extRemes (version 2.0-9)

lr.test: Likelihood-Ratio Test

Description

Conduct the likelihood-ratio test for two nested extreme value distribution models.

Usage

lr.test(x, y, alpha = 0.05, df = 1, ...)

Arguments

x,y

Each can be either an object of class “fevd” (provided the fit method is MLE or GMLE) or a single numeric giving the negative log-likelihod value for each model. x should be the model with fewer parameters, but if both x and y are “fevd” objects, then the order does not matter (it will be determined from which model has more parameters).

alpha

single numeric between 0 and 1 giving the significance level for the test.

df

single numeric giving the degrees of freedom. If both x and y are “fevd” objects, then the degrees of freedom will be calculated, and this argument ignored. Otherwise, if either or both of x and y are single numerics, then it must be provided or the test may be invalid.

…

Not used.

Value

A list object of class “htest” is returned with components:

statistic

The test statistic value (referred to as D above).

parameter

numeric vector giving the chi-square critical value (c.alpha described above), the significance leve (alpha) and the degrees of freedom.

alternative

character string stating “greater” indicating that the alternative decision is determined if the statistic is greater than c.alpha.

p.value

numeric giving the p-value for the test. If the p-value is smaller than alpha, then the decision is to reject the null hypothesis in favor of the model with more parameters.

method

character string saying “Likelihood-ratio Test”.

data.name

character vector of length two giving the names of the datasets used for the test (if “fevd” objects are passed) or the negative log-likelihood values if numbers are passed, or the names of x and y. Although the names may differ, the models should have been fit to the same data set.

Details

When it is desired to incorporate covariates into an extreme value analysis, one method is to incorporate them into the parameters of the extreme value distributions themselves in a regression-like manner (cf. Coles, 2001 ch 6; Reiss and Thomas, 2007 ch 15). In order to justify whether or not inclusion of the covariates into the model is significant or not is to apply the likelihood-ratio test (of course, the test is more general than that, cf. Coles (2001) p 35).

The test is only valid for comparing nested models. That is, the parameters of one model must be a subset of the parameters of the second model.

Suppose the base model, m0, is nested within the model m1. Let x be the negative log-likelihood for m0 and y for m1. Then the likelihood-ratio statistic (or deviance statistic) is given by (Coles, 2001, p 35; Reiss and Thomas, 2007, p 118):

D = -2*(y - x).

Letting c.alpha be the (1 - alpha) quantile of the chi-square distribution with degrees of freedom equal to the difference in the number of model parameters, the null hypothesis that D = 0 is rejected if D > c.alpha (i.e., in favor of model m1).

References

Coles, S. (2001) An introduction to statistical modeling of extreme values, London, U.K.: Springer-Verlag, 208 pp.

Reiss, R.-D. and Thomas, M. (2007) Statistical Analysis of Extreme Values: with applications to insurance, finance, hydrology and other fields. Birkh\"auser, 530pp., 3rd edition.

See Also

fevd, taildep.test

Examples

# NOT RUN {
data(PORTw)
fit0 <- fevd(PORTw$TMX1, type="Gumbel") 
fit1 <- fevd(PORTw$TMX1)
fit2 <- fevd(TMX1, PORTw, scale.fun=~STDTMAX)
lr.test(fit0, fit1)
lr.test(fit1, fit2)

# }