On a new approach to the multi-sample goodness-of-fit problem

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.