Abstract
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.
Note
Originally published in Diskretnaya Matematika (2020) 32,№4, 38–51 (in Russian).
References
[1] Dyakonova E. E., Mikhailov V. G., “Sufficient conditions for asymptotic normality of decomposable statistics in an inhomogeneous allocation scheme”, Discrete Math. Appl., 2:3 (1992), 325–335.10.1515/dma.1992.2.3.325Suche in Google Scholar
[2] Zubkov A. M., “Recurrent formulae for distributions of functions of discrete random variables”, Obozr. Prikl. Prom. Mat., 3:4 (1996), 567–573 (In Russian).Suche in Google Scholar
[3] Zubkov A. M., “Computational algorithms for distributions of sums of random variables”, Tr. Diskr. Mat., 5, Fizmatlit, Moscow, 2002, 51–60 (In Russian).Suche in Google Scholar
[4] Ivchenko G., Medvedev Yu., Mathematical Statistics, URSS, 1994 (In Russian), 304 pp.Suche in Google Scholar
[5] Ivanov V. A., Ivchenko G. I.,Medvedev Yu. I., “Discrete problems in probability theory”, J.Math. Sci. (N. Y.), 31:2 (1985), 2759–2795.10.1007/BF02116601Suche in Google Scholar
[6] Ivchenko G. I.,Medvedev Yu. I., “Decomposable statistics and hypothesis testing. The case of small samples”, Theory Probab. Appl., 23:4 (1979), 764–775.10.1137/1123092Suche in Google Scholar
[7] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Random Allocations, Winston & Sons, Washington, DC, 1978, xi+262 pp.Suche in Google Scholar
[8] Cramér, H., Mathematical Methods of Statistics, Princeton Univ. Press, Princeton, 1946, 575 pp.10.1515/9781400883868Suche in Google Scholar
[9] Medvedev Yu. I., “Decomposable statistics in a polynomial scheme. I”, Theory Probab. Appl., 22:1 (1977), 1–15.10.1137/1122001Suche in Google Scholar
[10] Medvedev Yu. I., “Decomposable statistics in a polynomial scheme. II”, Theory Probab. Appl., 22:3 (1978), 607–615.10.1137/1122075Suche in Google Scholar
[11] Mikhailov V. G., “Asymptotic normality of decomposable statistics of the frequencies of m-chains”, Discrete Math. Appl., 1:3 (1991), 335–347.10.1515/dma.1991.1.3.335Suche in Google Scholar
[12] Selivanov B. I., “On calculation of exact distributions of decomposable statistics in the multinomial scheme”, Discrete Math. Appl., 16:4 (2006), 359–369.10.1515/156939206778609714Suche in Google Scholar
[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), 569-597.10.1080/17442508.2011.566337Suche in Google Scholar
[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), 1–40.10.18637/jss.v026.i02Suche in Google Scholar
[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.Suche in Google Scholar
[16] Filina M. V., Zubkov A. M., “Exact computation of Pearson statistics distribution and some experimental results”, Austrian J. Statist., 37:1 (2008), 129–135.10.17713/ajs.v37i1.294Suche in Google Scholar
[17] Filina M. V., Zubkov A. M., “Tail properties of Pearson statistics distributions”, Austrian J. Statist., 40:1 & 2 (2011), 47–54.Suche in Google Scholar
[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, 476–484.10.1007/978-3-319-71504-9_39Suche in Google Scholar
[19] Good I. J., Gover T.N., Mitchell G. J., “Exact distributions for χ2 and for likelihood-ratio statistic for the equiprobable multinomial distribution”, J. Amer. Statist. Assoc., 65 (1970), 267–283.10.1080/01621459.1970.10481078Suche in Google Scholar
[20] Holzman G. I., Good I. J., “The Poisson and chi-squared approximation as compared with the true upper-tail probability of Pearson’s χ2 for equiprobable multinomials”, J. Statist. Plann. Inference, 13:3 (1986), 283–295.10.1016/0378-3758(86)90140-0Suche in Google Scholar
[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.Suche in Google Scholar
[22] Marsaglia G., Tsang W.W., Wang J., “Evaluating Kolmogorov’s distribution”, J. Statist. Software, 8:18 (2003), 1–4.10.18637/jss.v008.i18Suche in Google Scholar
[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), 1833–1840.10.1016/j.cam.2011.10.015Suche in Google Scholar
[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.10.1007/978-1-4612-4578-0Suche in Google Scholar
[25] Simard R., L’Ecuyer P., “Computing the two-sided Kolmogorov-Smirnov distribution”, J. Statist. Software, 39:11 (2011), 1–18.10.18637/jss.v039.i11Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Contents
- Group service system with three queues and load balancing
- Formulas for the numbers of sequences containing a given pattern given number of times
- On a generalization of class of negative binomial distributions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- On synthesis of reversible circuits consisting of NOT, CNOT, 2-CNOT gates with small number of additional inputs
- Computation of distributions of statistics by means of Markov chains
Artikel in diesem Heft
- Contents
- Group service system with three queues and load balancing
- Formulas for the numbers of sequences containing a given pattern given number of times
- On a generalization of class of negative binomial distributions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- On synthesis of reversible circuits consisting of NOT, CNOT, 2-CNOT gates with small number of additional inputs
- Computation of distributions of statistics by means of Markov chains