zbMATH — the first resource for mathematics

Probability, minimax approximation, and Nash-equilibrium. Estimating the parameter of a biased coin. (English) Zbl 1415.65024
Summary: This paper deals with the application of approximation theory techniques to study a classical problem in probability: estimating the parameter of a biased coin. For this purpose, a minimax estimation problem is considered, and the characterization of the optimal estimator is shown together with the weak asymptotics of such optimal choices as the number of coin tosses approaches infinity. In addition, a number of numerical examples and graphs are provided. At the same time, the problem is also discussed from the game theory viewpoint, as a non-cooperative, two-player game, and the existence of a Nash-equilibrium is established. The particular case of $$n=2$$ tosses is completely solved.

MSC:
 65C50 Other computational problems in probability (MSC2010) 41A05 Interpolation in approximation theory 41A10 Approximation by polynomials 91A05 2-person games
Full Text:
References:
 [1] D. A. BLACKWELL ANDM. A. GIRSHICK, Theory of Games and Statistical Decisions, Dover, New York, 2012. [2] C.DEBOOR, Polynomial Interpolation, in Proc. of the International Conference of Mathematicians 1978, O. Lehto, ed., Acad. Sci. Fennica, Helsinki, 1980, pp. 917–922. [3] C.DEBOOR ANDA. PINKUS, Proof of the conjectures of Bernstein and Erd˝os concerning the optimal nodes for polynomial interpolation, J. Approx. Theory, 24 (1978), pp. 289–303. [4] L. BRUTMAN, On the Lebesgue function for polynomial interpolation, SIAM J. Numer. Anal., 15 (1978), pp. 699–704. · Zbl 0391.41002 [5] O. CANDOGAN, A. OZDAGLAR,ANDP. A. PARRILLO, Dynamics in near-potential games, Games Econom. Behav., 82 (2013), pp. 66–90. ETNA Kent State University and Johann Radon Institute (RICAM) 128D. BENKO, D. COROIAN, P. DRAGNEV, AND R. ORIVE · Zbl 1282.91048 [6] C. DASKALAKIS, P. W. GOLDBERG,ANDC. H. PAPADIMITRIOU, The complexity of computing a Nash equilibrium, SIAM J. Comput., 39 (2009), pp. 195–259. · Zbl 1185.91019 [7] D. FUDENBERG ANDJ. TIROLE, Game Theory, MIT Press, Cambridge, 1991. [8] I. L. GLICKSBERG, A further generalization of the Kakutani fixed point theorem with application to Nashequilibrium, Proc. Amer. Math. Soc., 3 (1952), pp. 170–174. · Zbl 0046.12103 [9] O. HOLTZ, F. NAZAROV,ANDY. PERES, New coins from old, smoothly, Constr. Approx., 33 (2011), pp. 331– 363. · Zbl 1238.41007 [10] R. KAAS ANDJ. M. BUHRMAN, Mean, median and mode in binomial distributions, Statist. Neerlandica, 34 (1980), pp. 13–18. · Zbl 0444.62021 [11] S. KAKUTANI, A generalization of Brouwer’s fixed point theorem, Duke Math. J., 8 (1941), pp. 457–159. · Zbl 0061.40304 [12] T. A. KILGORE, A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm, J. Approx. Theory, 24, (1978), pp. 273–288. · Zbl 0428.41023 [13] E. L. LEHMANN ANDG. CASELLA, Theory of Point Estimation, 2nd ed., Springer, New York, 1998. [14] G. MASTROIANNI ANDG. MILOVANOVI ´C, Interpolation Processes. Basic Theory and Applications, Springer, Berlin, 2008. [15] R. D. MCKELVEY ANDT. R. PALFREY, A statitistical theory of equilibrium games, Japan Econ. Rev., 47 (1996), pp. 186–209. [16] S. NACU, Y. PERES, Fast simulation of new coins from old, Ann. Appl. Probab., 15 (2005), pp. 93–115. · Zbl 1072.65007 [17] J. F. NASH, Equilibrium points in N -person games, Proc. Nat. Acad. Sci. U. S. A., 36 (1950), pp. 48–49. · Zbl 0036.01104 [18] , Non-cooperative games, Ann. of Math. (2), 54, (1951), pp. 286–295. · Zbl 0045.08202 [19] J.VONNEUMANN, Various techniques used in connection with random digits, J. Res. Nat. Bur. Stand. Appl. Math. Series., 3 (1951), pp. 36–38. [20] M. J. OSBORNE ANDA. RUBINSTEIN, A Course in Game Theory, MIT Press, Cambridge, 1994. [21] J. PFANZAGL, Parametric Statistical Theory, de Gruyter, Berlin, 1994. [22] S. J. SMITH, Lebesgue constants in polynomial interpolation, Ann. Math. Inform., 33 (2006), pp. 109–123. · Zbl 1135.41300
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.