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Nuisance parameters, goodness-of-fit problems, and Kolmogorov-type statistics. (English) Zbl 0683.62026
Goodness-of-fit, Debrecen/Hung. 1984, Colloq. Math. Soc. János Bolyai 45, 21-58 (1987).
[For the entire collection see Zbl 0606.00025.] {\it A. Kolmogorov} [Giorn. Ist. Ital. Atturi 4, 83-91 (1933; Zbl 0006.17402)] in treating the GOF (goodness-of-fit) hypothesis $``H\sb 0:$ $F=F\sb 0''$, introduced the statistic $D\sb n=\sup \vert F\sb n(z)-F\sb 0(z)\vert,$ where $F\sb 0(\cdot)$ is a completely specified continuous distribution, and $F\sb n(\cdot)$ is the EDF (empirical distribution function) of the data $Z=(X\sb 1,...,X\sb n)$. $D\sb n$ 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 $``H\sb 0:F\in \Omega ''',$ where $\Omega '$ is a family of cpfs parametrized by a nuisance parameter. For example, $\Omega '$ 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.

62G10Nonparametric hypothesis testing