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 $100(1-\alpha )\mathrm{\%}$ quantile of a standard normal distribution. The corresponding P-value is thus given by $$\mathrm{\Phi }(T_{k,n})$$ with $\mathrm{\Phi }$ 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)
# }