that \(\hat \varphi_n\) is piecewise linear, with knots only at (some of the) observation points. Therefore, the infinite-dimensional optimization problem of finding the function \(\hat \varphi_n\) boils down to a finite dimensional problem of finding the vector \((\hat \varphi_n(Y_1),\ldots,\hat \varphi(Y_n))\). How to solve this problem is described in Rufibach (2006, 2007) and in a more general setting in Duembgen, Huesler, and Rufibach (2010). The distribution function based on \(\hat f_n\) is defined as

$$\hat F_n(x) = \int_{Y_1}^x \hat f_n(t) d t$$

for \(x\) a real number. The definition of \(\hat F_n\) is justified by the fact that \(\hat F_n(Y_1) = 0\).

References

Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40--68.

Duembgen, L., Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.

Mueller, S. and Rufibach K. (2009). Smooth tail index estimation. J. Stat. Comput. Simul., 79, 1155--1167.

Mueller, S. and Rufibach K. (2008). On the max--domain of attraction of distributions with log--concave densities. Statist. Probab. Lett., 78, 1440--1444.

Rufibach K. (2006) Log-concave Density Estimation and Bump Hunting for i.i.d. Observations. PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006.
Available at https://biblio.unibe.ch/download/eldiss/06rufibach_k.pdf.

Rufibach, K. (2007) Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul., 77, 561--574.

See Also

Package logcondens.

Examples

# generate ordered random sample from GPD
set.seed(1977)
n <- 20
gam <- -0.75
x <- rgpd(n, gam)

# compute known endpoint
omega <- -1 / gam

# estimate log-concave density, i.e. generate dlc object
est <- logConDens(x, smoothed = FALSE, print = FALSE, gam = NULL, xs = NULL)

# plot distribution functions
s <- seq(0.01, max(x), by = 0.01)
plot(0, 0, type = 'n', ylim = c(0, 1), xlim = range(c(x, s))); rug(x)
lines(s, pgpd(s, gam), type = 'l', col = 2)
lines(x, 1:n / n, type = 's', col = 3)
lines(x, est$Fhat, type = 'l', col = 4)
legend(1, 0.4, c('true', 'empirical', 'estimated'), col = c(2 : 4), lty = 1)

# compute tail index estimators for all sensible indices k
falk.logcon <- falk(est)
falkMVUE.logcon <- falkMVUE(est, omega)
pick.logcon <- pickands(est)
genPick.logcon <- generalizedPick(est, c = 0.75, gam0 = -1/3)

# plot smoothed and unsmoothed estimators versus number of order statistics
plot(0, 0, type = 'n', xlim = c(0,n), ylim = c(-1, 0.2))
lines(1:n, pick.logcon[, 2], col = 1); lines(1:n, pick.logcon[, 3], col = 1, lty = 2)
lines(1:n, falk.logcon[, 2], col = 2); lines(1:n, falk.logcon[, 3], col = 2, lty = 2)
lines(1:n, falkMVUE.logcon[,2], col = 3); lines(1:n, falkMVUE.logcon[,3], col = 3, 
    lty = 2)
lines(1:n, genPick.logcon[, 2], col = 4); lines(1:n, genPick.logcon[, 3], col = 4, 
    lty = 2)
abline(h = gam, lty = 3)
legend(11, 0.2, c("Pickands", "Falk", "Falk MVUE", "Generalized Pickands'"), 
    lty = 1, col = 1:8)