Renext (version 3.1-4)

LRGumbel.test: Likelihood Ratio test for the Gumbel distribution

Description

Likelihood Ratio test of Gumbel vs. GEV

Usage

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

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 \(W := -2 \log \textrm{LR}\).

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.

Author

Yves Deville

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.

Examples

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