Printed in Great Britain Simultaneous tolerance intervals in regression!

Joint prediction intervals (based upon the original fitted model) for K future responses at each of K separate settings of the independent variable have been treated by Iieberman (1961). When K is unknown and possibly arbitrarily large, these results do not apply. A solution to the problem of arbitrary K is given in terms of tolerance intervals on the distributions of future observations, the intervals being (probabilistically) simultaneous in each possible value of the independent variable. Four alternative techniques are pro-posed and compared for their applicability in different situations. The first is the simul-taneous extension of the Wallis (1951) technique. The other three are based on Scheffe" simultaneous confidence principles. One gives intervals for a fixed central proportion P of the distribution which are simultaneous in all values of the independent variable; the other two give intervals simultaneous in the independent variable and different central propor-tions P. A numerical example is analysed, and some remarks are made on the applicability of the Scheffe ' techniques to the detection of outliers. 1