Geodesic
On calculation of exact distributions of decomposable statistics in the multinomial scheme
Diskretnaya Matematika, Tome 18 (2006) no. 3, pp. 85-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

We present several recursive procedures for calculating exact (non-limit) distributions of decomposable statistics in the multinomial scheme. In order to obtain recurrence relations, we make use of the generating functions of the statistics under consideration. In the general case, the method of calculating reduces to the algorithm suggested by A. Zubkov in 1996 which is presented in the language of Markov chains. 
@article{DM_2006_18_3_a5,
     author = {B. I. Selivanov},
     title = {On calculation of exact distributions of decomposable statistics in the multinomial scheme},
     journal = {Diskretnaya Matematika},
     pages = {85--94},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2006_18_3_a5/}
}
TY  - JOUR
AU  - B. I. Selivanov
TI  - On calculation of exact distributions of decomposable statistics in the multinomial scheme
JO  - Diskretnaya Matematika
PY  - 2006
SP  - 85
EP  - 94
VL  - 18
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2006_18_3_a5/
LA  - ru
ID  - DM_2006_18_3_a5
ER  - 
%0 Journal Article
%A B. I. Selivanov
%T On calculation of exact distributions of decomposable statistics in the multinomial scheme
%J Diskretnaya Matematika
%D 2006
%P 85-94
%V 18
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2006_18_3_a5/
%G ru
%F DM_2006_18_3_a5
B. I. Selivanov. On calculation of exact distributions of decomposable statistics in the multinomial scheme. Diskretnaya Matematika, Tome 18 (2006) no. 3, pp. 85-94. http://geodesic.mathdoc.fr/item/DM_2006_18_3_a5/

[1] Kramer G., Matematicheskie metody statistiki, IL, Moskva, 1948

[2] Van der Varden B. L., Matematicheskaya statistika, IL, Moskva, 1960

[3] Ivchenko G. I., Medvedev Yu. I., Matematicheskaya statistika, Vysshaya shkola, Moskva, 1984

[4] Good I. J., Gover T. N., Mitchell G. J., “Exact distributions for $\chi^2$ and for likelihood-ratio statistic for the equiprobable multinomial distribution”, J. Amer. Statist. Assoc., 65 (1970), 267–283 | DOI

[5] Holzman G. I., Good I. J., “The Poisson and chi-squared approximation as compared with the true upper-tail probabillity of Pearson's $\chi^2$ for equiprobable miltinomials”, J. Statist. Plann. Inference, 13:3 (1986), 283–295 | DOI | MR

[6] Selivanov B. I., “O proizvodyaschei funktsii odnoi statistiki v polinomialnoi skheme so schetnym mnozhestvom iskhodov”, Kombinatornyi analiz, 2 (1972), 11–16 | MR

[7] Zubkov A. M., “Rekurrentnye formuly dlya raspredelenii funktsionalov ot diskretnykh sluchainykh velichin”, Obozrenie prikladnoi i promyshlennoi matematiki, 3:4 (1996), 567–573 | MR | Zbl

[8] Selivanov B. I., “O kombinatornykh funktsiyakh, svyazannykh s ryadom Byurmana–Lagranzha. Sootnosheniya kvaziortogonalnosti”, Teoriya veroyatnostei i ee primeneniya, 10:1 (1998), 127–140 | MR | Zbl

[9] Mikhailov V. G., “Tsentralnaya predelnaya teorema dlya chisla nepolnykh dlinnykh povtorenii”, Teoriya veroyatnostei i ee primeneniya, 20:4 (1975), 880–884 | MR | Zbl

[10] Korolyuk V. S., Borovskikh Yu. V., Teoriya $U$-statistik, Naukova Dumka, Kiev, 1989

[11] Selivanov B. I., “O predelnykh raspredeleniyakh statistiki $\chi^2$”, Teoriya veroyatnostei i ee primeneniya, 29:1 (1984), 132–134 | Zbl

[12] Medvedev Yu. I., “Razdelimye statistiki v polinomialnoi skheme, I, II”, Teoriya veroyatnostei i ee primenenie, 22:1 (1977), 3–17 ; 3, 623–631 | MR | Zbl | MR | Zbl

[13] Medvedev Yu. I., “Nekotorye teoremy ob asimptoticheskom raspredelenii statistiki $\chi^2$”, Doklady AN SSSR, 192:5 (1970), 987–980

[14] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Sluchainye razmescheniya, Nauka, Moskva, 1976

[15] Good I. J., “Saddle-point methods for the multinomial distribution”, Ann. Math. Statist., 28:4 (1957), 861–881 | DOI | MR | Zbl

[16] Selivanov B. I., “O vremeni ozhidaniya v skheme sluchainykh razmeschenii okrashennykh chastits”, Diskretnaya matematika, 7:1 (1995), 134–144 | Zbl