Data driven smooth tests for composite hypotheses. (English) Zbl 0904.62055

Inglot, Tadeusz; Kallenberg, Wilbert C. M.; Ledwina, Teresa

Ann. Stat. 25, No. 3, 1222-1250 (1997).

Summary: The classical problem of testing goodness-of-fit of a parametric family is reconsidered. A new test for this problem is proposed and investigated. The new test statistic is a combination of the smooth test statistic and Schwarz’s selection rule [G. Schwarz, ibid. 6, 461-464 (1978; Zbl 0379.62005)]. More precisely, as the sample size increases, an increasing family of exponential models describing departures from the null model is introduced and Schwarz’s selection rule is presented to seleet among them. Schwarz’s rule provides the “right” dimension given by the data, while the smooth test in the “right” dimension finishes the job. Theoretical properties of the selection rules are derived under null and alternative hypotheses. They imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative.

MSC:

62G10	Nonparametric hypothesis testing
62G20	Asymptotic properties of nonparametric inference

Keywords:

Schwarz’s BIC criterion; data driven procedure; Neyman’s test; testing goodness-of-fit; smooth test

Citations:

Zbl 0379.62005