Data Driven Smooth Test for Exponentiality

Description

Performs data driven smooth test for composite hypothesis of exponentiality.

Usage

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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

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# 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