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The paper deals with the problem of testing independence of a pair of random variables $X=W+\Delta ,\ Y=W^*+\Delta Z$ by locally most powerful rank tests in a neighborhood of the point $\Delta =0$. The corresponding tests for double-exponentially and for normally distributed random variables $W$ and $W^*$ are introduced. The power-functions of the $U$-test in a neighborhood of the points $\Delta =\rho =0$ for both cases are given numerically.
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