On an independence test approach to the goodness-of-fit problem

Let X1,…,XnX1,…,Xn be independent and identically distributed random variables with distribution FF. Assuming that there are measurable functions f:R2→Rf:R2→R and g:R2→Rg:R2→R characterizing a family FF of distributions on the Borel sets of RR in the way that the random variables f(X1,X2),g(X1,X2)f(X1,X2),g(X1,X2) are independent, if and only if F∈FF∈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(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 FF is the family of gamma distributions or the family of inverse Gaussian distributions.