Locally most powerful rank tests for testing randomness and symmetry

Abstract

Let X i, 1 ≤ i ≤ N, be N independent random variables (i.r.v.) with distribution functions (d.f.) F i(x,Θ), 1 ≤ i ≤ N, respectively, where Θ is a real parameter. Assume furthermore that F i(·,0) = F(·) for 1 ≤ i ≤ N. Let R = (R 1,R N) and R +,...,R +N be the rank vectors of X = (X 1,X N) and |X|=(|X 1|,...,|X N|), respectively, and let V = (V 1,V N) be the sign vector of X. The locally most powerful rank tests (LMPRT) S = S(R) and the locally most powerful signed rank tests (LMPSRT) S = S(R +, V) will be found for testing Θ = 0 against Θ > 0 or Θ < 0 with F being arbitrary and with F symmetric, respectively.