Geodesic
Computation of distributions of statistics by means of Markov chains
Diskretnaya Matematika, Tome 32 (2020) no. 4, pp. 38-51.

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An approach to the construction of efficient algorithms for the exact computation of distributions of statistics by means of the Markov chains is described. The Pearson statistic, the number of empty cells for random allocations of particles, and the Kolmogorov – Smirnov statistic are considered as examples. Possibilities of extending the approach are discussed, in particular to the computation of the joint distributions of statistics.
Keywords: prelimit distributions of statistics, exact computation of distributions, Markov chains, Pearson statistics, number of empty cells, Kolmogorov – Smirnov statistics. 
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A. M. Zubkov; M. V. Filina. Computation of distributions of statistics by means of Markov chains. Diskretnaya Matematika, Tome 32 (2020) no. 4, pp. 38-51. http://geodesic.mathdoc.fr/item/DM_2020_32_4_a2/

[1] Dyakonova E.E., Mikhailov V.G., “O dostatochnykh usloviyakh asimptoticheskoi normalnosti razdelimykh statistik v neodnorodnoi skheme razmescheniya”, Diskretnaya matematika, 3:1 (1991), 145–154

[2] Zubkov A. M., “Rekurrentnye formuly dlya raspredelenii funktsionalov ot diskretnykh sluchainykh velichin”, Obozr. prikl. promyshl. matem., 3:4 (1996), 567–573 | Zbl

[3] Zubkov A. M., “Metody rascheta raspredelenii summ sluchainykh velichin”, Trudy po diskretnoi matematike, 5, Fizmatlit, M., 2002, 51–60

[4] Ivchenko G.I., Medvedev Yu.I., Matematicheskaya statistika, Vysshaya shkola, M., 1992, 304 pp. | MR

[5] Ivanov V.A., Ivchenko G.I., Medvedev Yu.I., “Diskretnye zadachi teorii veroyatnostei”, Itogi nauki i tekhniki. Ser. Teoriya veroyatnostei. Matematicheskaya statistika. Teoreticheskaya kibernetika, 22, VINITI, M., 1984, 3–60

[6] Ivchenko G.I., Medvedev Yu.I., “Razdelimye statistiki i proverka gipotez. Sluchai malykh vyborok”, Teoriya veroyatn. i ee primen., XXIII:4 (1978), 796–806 | Zbl

[7] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Sluchainye razmescheniya, Nauka, M., 1976, 224 pp.

[8] Kramer G., Matematicheskie metody statistiki, Mir, M., 1975, 648 pp.

[9] Medvedev Yu.I., “Razdelimye statistiki v polinomialnoi skheme. I”, Teoriya veroyatn. i ee primen., XXII:1 (1977), 3–17 | Zbl

[10] Medvedev Yu.I., “Razdelimye statistiki v polinomialnoi skheme. II”, Teoriya veroyatn. i ee primen., 22:2 (1977), 623–631 | Zbl

[11] Mikhailov V.G., “Ob asimptoticheskoi normalnosti simmetricheskikh razdelimykh statistik v neodnorodnoi skheme”, Diskretnaya matematika, 1:4 (1989), 92–103

[12] Selivanov B. I., “O vychislenii dopredelnykh raspredelenii razdelimykh statistik polinomialnoi skhemy”, Diskretnaya matematika, 18:3 (2006), 85–94 | Zbl

[13] Arbenz P., Embrechts P., Puccetti G., “The GAEP algorithm for the fast computation of the distribution of a function of dependent random variables”, Stochastics, 84:5-6 (2012), 569-597 | DOI | MR | Zbl

[14] Brown J.R., Harvey M.E., “Rational arithmetic Mathematica functions to evaluate the two-sided one sample K-S cumulative sampling distribution”, J. Statist. Software, 26:2 (2008), 1–40 | DOI

[15] Facchinetti S., “A procedure to find exact critical values of Kolmogorov-Smirnov test”, Statistica Applicata — Italian J. Appl. Statist., 21:3-4 (2009), 337–359

[16] Filina M. V., Zubkov A. M., “Exact computation of Pearson statistics distribution and some experimental results”, Austrian J. Statist., 37:1 (2008), 129–135

[17] Filina M. V., Zubkov A. M., “Tail properties of Pearson statistics distributions”, Austrian J. Statist., 40:1 2 (2011), 47–54 | MR

[18] Filina M., Zubkov A., “Algorithm of exact computation of decomposable statistics distributions and its applications”, Analytical and Computational Methods in Probability Theory. ACMPT 2017, Lect. Notes Comput. Sci., 10684, 2017, 476–484 | DOI | Zbl

[19] 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

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

[21] Marhuenda M. A., Marhuenda Y., Morales D., “On the computation of the exact distribution of power divergence test statistics”, Kybernetika, 39:1 (2003), 55–74 | MR | Zbl

[22] Marsaglia G., Tsang W.W., Wang J., “Evaluating Kolmogorov's distribution”, J. Statist. Software, 8:18 (2003), 1–4 | DOI

[23] Puccetti G., Rüschendorf L., “Computation of sharp bounds on the distribution of a function of dependent risks”, J. Comput. Appl. Math., 236:7 (2012), 1833–1840 | DOI | MR | Zbl

[24] Read T.R.C., Cressie N.A.C., Goodness-of-Fit Statistics for Discrete Multivariate Data, Springer-Verlag, N.-Y., 1988, viii+211 pp. | MR | Zbl

[25] Simard R., L'Ecuyer P., “Computing the two-sided Kolmogorov-Smirnov distribution”, J. Statist. Software, 39:11 (2011), 1–18 | DOI