Confidence intervals for a shift. [Maritz (1979)]

(i) Let X1,..., Xm; Y1,..., Yn be independently distributed according to continuous distributions F(x) and G(y) = F(y − ) respectively. Without any further assumptions concerning F, confidence intervals for can be obtained from permutation tests of the hypotheses H(0) : = 0. Specifically, consider the point (z1,...,zm+n) = (x1,..., xm, y1 − ,... , yn − ) and the m+n m



permutations i1 < ··· < im; im+1 < ··· < im+n of the integers 1,..., m + n. Suppose that the hypothesis H() is accepted for the k of these permutations which lead to the smallest values of

m

+n j=m+1 zi j /n −m j=1 zi j /m

, where k = (1 − α)



m + n m



.

Then the totality of values for which H() is accepted constitute an interval, and these intervals are confidence intervals for at confidence level 1 − α.

(ii) Let Z1,..., ZN be independently distributed, symmetric about θ, with distribution F(z − θ), where F(z) is continuous and symmetric about 0. Without any further assumptions about F, confidence intervals for θ can be obtained by considering the 2N points Z

1,..., Z

N where Z

i = ±(Zi − θ0), and accepting H(θ0) : θ = θ0 for the k of these points which lead to the smallest values of

|

Z

i|, where k = (1 − α)2N .

[(i): A point is in the acceptance region for H() if (yj − )
n −
xi m =|¯y − ¯x − |
is exceeded by at least m+n n 
− k of the quantities | ¯y − ¯x − γ|, where (x
1,..., x
m, y
1,..., y
n) is a permutation of (x1,..., xm, y1,..., yn), the quantity γ is determined by this permutation, and |γ| ≤ 1. The desired result now follows from the following facts (for an alternative proof, see Section 14):

(a) The set of ’s for which (y¯ − ¯x − )2 ≤ (y¯ − ¯x − γ)2 is, with probability one, an interval containing y¯ − ¯x.

(b) The set of ’s for which (y¯ − ¯x − )2 is exceeded by a particular set of at least m+n m 
− k of the quantities (y¯ − ¯x − γ)2 is the intersection of the corresponding intervals

(a) and hence is an interval containing y¯ − ¯x.

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

(b) and, since they have a nonempty intersection, also an interval.]
Section 5.10