| Title:
|
On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$ (English) |
| Author:
|
Saran, Jagdish |
| Author:
|
Sen, Kanwar |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
27 |
| Issue:
|
6 |
| Year:
|
1982 |
| Pages:
|
417-425 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
The contents of the paper is concerned with the two-sample problem where $F_m(x)$ and $G_n(x)$ are two empirical distribution functions. The difference $F_m(x)-G_n(x)$ changes only at an $x_i, i=1,2,\ldots, m+n$, corresponding to one of the observations. Let $R^+_{mn}(j)$ denote the subscript $i$ for which $F_m(x_i)-G_n(x_i)$ achieves its maximum value $D^+_{mn}$ for the $j$th time $(j=1,2,\ldots)$. The paper deals with the probabilities for $R^+_{mn}(j)$ and for the vector $(D^+_{mn}, R^+_{mn}(j))$ under $H_0 : F=G$, thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model. (English) |
| Keyword:
|
points of maximal deviation |
| Keyword:
|
two-sample Smirnov statistic |
| Keyword:
|
empirical distribution functions |
| Keyword:
|
joint distribution |
| Keyword:
|
random walk model |
| MSC:
|
62E15 |
| MSC:
|
62G10 |
| MSC:
|
62G30 |
| idZBL:
|
Zbl 0514.62025 |
| idMR:
|
MR0678111 |
| DOI:
|
10.21136/AM.1982.103988 |
| . |
| Date available:
|
2008-05-20T18:20:25Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103988 |
| . |
| Reference:
|
[1] M. T. L. Bizley: Derivation of a new formula for the number of minimal lattice paths from $(0, 0)$ to $(km, kn)$ having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line.J. Inst. Actuar. 80 (1954), 55-62. MR 0061567, 10.1017/S002026810005424X |
| Reference:
|
[2] M. Dwass: Simple random walk and rank order statistics.Ann. Math. Statist. 38 (1967), 1042-1053. Zbl 0162.50204, MR 0215463, 10.1214/aoms/1177698773 |
| Reference:
|
[3] N. L. Geller: Two limiting distributions for two-sample Kolmogorov-Smirnov type statistics.Report No. 5, Centre for System Science, Case Western Reserve University, Cleveland, Ohio, (1971). |
| Reference:
|
[4] I. Očka: Simple random walk and rank order statistics.Apl. mat. 22 (1977), 272-290. MR 0438583 |
| Reference:
|
[5] K. Sarkadi: On Galton's rank order test.Publ. Math. Inst. Hungar. Acad. Sci. 6 (1961), 125-128. |
| Reference:
|
[6] Z. Šidák: Applications of Random Walks in Nonparametric Statistics.Bull. Internat. Statist. Inst., Proc. of the 39th session, vol. 45 (1973), book 3, 34-42. MR 0356357 |
| Reference:
|
[7] G. P. Steck: The Smirnov two-sample tests as rank tests.Ann. Math. Statist. 40 (1969), 1449-1466. Zbl 0186.52304, MR 0246473, 10.1214/aoms/1177697516 |
| Reference:
|
[8] G. P. Steck G. J. Simmons: On the distributions of $R_{mn}^+$ and $(D_{mn}^+, R_{mn}^+)$.Studia Sci. Math. Hung. 8(1973), 79-89. MR 0339374 |
| Reference:
|
[9] I. Vincze: Einige Zweidimensionale Verteilungs und Grenzverteilungssätze in der Theorie der geordneten Stichproben.Publ. Math. Inst. Hungar. Acad. Sci. 2 (1957), 183 - 209. MR 0105172 |
| . |
