Locally most powerful rank tests for testing randomness and symmetry

Let X_i, 1 i N, be N independent random variables (i.r.v.) with distribution functions (d.f.) F_i(x,), 1 i N, respectively, where is a real parameter. Assume furthermore that F_i(·,0) = F(·) for 1 i N.Let R = (R₁,R_N) and R⁺,...,R_N⁺be the rank vectors of X = (X₁,X_N) and |X|=(|X₁|,...,|X_N|), respectively, and let V = (V₁,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 = 0 against > 0 or < 0 with F being arbitrary and with F symmetric, respectively.