Likelihood Ratio test of exponentiality vs. GPD

Likelihood Ratio test of exponentiality vs. GPD.

Usage

LRExp.test(x,
              alternative = c("lomax", "GPD", "gpd", "maxlo"),
              method = c("num", "sim", "asymp"),
              nSamp = 15000,
              simW = FALSE)

Arguments

x

Numeric vector of positive sample values. For the POT context this should be the vector of excesses over the threshold.

alternative

Character string describing the alternative distribution.

method

Method used to compute the \(p\)-value.

nSamp

Number of samples for a simulation, if method is "sim".

simW

Logical. If this is set to TRUE and method is "sim", the simulated values are returned as an element W in the list.

Value

A list of results with elements statistic, p.value

and method. Other elements are

alternative

Character describing the alternative hypothesis.

W

If simW is TRUE and method is "sim" only. A vector of nSamp simulated values of the statistic \(W := -2 \log \textrm{LR}\).

Details

The Lomax and maxlo alternatives correspond to a GPD alternative with positive shape parameter \(\xi > 0\) (Lomax) and GPD with \(\xi < 0\) (maxlo).

The asymptotic distribution of the Likelihood-ratio statistic is known. For the GPD alternative, this is a chi-square distribution with one df. For the Lomax alternative, this is the distribution of a product \(BC\) where \(B\) and \(C\) are two independent random variables following a Bernoulli distribution with probability parameter \(p = 0.5\) and a chi-square distribution with one df.

When method is "num", a numerical approximation of the distribution is used. This method is not unlike that used by Kozubowski et al., but a different approximation is used. However, if x has a length \(n > 500\), the method is turned to "asymp".
When method is "sim", nSamp samples of the exponential distribution with the same size as x are drawn and the LR statistic is computed for each sample. The \(p\)-value is simply the estimated probability that a simulated LR is greater than the observed LR.
Finally when method is "asymp", the asymptotic distribution is used.

Note

For the Lomax alternative, the distribution of the test statistic has mixed type: it can take any positive value as well as the value \(0\) with a positive probability mass. The probability mass is the probability that the ML estimate of the GPD shape parameter is negative, and a good approximation of it is provided by the pGreenwood1 function. Note that this probability converges to its limit \(0.5\) very slowly, which suggests that the asymptotic distribution provides poor results for medium sample sizes, say \(< 100\).

Similarly for a maxlo alternative, the distribution of the test statistic has mixed type: it can take any positive value as well as the value \(0\) with a positive probability mass approximately given by 1 -pGreenwood1(n) where \(n\) is the sample size.

Author

Yves Deville

References

T.J. Kozubowski, A. K. Panorska, F. Qeadan, A. Gershunov and D. Rominger (2009) "Testing Exponentiality Versus Pareto Distribution via Likelihood Ratio" Comm. Statist. Simulation Comput. 38(1), pp. 118-139.

The approximation method used is described in the Renext Computing Details report.

Examples

set.seed(1234)
x <- rGPD(n = 50, loc = 0, scale = 1, shape = 0.1)
LRExp.test(x, method = "num")$p.value
#> [1] 0.403011
LRExp.test(x, method = "asymp")$p.value
#> [1] 0.5
if (FALSE) {
## requires much time
LRExp.test(x, method = "sim")$p.value
}