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Likelihood Ratio test for the Gumbel distribution

LRGumbel.test.Rd

Likelihood Ratio test of Gumbel vs. GEV

Usage

LRGumbel.test(x,
                 alternative = c("frechet", "GEV"),
                 method = c("num", "sim", "asymp"),
                 nSamp = 1500,
                 simW = FALSE)

Arguments

x

Numeric vector of sample values.

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 asymptotic distribution of the Likelihood-ratio statistic is known. For the GEV alternative, this is a chi-square distribution with one df. For the Fréchet alternative, this is the distribution of a product \(XY\) where \(X\) and \(Y\) 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.

  • When method is "sim", nSamp samples of the Gumbel 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. This method requires more computation time than the tow others.

  • Finally when method is "asymp", the asymptotic distribution is used.

Note

For the Fréchet 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 GEV shape parameter is negative.

When method is "sim", the computation can be slow because each of the nSamp simulated values requires two optimisations. The "asymp" method provides an acceptable precision for \(n \geq 50\), and may even be used for \(n \geq 30\).

Author

Yves Deville

Examples

set.seed(1234)
x <- rgumbel(60)
res <- LRGumbel.test(x)