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Abstract For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn 3, 4 is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates. This result is based on an inequality for the probabilities of rectangles in normal distributions, which permits one to factor out the probability for any single coordinate.
Zbyněk Šidák (1967) studied this question.