General chi-square goodness-of-fit tests with data-dependent cells

SummaryThe goodness-of-fit of a parametric model for non-categorical data can be tested using the x2 statistic calculated after grouping the data into a finite number of disjoint cells. Work of Watson, Čebyšev, Moore and others shows that the classical limit distributions still hold even for certain methods of grouping that depend on the data themselves. These results are generalised to cover a much wider class of methods of grouping; the parameters can be estimated from either the grouped or the ungrouped data. The proofs use a Central Limit Theorem for Empirical Measures due to Dudley. The grouping cells are allowed to come from any Donsker class for the underlying sampling distribution.

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