# ddst.exp.test: Data Driven Smooth Test for Exponentiality In ddst: Data Driven Smooth Tests

## Description

Performs data driven smooth test for composite hypothesis of exponentiality.

## Usage

 ```1 2``` ```ddst.exp.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) = exp(-z/gamma)\$ for z >= 0 and 0 otherwise.

Modelling alternatives similarly as in Kallenberg and Ledwina (1997 a,b), e.g., and estimating \$gamma\$ by \$tilde gamma= 1/n sum_i=1^n Z_i\$ yields the efficient score vector \$l^*(Z_i;tilde gamma)=(phi_1(F(Z_i;tilde gamma)),...,phi_k(F(Z_i;tilde gamma)))\$, where \$phi_j\$'s are jth degree orthonormal Legendre polynomials on [0,1] or cosine functions \$sqrt(2) cos(pi j x), j>=1\$, while \$F(z;gamma)\$ is the distribution function pertaining to \$f(z;gamma)\$.

The 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 computed in a numerical way in case of cosine basis. In the implementation the default value of c in \$T^*\$ is set to be 100.

Therefore, \$T^*\$ practically coincides with S1 considered in Kallenberg and Ledwina (1997 a).

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

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

Kallenberg, W.C.M., Ledwina, T. (1997 b). Data driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc. 92, 1094–1104.

## Examples

 ```1 2 3 4 5 6 7 8``` ```# H0 is true z = rexp(80,4) ddst.exp.test (z, compute.p = TRUE) # H0 is false z = rchisq(80,4) (t = ddst.exp.test (z, compute.p = TRUE)) t\$p.value ```

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