gp_test: Bootstrap test for the generalized Pareto distribution

Description

Test of fit for the generalized Pareto distribution (gPd) with unknown parameters by Villasenor-Alva and Gonzalez-Estrada (2009).

Usage

gp_test(x, B = 999)

Arguments

numeric data vector containing a random sample of positive real numbers.

number of bootstrap samples used to approximate p-values. Default is B=999.

Value

A list with class "htest" containing the following components.

p.value

an approximated p-value of the test using parametric bootstrap.

method

the character string "Bootstrap test of fit for the generalized Pareto distribution".

data.name

a character string giving the name of the data set.

pvalues

approximated p-values of the tests for $H_0^-$ and $H_0^+$

Details

This bootstrap test for the null hypothesis $H_0:$ a random sample has a gPd with unknown shape parameter $\gamma$ 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.

The gPd function with unknown shape and scale parameters $\gamma$ and $\sigma$ is 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 $\gamma = 0$, F(x) becomes 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. http://dx.doi.org/10.1016/j.csda.2009.04.001

Examples

Run this code

# NOT RUN {
# Testing the gPd hypothesis on the excesses above the threshold 0.165 ppm of the ozone
# levels given in the o3 data set
data(o3)
o3levels <- o3$ozone_level - 0.165  # ozone levels minus the threshold 0.165 ppm     
gp_test(o3levels)                   # testing the gPd hypothesis
# }

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