trTest function

Test for truncated Pareto-type tails

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

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.

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.

Returns

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.

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.

Author(s)

Tom Reynkens.

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)