smoothtail (version 1.0)

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 $\gamma \in [-1,0]$, this package 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 $\gamma \in [-1,0]$.

Details

ll{ Package: smoothtail Type: Package Version: 1.0 Date: 2006-11-27 License: GPL version 2 or newer } 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 $\gamma \in [-1,0]$. The functions for estimation of the tail index are: pickands falk falkMVUE 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. This package depends on the package logcondens for estimation of a log--concave density. 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 (2006), 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 (2006a, b). 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. (2006). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. Preprint, IMSV, University of Bern. Mueller, S. and Rufibach K. (2006a). Smooth tail index estimation. Preprint. Mueller, S. and Rufibach K. (2006b). On the max--domain of attraction of distributions with log--concave densities. Submitted. Rufibach, K. (2006a). Computing maximum likelihood estimators of a log--concave density function. To appear in Journal of Statistical Computation and Simulation. Rufibach K. (2006b). 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.

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, for illustration purposes
est <- activeSetLogCon(x)

# 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
falk.logcon <- falk(x)
falkMVUE.logcon <- falkMVUE(x, omega)
pick.logcon <- pickands(x)

# 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)
abline(h = gam, lty = 3)
legend(11, 0.2, c("Pickands", "Falk", "Falk MVUE"), lty = 1, col = 1:6)