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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.]

A. Kolmogorov [Giorn. Ist. Ital. Atturi 4, 83-91 (1933; Zbl 0006.17402)] in treating the GOF (goodness-of-fit) hypothesis ``H 0 : F=F 0 '' , introduced the statistic D n =sup|F n (z)-F 0 (z)|, where F 0 (·) is a completely specified continuous distribution, and F n (·) is the EDF (empirical distribution function) of the data Z=(X 1 ,···,X n ). D 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 0 :FΩ ''' , where Ω ' is a family of cpfs parametrized by a nuisance parameter. For example, Ω ' 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.

MSC:
62G10Nonparametric hypothesis testing