Asymptotic normality of multivariate linear rank statistics under general alternatives

James A. Koziol

Aplikace matematiky (1979)

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

Abstract

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Let X j , 1 ≤ j ≤ N , be independent random p -vectors with respective continuous cumulative distribution functions F j 1 ≤ j ≤ N . Define the p -vectors R j by setting R i j equal to the rank of X i j among X i j , ... , X i N , 1 ≤ i ≤ p , 1 ≤ j ≤ N . Let a ( N ) ( . ) denote a multivariate score function in R p , and put S = ∑ j = 1 N c j a ( 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.

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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  1. Chernoff H., and Savage I. R., 10.1214/aoms/1177706436, Ann. Math. Stat. 29 (1958), 972-994. (1958) MR0100322DOI10.1214/aoms/1177706436
  2. Dupač V., A contribution to the asymptotic normality of simple linear rank statistics, In Nonparametric Techniques in Statistical Inference (M. L. Prui, Ed.), pp. 75-88, University Press, Cambridge, 1970. (1970) MR0283930
  3. Hájek J., 10.1214/aoms/1177698394, Ann. Math. Stat. 39 (1968), 325-246. (1968) MR0222988DOI10.1214/aoms/1177698394
  4. Hoeffding W., 10.1214/aos/1193342381, Ann. Stat. 1 (1973), 54-66. (1973) Zbl0255.62015MR0362689DOI10.1214/aos/1193342381
  5. Natanson I. P., Theory of Functions of a Real Variable 1, Frederick Ungar, New York, 1961. (1961) MR0067952
  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
  7. Puri M. L., Sen P. K., Nonparametric Methods in Multivariate Analysis, John Wiley, New York, 1971. (1971) Zbl0237.62033MR0298844
  8. Sen P. K., Puri M. L., 10.1214/aoms/1177698790, Ann. Math. Stat. 38 (1968), 1216-1228. (1968) MR0212954DOI10.1214/aoms/1177698790

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