Limiting Distribution of the Studentized Largest Observation1
Simeon M. Berman · Scandinavian Actuarial Journal · 1962
The distribution of the “studentized” largest observation, i.e. the largest observation minus the sample mean divided by the sample standard deviation, is fundamental in the theory of rejection of outlying observations. For samples up to size 25 from a normal population, this distribution has been tabulated by Grubbs [5]. The object of the present work is to derive the limiting distribution of the studentized largest observation for a general population. Empirical observations of Gumbel seem to indicate that if the largest observation has a limiting distribution, then the studentized largest observation has the same one [6]. Here this will be shown to hold under general conditions. This solves the problem raised by Borenius [1, p. 151].