| Title:
|
Asymptotic normality of multivariate linear rank statistics under general alternatives (English) |
| Author:
|
Koziol, James A. |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
24 |
| Issue:
|
5 |
| Year:
|
1979 |
| Pages:
|
326-347 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
asymptotic normality of multivariate linear rank statistics |
| Keyword:
|
general alternatives |
| MSC:
|
62E20 |
| MSC:
|
62G10 |
| MSC:
|
62H10 |
| idZBL:
|
Zbl 0436.62022 |
| idMR:
|
MR0547037 |
| DOI:
|
10.21136/AM.1979.103814 |
| . |
| Date available:
|
2008-05-20T18:12:35Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103814 |
| . |
| Reference:
|
[1] Chernoff H., and Savage I. R.: Asymptotic normality and efficiency of certain nonparametric test statistics.Ann. Math. Stat. 29 (1958), 972-994. MR 0100322, 10.1214/aoms/1177706436 |
| Reference:
|
[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. MR 0283930 |
| Reference:
|
[3] Hájek J.: Asymptotic normality of simple linear rank statistics under alternatives.Ann. Math. Stat. 39 (1968), 325-246. MR 0222988, 10.1214/aoms/1177698394 |
| Reference:
|
[4] Hoeffding W.: On the centering of a simple linear rank statistic.Ann. Stat. 1 (1973), 54-66. Zbl 0255.62015, MR 0362689, 10.1214/aos/1193342381 |
| Reference:
|
[5] Natanson I. P.: Theory of Functions of a Real Variable 1.Frederick Ungar, New York, 1961. MR 0067952 |
| Reference:
|
[6] Patel K. M.: Hájek-Šidák approach to the asymptotic distribution of multivariate rank order statistics.J. Multivariate Analysis 3 (1973), 57-70. Zbl 0254.62030, MR 0326911, 10.1016/0047-259X(73)90011-0 |
| Reference:
|
[7] Puri M. L., Sen P. K.: Nonparametric Methods in Multivariate Analysis.John Wiley, New York, 1971. Zbl 0237.62033, MR 0298844 |
| Reference:
|
[8] Sen P. K., Puri M. L.: On the theory of rank order tests for location in the multivariate one sample problem.Ann. Math. Stat. 38 (1968), 1216-1228. MR 0212954, 10.1214/aoms/1177698790 |
| . |
