ReIns (version 1.0.7)

trTest: Test for truncated Pareto-type tails

Description

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

Usage

trTest(data, alpha = 0.05, plot = TRUE, main = "Test for truncation", ...)

Arguments

data

Vector of \(n\) observations.

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. qnorm(1-alpha/2).

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 Pareto tails vs. \(H_1: X\) has truncated Pareto tails. Let $$E_{k,n}(\gamma) = 1/k \sum_{j=1}^k (X_{n-k,n}/X_{n-j+1,n})^{1/\gamma},$$ with \(X_{i,n}\) the \(i\)-th order statistic. The test statistic is then $$T_{k,n}=\sqrt{12k} (E_{k,n}(H_{k,n})-1/2) / (1-E_{k,n}(H_{k,n}))$$ which is asymptotically standard normally distributed. We reject \(H_0\) on level \(\alpha\) if $$T_{k,n}<-z_{\alpha}$$ where \(z_{\alpha}\) is the \(100(1-\alpha)\%\) quantile of a standard normal distribution. The corresponding P-value is thus given by $$\Phi(T_{k,n})$$ with \(\Phi\) the CDF of a standard normal distribution.

See Beirlant et al. (2016) or Section 4.2.3 of Albrecher et al. (2017) for more details.

References

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.

See Also

trHill, trTestMLE

Examples

# NOT RUN {
# Sample from truncated Pareto distribution.
# truncated at 95% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.95, shape=shape))

# Test for truncation
trTest(X)


# Sample from truncated Pareto distribution.
# truncated at 99% quantile
shape <- 2
X <- rtpareto(n=1000, shape=shape, endpoint=qpareto(0.99, shape=shape))

# Test for truncation
trTest(X)
# }