LRExp.test: Likelihood Ratio test of exponentiality vs. GPD

In Renext: Renewal Method for Extreme Values Extrapolation

Likelihood Ratio test of exponentiality vs. GPD

Description

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.

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.

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}.

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(s)

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.

See Also

Lomax, Maxlo, GPD for the alternatives used here.

Examples

set.seed(1234)
x <- rGPD(n = 50, loc = 0, scale = 1, shape = 0.1)
LRExp.test(x, method = "num")$p.value
LRExp.test(x, method = "asymp")$p.value
## Not run: 
## requires much time
LRExp.test(x, method = "sim")$p.value

## End(Not run)

Renext documentation built on Aug. 30, 2023, 1:06 a.m.