On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$

Jagdish Saran; Kanwar Sen

On the distributions of $R_{m n}^{+} (j)$ and $(D_{m n}^{+}, R_{m n}^{+} (j))$

Jagdish Saran; Kanwar Sen

Aplikace matematiky (1982)

Volume: 27, Issue: 6, page 417-425
ISSN: 0862-7940

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Abstract

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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, ..., m + n

, corresponding to one of the observations. Let

R_{m n}^{+} (j)

denote the subscript

i

for which

F_{m} (x_{i}) - G_{n} (x_{i})

achieves its maximum value

D_{m n}^{+}

for the

j

th time

(j = 1, 2, ...)

. The paper deals with the probabilities for

R_{m n}^{+} (j)

and for the vector

(D_{m n}^{+}, R_{m n}^{+} (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.

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Saran, Jagdish, and Sen, Kanwar. "On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$." Aplikace matematiky 27.6 (1982): 417-425. <http://eudml.org/doc/15262>.

@article{Saran1982,
abstract = {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.},
author = {Saran, Jagdish, Sen, Kanwar},
journal = {Aplikace matematiky},
keywords = {points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model; points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model},
language = {eng},
number = {6},
pages = {417-425},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the distributions of $R^+_\{mn\}(j)$ and $(D^+_\{mn\}, R^+_\{mn\}(j))$},
url = {http://eudml.org/doc/15262},
volume = {27},
year = {1982},
}

TY - JOUR
AU - Saran, Jagdish
AU - Sen, Kanwar
TI - On the distributions of $R^+_{mn}(j)$ and $(D^+_{mn}, R^+_{mn}(j))$
JO - Aplikace matematiky
PY - 1982
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 27
IS - 6
SP - 417
EP - 425
AB - 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.
LA - eng
KW - points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model; points of maximal deviation; two-sample Smirnov statistic; empirical distribution functions; joint distribution; random walk model
UR - http://eudml.org/doc/15262
ER -

References

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G. P. Steck G. J. Simmons, On the distributions of $R_{m n}^{+}$ and $(D_{m n}^{+}, R_{m n}^{+})$ , Studia Sci. Math. Hung. 8(1973), 79-89. (1973) MR0339374
I. Vincze, Einige Zweidimensionale Verteilungs und Grenzverteilungssätze in der Theorie der geordneten Stichproben, Publ. Math. Inst. Hungar. Acad. Sci. 2 (1957), 183 - 209. (1957) MR0105172

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