ddst (version 1.4)

ddst.extr.test: Data Driven Smooth Test for Extreme Value Distribution

Description

Performs data driven smooth test for composite hypothesis of extreme value distribution.

Usage

ddst.extr.test(x, base = ddst.base.legendre, c = 100, B = 1000, compute.p = F,  Dmax = 5, ...)

Arguments

x

a (non-empty) numeric vector of data values.

base

a function which returns orthogonal system, might be ddst.base.legendre for Legendre polynomials or ddst.base.cos for cosine system, see package description.

c

a parameter for model selection rule, see package description.

B

an integer specifying the number of replicates used in p-value computation.

compute.p

a logical value indicating whether to compute a p-value.

Dmax

an integer specifying the maximum number of coordinates, only for advanced users.

...

further arguments.

Value

An object of class htest

statistic

the value of the test statistic.

parameter

the number of choosen coordinates (k).

method

a character string indicating the parameters of performed test.

data.name

a character string giving the name(s) of the data.

p.value

the p-value for the test, computed only if compute.p=T.

Details

Null density is given by $f(z;gamma)=1/gamma_2 exp((z-gamma_1)/gamma_2- exp((z-gamma_1)/gamma_2))$, z in R.

We model alternatives similarly as in Kallenberg and Ledwina (1997) and Janic-Wroblewska (2004) using Legendre's polynomials or cosines. The parameter $gamma=(gamma_1,gamma_2)$ is estimated by $tilde gamma=(tilde gamma_1,tilde gamma_2)$, where $tilde gamma_1=-1/n sum_i=1^n Z_i + varepsilon G$, where $varepsilon approx 0.577216 $ is the Euler constant and $ G = tilde gamma_2 = [n(n-1) ln2]^-1sum_1<= j="" <="" i="" -="" z_n:j^o)="" $<="" em=""> while $Z_n:1^o <= ...="" <="Z_n:n^o$ are ordered variables $-Z_1,...,-Z_n$, cf Hosking et al. (1985). The above yields auxiliary test statistic $W_k^*(tilde gamma)$ described in details in Janic and Ledwina (2008), in case when Legendre's basis is applied.

The related matrix $[I^*(tilde gamma)]^-1$ does not depend on $tilde gamma$ and is calculated for succeding dimensions k using some recurrent relations for Legendre's polynomials and numerical methods for cosine functions. In the implementation the default value of c in $T^*$ was fixed to be 100. Hence, $T^*$ is Schwarz-type model selection rule. The resulting data driven test statistic for extreme value distribution is $W_T^*=W_T^*(tilde gamma)$.

For more details see: http://www.biecek.pl/R/ddst/description.pdf.

References

Hosking, J.R.M., Wallis, J.R., Wood, E.F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27, 251--261.

Janic-Wroblewska, A. (2004). Data-driven smooth test for extreme value distribution. Statistics 38, 413--426.

Janic, A. and Ledwina, T. (2008). Data-driven tests for a location-scale family revisited. J. Statist. Theory. Pract. Special issue on Modern Goodness of Fit Methods. accepted..

Kallenberg, W.C.M., Ledwina, T. (1997). Data driven smooth tests for composite hypotheses: Comparison of powers. J. Statist. Comput. Simul. 59, 101--121.

Examples

library(evd)

# for given vector of 19 numbers
z = c(13.41, 6.04, 1.26, 3.67, -4.54, 2.92, 0.44, 12.93, 6.77, 10.09, 
   4.10, 4.04, -1.97, 2.17, -5.38, -7.30, 4.75, 5.63, 8.84)
ddst.extr.test(z, compute.p=TRUE)

# H0 is true
x = -qgumbel(runif(100),-1,1)
ddst.extr.test (x, compute.p = TRUE)

# H0 is false
x = rexp(80,4)
ddst.extr.test (x, compute.p = TRUE)