Unlimited simultaneous discrimination intervals in regression

Abstract

The discrimination problem can be described as follows: The statistician has n pairs of values ( x 1 , Y 1 ), ( x 2 , Y 2 ),…, ( x n , Y n ) from which he estimates the regression line α+β x . He now observes K additional observations Y 1 * , Y 2 * ,…, Y K * for which the corresponding independent variable values x 1 * , x 2 * ,…, x K * 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 (1968) and another solution was given by Miller (1966). When K is unknown and pomibly arbitrarily large, these results do not apply. A solution to this problem of arbitrary K is given in term of unlimited simultaneous discriminatdon intervals . Unlimited simultaneous discrimination intervals [D Y − ( P ), D Y + .( P )] are presented which are based upon the same estimated linear regression and which have the property that at least l00 P per cent of the discrimination intervals will contain the true x 's with confidence 1 −α. 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 simultaneons discrimination intervals is given.