Unlimited simultaneous discrimination intervals in regression.

The discrimination problem can be described as follows: The statistician has n pairs of values (xl, YI), (x2, Y2), ... , (xn, YJ) from which he estimates the regression line ac +,fx. He now observes K additional observations Y*, 4Y*, ... , YK for which the corresponding independent variable values x1', 4,..., XK are unknown. The statistician wishes to estimate these values of x and bracket them by means of simultaneous confidence intervals. This problem was first treated by Mandel (1958) and another solution was given by Miller (1966). When K is unknown and possibly arbitrarily large, these results do not apply. A solution to this problem of arbitrary K is given in terms of unlimited simultaneous discrimination intervals. Unlimited simultaneous discrimination intervals [D(P), D+*(P)] are presented which are based upon the same estimated linear regression and which have the property that at least lOOP per cent of the discrimination intervals will contain the true x's with confidence 1-ca. In this paper two techniques for obtaining unlimited simultaneous discrimination intervals are given. The first method is a procedure obtained through the Bonferroni inequality, while the second technique is based upon an idea of Lieberman & Miller (1963). A numerical example is analyzed. A general discussion and comparison of the two methods for finding unlimited simultaneous discrimination intervals is given.