Risk function and optimality of statistical procedures for identification of network structures

P.A. Koldanov

National Research University Higher School of Economics, Nizhny Novgorod, 603025 Russia

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Abstract

Identification of network structures using the finite-size sample has been considered. The concepts of random variables network and network model, which is a complete weighted graph, have been introduced. Two types of network structures have been investigated: network structures with an arbitrary number of elements and network structures with a fixed number of elements of the network model. The problem of identification of network structures has been investigated as a multiple testing problem. The risk function of statistical procedures for identification of network structures can be represented as a linear combination of expected numbers of incorrectly included elements and incorrectly non-included elements. The sufficient conditions of optimality for statistical procedures for network structures identification with an arbitrary number of elements have been given. The concept of statistical uncertainty of statistical procedures for identification of network structures has been introduced.

Keywords: random variables network, network model, network structure, procedure for identification of network structure, additive loss function, risk function, unbiasedness, optimality, statistical uncertainty

Acknowledgments. This work was supported in part by the Laboratory of Algorithms and Technologies for Network Analysis of National Research University Higher School of Economics and by the Russian Foundation for Basic Research (project no. 18-07-00524).

References

1. Jordan M.I. Graphical models.  Stat. Sci., 2004, vol. 19, no. 1, pp. 140–155. doi: 10.1214/088342304000000026.

2. Lauritzen S.L. Graphical Models. Oxford, Oxford Univ. Press, 1996. 298 p.

3. Anderson T.W. An Introduction to Multivariate Statistical Analysis. New York, John Wiley & Sons, 2003. 752 p.

4. Drton M., Perlman M.D. Model selection for Gaussian concentration graph.  Biometrika, 2004, vol. 91, no. 3, pp. 591–602. doi: 10.1093/biomet/91.3.591.

5. Drton M., Perlman M. Multiple testing and error control in Gaussian graphical model selection.  Stat. Sci., 2008, vol. 22, no. 3, pp. 430–449. doi: 10.1214/088342307000000113.

6. Boginski V., Butenko S., Pardalos P.M. On structural properties of the market graph. In:  Innovations in Financial and Economic Networks. Cheltenham, Edward Elgar Publ., 2003, pp. 29–45.

7. Mantegna R.N. Hierarchical structure in financial markets.  Eur. Phys. J. B, 1999, vol. 11, no. 1, pp. 193–197. doi: 10.1007/s100510050929.

8. Koldanov A.P., Koldanov P.A. , Kalyagin V.A., Pardalos P.M. Statistical procedures for the market graph construction.  Comput. Stat. Data Anal., 2013, vol. 68, pp. 17–29. doi: 10.1016/j.csda.2013.06.005.

9. Koldanov P.A. Risk function of statistical procedures for network structures identification.  Vestn. TVGU. Ser. Prikl. Mat., 2017, no. 3, pp. 45–59. doi: 10.26456/vtpmk178. (In Russian)

10. Lehmann E.L. A theory of some multiple decision problems, I.  Ann. Math. Stat., 1957, vol. 28, no. 1, pp. 1–25.

11. Wald A. Statistical Decision Functions. New York, John Wiley & Sons, 1950. 179 p.

12. Koldanov P., Koldanov A., Kalyagin V., Pardalos P.M. Uniformly most powerful unbiased test for conditional independence in Gaussian graphical model.  Stat. Probab. Lett., 2017, vol. 122, pp. 90–95. doi: 10.1016/j.spl.2016.11.003.

13. Kalyagin V.A., Koldanov A.P., Koldanov P.A., Pardalos P.M., Zamaraevand V.A. Measures of uncertainty in market network analysis,  Phys. A, 2014, vol. 413, no. 1, pp. 59–70. doi: 10.1016/j.physa.2014.06.054.

Recieved

October 10, 2017

For citation: Koldanov P.A. Risk function and optimality of statistical procedures for identification of network structures. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 317–326.

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