One optional observation inflates $α$ by $100/\sqrt{n}$ per cent
Abstract
Description
For one-sample level $α$ tests $ψ_m$ 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 $ψ_{n},...,ψ_{n+k}$ would reject. For $k=1$ and usual tests at usual levels $α$, 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.
17 pages. Introduction sligthly extended, typos corrected, references linked. To appear in Metrika
17 pages. Introduction sligthly extended, typos corrected, references linked. To appear in Metrika