Let Z1,..., Zn be identically independently distributed according to a continuous distribution D, of which it is assumed only that it is symmetric about some (unknown) point. For testing the hypothesis H : D(0) = 1 2 , the sign test maximizes the minimum power against the alternatives K : D(0) ≤ q(q < 1 2 ). [A pair of least favorable distributions assign probability 1, respectively, to the distributions F ∈ H, G ∈ K with densities f (x) = 1 − 2q 2(1 − q)

 q 1 − q

[|x|]

, g(x) = (1 − 2q)

 q 1 − q

|[x]|

where for all x (positive, negative, or zero) [x] denotes the largest integer ≤ x.]