A conventional statistical problem with irregular minimax behavior

Summary The problem of testing the mean vector of the two dimensional circularly symmetrical normal distribution with unit variances, where the data consists of just one sample point inR², is examined for stability of -maximin criteria. If the null hypothesisH₀ is the one point set containing the origin and the alternative set equal to the whole ofR²–H₀, then the -maximin is not unique. If a zone of indifference _I containingH₀ is introduced, then the problem of testingH₀ againstR² – _I can turn out to have a unique -maximin test. In the present paper we show a class of such _I for which this is the case. We show further that, given any -maximin test for testingH₀ againstR²–H₀, there is a decreasing sequence of _I, with intersection equal toH₀, for which the corresponding sequence of -maximin tests forH₀ againstR² – _I approaches a limit (in the usual weak star topology) which is not equivalent to .