Goodness-of-fit, Debrecen/Hung. 1984, Colloq. Math. Soc. János Bolyai 45, 21-58 (1987).
[For the entire collection see Zbl 0606.00025.]
A. Kolmogorov [Giorn. Ist. Ital. Atturi 4, 83-91 (1933; Zbl 0006.17402)] in treating the GOF (goodness-of-fit) hypothesis , introduced the statistic where is a completely specified continuous distribution, and is the EDF (empirical distribution function) of the data . is called the K-S (Kolmogorov-Smirnov) statistic.
In this paper one is concerned with cases in which the hypothesized cpf is not completely specified. The hypotheses here are of the form where is a family of cpfs parametrized by a nuisance parameter. For example, could be a family of normals, or exponentials or Paretos.
Since the hypothesized cpf is not completely specified, the K-S statistic cannot be used without some modifications. The object of this paper is to extend the methodology to a variety of families of cpfs; to several families of stochastic process laws; and to censored data problems. Further, the authors attempt to present a general framework, within which K-S type tests for nuisance parameter problems can be constructed.