ReIns (version 1.0.10)

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", ...)

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.

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.

Author

Tom Reynkens.

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

# 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)