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

## Description

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

## Usage

 ```1 2``` ```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.

## 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 <= n(Z_n:i^o - Z_n:j^o) \$ 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.

## 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`.

## Author(s)

Przemyslaw Biecek and Teresa Ledwina

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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) ```

ddst documentation built on May 29, 2017, 9:34 p.m.