Asymptotic normality of multivariate linear rank statistics under general alternatives

Aplikace matematiky (1979)

• Volume: 24, Issue: 5, page 326-347
• ISSN: 0862-7940

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Abstract

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Let ${X}_{j},1\le j\le N$, be independent random $p$-vectors with respective continuous cumulative distribution functions ${F}_{j}1\le j\le N$. Define the $p$-vectors ${R}_{j}$ by setting ${R}_{ij}$ equal to the rank of ${X}_{ij}$ among ${X}_{ij},...,{X}_{iN},1\le i\le p,1\le j\le N$. Let ${a}^{\left(N\right)}\left(.\right)$ denote a multivariate score function in ${R}_{p}$, and put $S={\sum }_{j=1}^{N}{c}_{j}{a}^{\left(N\right)}\left({R}_{j}\right)$, 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.

How to cite

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Koziol, James A.. "Asymptotic normality of multivariate linear rank statistics under general alternatives." Aplikace matematiky 24.5 (1979): 326-347. <http://eudml.org/doc/15110>.

@article{Koziol1979,
abstract = {Let $X_j, 1\le j\le N$, be independent random $p$-vectors with respective continuous cumulative distribution functions $F_j 1\le j\le N$. Define the $p$-vectors $R_j$ by setting $R_\{ij\}$ equal to the rank of $X_\{ij\}$ among $X_\{ij\}, \ldots , X_\{iN\}, 1\le i \le p, 1\le j \le N$. 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.},
author = {Koziol, James A.},
journal = {Aplikace matematiky},
keywords = {asymptotic normality of multivariate linear rank statistics; general alternatives; asymptotic normality of multivariate linear rank statistics; general alternatives},
language = {eng},
number = {5},
pages = {326-347},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic normality of multivariate linear rank statistics under general alternatives},
url = {http://eudml.org/doc/15110},
volume = {24},
year = {1979},
}

TY - JOUR
AU - Koziol, James A.
TI - Asymptotic normality of multivariate linear rank statistics under general alternatives
JO - Aplikace matematiky
PY - 1979
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 24
IS - 5
SP - 326
EP - 347
AB - Let $X_j, 1\le j\le N$, be independent random $p$-vectors with respective continuous cumulative distribution functions $F_j 1\le j\le N$. Define the $p$-vectors $R_j$ by setting $R_{ij}$ equal to the rank of $X_{ij}$ among $X_{ij}, \ldots , X_{iN}, 1\le i \le p, 1\le j \le N$. 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.
LA - eng
KW - asymptotic normality of multivariate linear rank statistics; general alternatives; asymptotic normality of multivariate linear rank statistics; general alternatives
UR - http://eudml.org/doc/15110
ER -

References

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6. Patel K. M., 10.1016/0047-259X(73)90011-0, J. Multivariate Analysis 3 (1973), 57-70. (1973) Zbl0254.62030MR0326911DOI10.1016/0047-259X(73)90011-0
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8. Sen P. K., Puri M. L., 10.1214/aoms/1177698790, Ann. Math. Stat. 38 (1968), 1216-1228. (1968) Zbl0155.26404MR0212954DOI10.1214/aoms/1177698790

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