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locally most powerful rank tests; randomness; symmetry
Let $X_i$, $1\le i \le N$, be $N$ independent random variables (i.r.v.) with distribution functions (d.f.) $F_i (x,\Theta )$, $1\le i \le N$, respectively, where $\Theta $ is a real parameter. Assume furthermore that $F_i(\cdot ,0)=F(\cdot )$ for $1\le i \le N$. Let $R=(R_1,\ldots ,R_N)$ and $R^+=(R_1^+,\ldots ,R_N^+)$ be the rank vectors of $X = (X_1,\ldots ,X_N)$ and $|X| = (|X_1|,\ldots ,|X_N|)$, respectively, and let $V = (V_1,\ldots ,V_N)$ 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 $\Theta = 0$ against $\Theta >0$ or $\Theta <0$ with $F$ being arbitrary and with $F$ symmetric, respectively.
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