On the asymptotic behaviour of location-scale invariant Bickel-Rosenblatt tests. (English) Zbl 1098.62055

Tenreiro, Carlos

J. Stat. Plann. Inference 137, No. 1, 103-116 (2007); erratum ibid. 139, No. 6, 2115 (2009).

Summary: Location-scale invariant Bickel-Rosenblatt goodness-of-fit tests (IBR tests) [P. J. Bickel and M. Rosenblatt, Ann. Stat. 1, 1071–1095 (1973; Zbl 0275.62033] are considered to test the hypothesis that \(f\), the common density function of the observed independent \(d\)-dimensional random vectors, belongs to a null location-scale family of density functions. The asymptotic behaviour of the test procedures for fixed and non-fixed bandwidths is studied by using a unifying approach. We establish the limiting null distribution of the test statistics, the consistency of the associated tests and we derive its asymptotic power against sequences of local alternatives. These results show the asymptotic superiority, for fixed and local alternatives, of IBR tests with fixed bandwidth over IBR tests with non-fixed bandwidth.

MSC:

62G10	Nonparametric hypothesis testing
62E20	Asymptotic distribution theory in statistics
62G20	Asymptotic properties of nonparametric inference

Keywords:

goodness-of-fit test; kernel density estimator; local power analysis

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