# gpd.test: Bootstrap goodness-of-fit test for the generalized Pareto... In gPdtest: 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

 `1` ```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`.

## 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 a 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 - [ 1 + gamma x / sigma ]^(-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:

1-exp(-x/sigma)

## 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.

## Author(s)

Elizabeth Gonzalez Estrada [email protected], Jose A. Villasenor Alva

## 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.

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

## Examples

 ```1 2``` ```x <- rgp(20,shape = 1) ## Random sample of size 20 gpd.test(x) ## Testing the gPd hypothesis on x ```

### Example output

```\$boot.test

Bootstrap test for the generalized Pareto distribution

data:  x
p-value = 0.8859

\$p.values
p.value R-statistic
H_0^-: x has a gPd with negative shape parameter  0.04604605   0.9441624
H_0^+: x has a gPd with positive shape parameter  0.88588589   0.9887903
```

gPdtest documentation built on May 29, 2017, 11:14 p.m.