Locally most powerful rank tests for testing randomness and symmetry

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Let Xi, 1 ≤ i ≤ N, be N independent random variables (i.r.v.) with distribution functions (d.f.) Fi(x,Θ), 1 ≤ i ≤ N, respectively, where Θ is a real parameter. Assume furthermore that Fi(·,0) = F(·) for 1 ≤ i ≤ N.

Let R = (R1,…,RN) and R+ = (R+1,…, R+N) be the rank vectors of X = (X1,…,XN) and |X| = (|X1…,|XN|), respectively, and let V = (V1,…,VN) 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.

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