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", ...)
data |
Vector of n observations. |
alpha |
The used significance level, default is |
plot |
Logical indicating if the P-values should be plotted as a function of k, default is |
main |
Title for the plot, default is |
... |
Additional arguments for the |
We want to test H_0: X has non-truncated Pareto tails vs. H_1: X has truncated Pareto tails. Let
E_{k,n}(γ) = 1/k ∑_{j=1}^k (X_{n-k,n}/X_{n-j+1,n})^{1/γ},
with X_{i,n} the i-th order statistic. The test statistic is then
T_{k,n}=√{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 α if
T_{k,n}<-z_{α}
where z_{α} is the 100(1-α)\% quantile of a standard normal distribution. The corresponding P-value is thus given by
Φ(T_{k,n})
with Φ 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.
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. |
Pval |
Corresponding P-values. |
Reject |
Logical vector indicating if the null hypothesis is rejected for a certain value of |
Tom Reynkens.
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.
# 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)
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