3.10

(a) If X is a random variable with a continuous distribution function, symmetric about zero, then sign X and |X| are stochastically independent.

(b) Prove that the linear signed rank statistic (3.89) is a nondecreasing step function of  with probability 1, provided the score function '+ is nondecreasing on (0, 1) and '+(0) = 0. (See van Eeden (1972) or Jurecková and Sen (1996), Section 6.4.)