Abstract
Computationally efficient algorithm realizing exact computation of decomposable statistics distributions for multinomial scheme is described. The algorithm is based on the embedding the scheme into nonhomogeneous Markov chain. It was used to compute the Pearson statistics distribution and distributions of some statistics for the random allocation of particles into cells. Comparisons of exact numerical values of distribution functions of statistics with usually used approximations from corresponding limit theorems show that exact tail probabilities may be considerably larger than that of approximating distributions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cramér, H.: Mathematical Methods of Statistics. Princeton Univercity Press, Princeton (1945)
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. Stat. Plann. Infer. 13(3), 283–295 (1986)
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. Am. Stat. Assoc. 65, 267–283 (1970)
Read, T.R.C., Cressie, N.A.C.: Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, New York (1988)
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), 569–597 (2012)
Puccetti, G., Rüschendorf, L.: Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236(7), 1833–1840 (2012)
Marhuenda, M.A., Marhuenda, Y., Morales, D.: On the computation of the exact distribution of power divergence test statistics. Kybernetika 39(1), 55–74 (2003)
Selivanov, B.I.: On calculation of exact distributions of decomposable statistics in the multinomial scheme. Discrete Math. and Appl. 16(4), 359–369 (2006)
Kolchin, V.F.: Random Mappings. Optimization Software Inc, NY (1986)
Kolchin, V.F., Sevast’yanov, B.A., Chistyakov, V.P.: Random Allocations, vol. H. Winston & Sons, Washington DC (1978)
Zubkov, A.M.: Recurrent formulas for distributions of functionals of discrete random variables. Obozr. prikl. promyshl. matem. 3(4), 567–573 (1996). (In Russian)
Zubkov, A.M.: Methods to compute the distributions of sums of random variables. Trudy po diskretnoi matematike 5, 51–60 (2002). (In Russian)
Filina, M.V., Zubkov, A.M.: Exact computation of Pearson statistics distribution and some experimental results. Austrian J. Stat. 37(1), 129–135 (2008)
Filina, M.V., Zubkov, A.M.: Tail properties of Pearson statistics distributions. Austrian J. Stat. 40(1 & 2), 47–54 (2011)
Filina M.V., Zubkov A.M.: Some remarks on the noncentral Pearson statistics distributions. In: Proceeding of XI International Conference on Computer Data Analysis and Modeling, pp. 155–158. Publishing center of BSU, Minsk (2016)
Acknowledgments
This work is supported by the Russian Science Foundation under grant â„–Â 14-50-00005.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Filina, M., Zubkov, A. (2017). Algorithm of Exact Computation of Decomposable Statistics Distributions and Its Applications. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_39
Download citation
DOI: https://doi.org/10.1007/978-3-319-71504-9_39
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71503-2
Online ISBN: 978-3-319-71504-9
eBook Packages: Computer ScienceComputer Science (R0)