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

Let X-1,..., X-n be independent and identically distributed random variables with distribution IF. Assuming that there are measurable functions f : R-2 -> R and g : R-2 -> R characterizing a family F of distributions on the Borel sets of R in the way that the random variables f (X-1, X-2), g (X-1, X-2) are independent, if and only if F is an element of F, we propose to treat the testing problem H : F is an element of F, K : F is not an element of F by applying a consistent nonparametric independence test to the bivariate sample variables (f (X-i, X-j), g(X-i, X-j)), 1 <= i, j <= n, i not equal 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 Cramer-von Mises in the special cases where F is the family of gamma distributions or the family of inverse Gaussian distributions. (C) 2015 Elsevier Inc. All rights reserved.