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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 ${H}_{0}:$ $F={F}_{0}^{\text{'}\text{'}}$, introduced the statistic ${D}_{n}=sup|{F}_{n}\left(z\right)-{F}_{0}\left(z\right)|,$ where ${F}_{0}\left(·\right)$ is a completely specified continuous distribution, and ${F}_{n}\left(·\right)$ is the EDF (empirical distribution function) of the data $Z=\left({X}_{1},···,{X}_{n}\right)$. ${D}_{n}$ 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 ${H}_{0}:F\in {{\Omega }}^{\text{'}\text{'}\text{'}},$ where ${{\Omega }}^{\text{'}}$ is a family of cpfs parametrized by a nuisance parameter. For example, ${{\Omega }}^{\text{'}}$ 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:
 62G10 Nonparametric hypothesis testing