trTestMLE: Test for truncated GPD tails

Description

Test between non-truncated GPD tails (light truncation) and truncated GPD tails (rough truncation).

Usage

trTestMLE(data, gamma, tau, alpha = 0.05, plot = TRUE, main = "Test for truncation", ...)

Value

A list with following components:

k: Vector of the values of the tail parameter $k$.
testVal: Corresponding test values.
critVal: Critical value used for the test, i.e. $\ln(1/\alpha)$.
Pval: Corresponding P-values.
Reject: Logical vector indicating if the null hypothesis is rejected for a certain value of k.

Arguments

data: Vector of $n$ observations.
gamma: Vector of $n-1$ estimates for the EVI obtained from trMLE.
tau: Vector of $n-1$ estimates for the $\tau$ obtained from trMLE.
alpha: The used significance level, default is 0.05.
plot: Logical indicating if the P-values should be plotted as a function of $k$, default is FALSE.
main: Title for the plot, default is "Test for truncation".
...: Additional arguments for the plot function, see plot for more details.

Author

Tom Reynkens.

Details

We want to test $H_0: X$ has non-truncated GPD tails vs. $H_1: X$ has truncated GPD tails. Let $\hat{\gamma}_k$ and $\hat{\tau}_k$ be the truncated MLE estimates for $\gamma$ and $\tau$. The test statistic is then $$T_{k,n}=k (1+\hat{\tau} (X_{n,n}-X_{-k,n}) )^{-1/\hat{\xi}_k}$$ which is asymptotically standard exponentially distributed. We reject $H_0$ on level $\alpha$ if $T_{k,n}>\ln (1/{\alpha)}$. The corresponding P-value is given by $\exp(-T_{k,n})$.

See Beirlant et al. (2017) for more details.

References

Beirlant, J., Fraga Alves, M. I. and Reynkens, T. (2017). "Fitting Tails Affected by Truncation". Electronic Journal of Statistics, 11(1), 2026--2065.

Examples

Run this code

# Sample from GPD truncated at 99% quantile
gamma <- 0.5
sigma <- 1.5
X <- rtgpd(n=250, gamma=gamma, sigma=sigma, endpoint=qgpd(0.99, gamma=gamma, sigma=sigma))

# Truncated ML estimator
trmle <- trMLE(X, plot=TRUE, ylim=c(0,2))

# Test for truncation
trTestMLE(X, gamma=trmle$gamma, tau=trmle$tau)

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