One optional observation inflates $α$ by $100/\sqrt{n}$ per cent

Details One optional observation inflates $α$ by $100/\sqrt{n}$ per cent One optional observation inflates $α$ by $100/\sqrt{n}$ per cent

2007-10-26

arxiv.org/abs/0710.5154v3

Mathematics

Statistics Theory

17 pages. Introduction sligthly extended, typos corrected, references linked. To appear in Metrika

Scientific paper

For one-sample level $\alpha$ tests $\psi_m$ based on independent observations $X_1,...,X_m$, we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests $\psi_{n},...,\psi_{n+k}$ would reject. For $k=1$ and usual tests at usual levels $\alpha$, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for $k=1$ in the special case of the Gauss test.

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