ReIns (version 1.0.7)

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", ...)

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.

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.

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.

See Also

trMLE, trDTMLE, trProbMLE, trEndpointMLE, trTestMLE, trTest

Examples

# NOT RUN {
# 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)
# }