# Article

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Keywords:
asymptotic normality of multivariate linear rank statistics; general alternatives
Summary:
Let $X_j, 1\leq j\leq N$, be independent random $p$-vectors with respective continuous cumulative distribution functions $F_j 1\leq j\leq N$. Define the $p$-vectors $R_j$ by setting $R_{ij}$ equal to the rank of $X_{ij}$ among $X_{ij}, \ldots, X_{iN}, 1\leq i \leq p, 1\leq j \leq N$. Let $a^{(N)}(.)$ denote a multivariate score function in $R_p$, and put $S= \sum ^N_{j=1} c_ja^{(N)}(R_j)$, the $c_j$ being arbitrary regression constants. In this paper the asymptotic distribution of $S$ is investigated under various sets of conditions on the constants, the score functions, and the underlying distribution functions. In particular, asymptotic normality of $S$ is established under the circumstance that the $F_j$ are merely continuous. In addition, under mild conditions, centering vectors for $S$ are found.
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