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Data-driven version of Neyman’s smooth test of fit. (English) Zbl 0805.62022
Summary: J. Neyman’s [Skand. Aktuarie Tidskr. 20, 149-199 (1937; Zbl 0018.03403)] smooth test for testing uniformity is a recognized goodness-of-fit procedure. As stated by V. N. LaRiccia [J. Am. Stat. Assoc. 86, 427-431 (1991)] the test can be viewed as a compromise between omnibus test procedures, with generally low power in all directions, and procedures whose power is focused in the direction of a specific alternative. The basic idea behind this test is to embed the null density into, say, a $$k$$-dimensional exponential family and then to construct an asymptotically optimal test for the parametric testing problem. The resulting procedure is Neyman’s test with $$k$$ components. The most difficult problem related with using this test is the choice of $$k$$. Recommendations in statistical literature are sometimes confusing. Some authors advocate a small number of components, whereas others show that in some situations a larger number of components is profitable. All existing suggestions concerning how to select $$k$$ exploit in fact some preliminary knowledge about a possible alternative.
In this article, a new data-driven method for selecting the number of components in Neyman’s test is proposed. The method consists of using G. Schwarz’s BIC procedure [Ann. Stat. 6, 461-464 (1978; Zbl 0379.62005)] to choose the dimension of the exponential model for the data and then using the chosen dimension as the number of components. So this novel procedure relies on fitting the model to the data and verifying whether the difference between the model and the uniform distribution is significant. Consistency of the method is proved. Presented simulations show that the test adapts well to the alternatives at hand. Simulated power of the data-driven version of Neyman’s test also performs well in comparison with that of other tests.

##### MSC:
 62F03 Parametric hypothesis testing 62G10 Nonparametric hypothesis testing
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