TY  - JOUR
A1  - Gaigall, Daniel
T1  - On a new approach to the multi-sample goodness-of-fit problem
JF  - Communications in Statistics - Simulation and Computation
N2  - Suppose we have k samples X₁,₁,…,X₁,ₙ₁,…,Xₖ,₁,…,Xₖ,ₙₖ with different sample sizes ₙ₁,…,ₙₖ and unknown underlying distribution functions F₁,…,Fₖ as observations plus k families of distribution functions {G₁(⋅,ϑ);ϑ∈Θ},…,{Gₖ(⋅,ϑ);ϑ∈Θ}, each indexed by elements ϑ from the same parameter set Θ, we consider the new goodness-of-fit problem whether or not (F₁,…,Fₖ) belongs to the parametric family {(G₁(⋅,ϑ),…,Gₖ(⋅,ϑ));ϑ∈Θ}. New test statistics are presented and a parametric bootstrap procedure for the approximation of the unknown null distributions is discussed. Under regularity assumptions, it is proved that the approximation works asymptotically, and the limiting distributions of the test statistics in the null hypothesis case are determined. Simulation studies investigate the quality of the new approach for small and moderate sample sizes. Applications to real-data sets illustrate how the idea can be used for verifying model assumptions.
KW  - Goodness-of-fit test
KW  - Multi-sample problem
KW  - Parametric bootstrap
Y1  - 2019
U6  - http://dx.doi.org/10.1080/03610918.2019.1618472
SN  - 1532-4141
VL  - 53
IS  - 10
SP  - 2971
EP  - 2989
PB  - Taylor & Francis
CY  - London
ER  - 
TY  - JOUR
A1  - Baringhaus, Ludwig
A1  - Gaigall, Daniel
T1  - On an independence test approach to the goodness-of-fit problem
JF  - Journal of Multivariate Analysis
N2  - Let X₁,…,Xₙ be independent and identically distributed random variables with distribution F. Assuming that there are measurable functions f:R²→R and g:R²→R characterizing a family F of distributions on the Borel sets of R in the way that the random variables f(X₁,X₂),g(X₁,X₂) are independent, if and only if F∈F, we propose to treat the testing problem H:F∈F,K:F∉F by applying a consistent nonparametric independence test to the bivariate sample variables (f(Xᵢ,Xⱼ),g(Xᵢ,Xⱼ)),1⩽i,j⩽n,i≠j. A parametric bootstrap procedure needed to get critical values is shown to work. The consistency of the test is discussed. The power performance of the procedure is compared with that of the classical tests of Kolmogorov–Smirnov and Cramér–von Mises in the special cases where F is the family of gamma distributions or the family of inverse Gaussian distributions.
KW  - Goodness-of-fit test
KW  - Independence test
KW  - Parametric bootstrap
KW  - Vapnik–Čhervonenkis class
KW  - Gamma distribution
Y1  - 2015
U6  - http://dx.doi.org/10.1016/j.jmva.2015.05.013
SN  - 0047-259X
VL  - 2015
IS  - 140
SP  - 193
EP  - 208
PB  - Elsevier
CY  - Amsterdam
ER  -