R: Likelihood-Ratio Test

lr.test {extRemes} | R Documentation

## 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.

### 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).

### 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.

### Author(s)

Eric Gilleland

### 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

    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)

[Package extRemes version 2.0-12 Index]