gPdtest (version 0.4)

gpd.test: Bootstrap goodness-of-fit test for the generalized Pareto distribution

Description

This function computes the bootstrap goodness-of-fit test by Villasenor-Alva and Gonzalez-Estrada (2009) for testing the null hypothesis $H_0:$ a random sample has a generalized Pareto distribution (gPd) with unknown shape parameter $gamma$, which is a real number.

Usage

gpd.test(x,J)

Arguments

x

numeric data vector containing a random sample from a distribution function with support on the positive real numbers.

J

number of bootstrap samples. This is an optional argument. Default J=999.

Value

A list with the following components.

boot.test

a list with class "htest" containing the p-value of the test, the name of the data set, and the character string "Bootstrap goodness-of-fit test for the generalized Pareto distribution".

p.values

the p-values of the tests of the hypotheses $H_0^-$ and $H_0^+$ described above.

Details

The bootstrap goodness-of-fit test for the gPd is an intersection-union test for the hypotheses $H_0^-:$ a random sample has a gPd with $gamma <0 $,="" and="" $h_0^+:$="" a="" random="" sample="" has="" gpd="" with="" $gamma="">=0$. Thus, heavy and non-heavy tailed gPd's are included in the null hypothesis. The parametric bootstrap is performed on $gamma$ for each of the two hypotheses.

We consider the distribution function of the gPd with shape and scale parameters $gamma$ and $sigma$ given by

$$F(x) = 1 - \left[ 1 + \frac{\gamma x}{ \sigma } \right] ^ { - 1 /\gamma}$$

where $gamma$ is a real number, $sigma > 0$ and $1 + gamma x / sigma > 0$. When $gamma = 0$, we have the exponential distribution with scale parameter $sigma$: $$F(x) = 1 -exp\left(-x/\sigma \right)$$

References

Villasenor-Alva, J.A. and Gonzalez-Estrada, E. (2009). A bootstrap goodness of fit test for the generalized Pareto distribution. Computational Statistics and Data Analysis,53,11,3835-3841.

See Also

gpd.fit for fitting a gPd to data, rgp for generating gPd random numbers.

Examples

x <-  rgp(20,shape = 1)   	## Random sample of size 20
gpd.test(x)                 ## Testing the gPd hypothesis on x