46. Confidence intervalsfor a shift. (i) Let XI" ' " Xm ; YI , . . . , y" be independently distributed according to continuous distributions F(x) and G(y) = F(y - 6) respectively. Without any further assumptions concerning F, confidence intervals for 6 can be obtained from permutation tests of the hypotheses H(6 0 ) : 6 = 6 0 , Specifically, consider the point (ZI" ' " zm+n) = (XI"' " Xm' YI - 6 , .. . ,Yn - 6) and the (m n) permutations i l < . . . < im; im+ 1 < . .. < i m +" of the integers 1, ... , m + n. Suppose that the hypothesis H(6) is accepted for the k of these permutations which lead to the smallest values of m+n m I L z;/n - L z;/m j-m+I j=1 where k = (1 - a)( m n). Then the totality of values 6 for which H( 6) is accepted constitute an interval, and these intervals are confidence intervals for 6 at confidence level 1 - Q . (ii) Let ZI' ... ' ZN be independently distributed, symmetric about (J, with distribution F( z - (J), where F( z) is continuous and symmetric about O. Without any further assumptions about F, confidence intervals for (J can be obtained by considering the 2N points ZI, . .. , Z;" where Z: = ±(Z; - (Jo), and accepting H«(Jo): (J = (Jo for the k of these points which lead to the smallest values of LIZ:I, where k = (1 -

a) 2N • [(i): A point is in the acceptance region for H( 6) if IL(Y;n- 6) _ L:;/=IY_X_6 1 is exceeded by at least (m n) - k of the quantities IY' - x' - y61, where (xi, . .. , X~" Y{,. .. , 1,;) is a permutation of (XI' .. . , xm' YI ' ... , Yn), the quantity y is determined by this permutation, and IYI s 1. The desired result now follows from the following facts (for an alternative proof, see Section 14):

(a) The set of 6's for which (ji - x - 6)2 (ji' - x' - y6)2 is, with probability one, an interval containing Y- X.

(b) The set of 6's for which (ji - x - 6)2 is exceeded by a particular set of at least (m;:; n) - k of the quantities tV - x' - yl1)2 is the intersection of the corresponding intervals

(a) and hence is an interval containing ji - x.

(c) The set of l1's of interest is the union of the intervals

(b) and, since they have a nonempty intersection, also an interval.]