Professor Ryszard Zieliński's contribution to Monte Carlo methods and random number generators. Uniform asymptotics in statistics

• Autorzy: Niemiro W.

Identyfikatory

DOI

10.14708/ma.v40i2.369

Warianty tytułu

EN

Professor Ryszard Zieliński's contribution to Monte Carlo methods and random number generators. Uniform asymptotics in statistics

• Języki publikacji: PL

Abstrakty

PL

The aim of the paper is to summarize contributions of Ryszard Zieliński to two important areas of research. First, we discuss his work related to Monte Carlo methods. Ryszard Zieliński was particularly interested in Monte Carlo optimization. About 10 of his papers concerned stochastic algorithms for seeking extrema. He examined methods related to stochastic approximation, random search and global optimization. We stress that Zielinski often considered computational problems from a statistical perspective. In several articles he explicitly indicated that optimization can be reformulated as a statistical estimation problem. We also discuss relation between the family of Simulated Annealing algorithms on the one hand and some procedures examined earlier by Ryszard Zieliński on the other. Another topic belonging to Monte Carlo methods, in which Ryszard Zieliński has achieved interesting results, is construction of random number generators and examination of their statistical properties. Zieliński proposed an aperiodic generator based on Weil sequences and showed how it can be efficiently implemented. Later he constructed an algorithm which uses several such generators and produces pseudo-random sequences with better statistical properties. The second area of Zieliński’s work discussed here is related to uniform limit theorems of mathematical statistics. We stress the methodological motivation behind the research in this direction. In Zieliński’s view, asymptotic results should hold uniformly with respect to the family of probability distributions under consideration. In his opinion, this requirement comes from the very nature of statistical models and the needs of practical applications. Zieliński examined uniform versions the Weak Law of Large Numbers, Strong Law of Large Numbers and Central Limit Theorem in several statistical models. Some results were rather unexpected. He also gave a necessary and sufficient condition for uniform consistency of sample quantiles. Two papers of Ryszard Zieliński were devoted to uniform consistency of smoothed versions of empirical cumulative distribution function. In one of them he proved a version of Dvoretzky-Kiefer-Wolfowitz inequality. The aim of the paper is to summarize contributions of Ryszard Zieliński to two important areas of research. First, we discuss his work related to Monte Carlo methods. Ryszard Zieliński was particularly interested in Monte Carlo optimization. About 10 of his papers concerned stochastic algorithms for seeking extrema. He examined methods related to stochastic approximation, random search and global optimization. We stress that Zielinski often considered computational problems from a statistical perspective. In several articles he explicitly indicated that optimization can be reformulated as a statistical estimation problem. We also discuss relation between the family of Simulated Annealing algorithms on the one hand and some procedures examined earlier by Ryszard Zieliński on the other. Another topic belonging to Monte Carlo methods, in which Ryszard Zieliński has achieved interesting results, is construction of random number generators and examination of their statistical properties. Zieliński proposed an aperiodic generator based on Weil sequences and showed how it can be efficiently implemented. Later he constructed an algorithm which uses several such generators and produces pseudo-random sequences with better statistical properties. The second area of Zieliński’s work discussed here is related to uniform limit theorems of mathematical statistics. We stress the methodological motivation behind the research in this direction. In Zieliński’s view, asymptotic results should hold uniformly with respect to the family of probability distributions under consideration. In his opinion, this requirement comes from the very nature of statistical models and the needs of practical applications. Zieliński examined uniform versions the Weak Law of Large Numbers, Strong Law of Large Numbers and Central Limit Theorem in several statistical models. Some results were rather unexpected. He also gave a necessary and sufficient condition for uniform consistency of sample quantiles. Two papers of Ryszard Zieliński were devoted to uniform consistency of smoothed versions of empirical cumulative distribution function. In one of them he proved a version of Dvoretzky-Kiefer-Wolfowitz inequality.

Słowa kluczowe

PL

stochastyczna optymalizacja; wyżarzanie symulowane; generator liczb pseudolosowych; jednostajne twierdzenia graniczne; jednostajna zgodność

EN

stochastic optimization; simulated annealing; random number generator; uniform limit theorems; uniform consistency

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2012
• Tom: Vol. 40, No. 2
• Strony: 92--106
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Niemiro W.

• Wydział Matematyki i Informatyki, Uniwersytet Mikołaja Kopernika i Wydział Matematyki, Informatyki i Mechaniki Uniwersytetu Warszawskiego ul. Chopina 12/18, 87-100 Torun i ul. Banacha 2, 02-097 Warszawa

Bibliografia

• [1] Borovkov. A.A. (1998). Mathematical statistics. Gordon and Breach.
• [2] Černý, V. (1985). Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications 45: 41–51. doi:10.1007/BF00940812.
• [3] Feller, W. (1978) Wstęp do rachunku prawdopodobieństwa. Tom II, Wydanie drugie, PWN Warszawa.
• [4] Franklin, J.N. (1963). Deterministic Simulation of Random Processes. Mathematics of Computation, Vol. 17, No. 81, 28-59
• [5] Ibragimov, I.A.; Haśminskii, R.Z. (1981) Statistical Estimation. Asymptotic Theory. Springer.
• [6] Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. (1983). Optimization by Simulated Annealing. Science 220 (4598): 671–680. doi:10.1126/science.220.4598.671. JSTOR 1690046. PMID 17813860.
• [7] Massart,P. (1990). The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality, Ann. Probab. 18, 1269–1283.
• [8] Matsumoto, M.; Nishimura, T. (1998). Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation 8 (1):3–30. doi:10.1145/272991.272995.
• [9] Niederreiter, H.G. (1992). Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics,1992. ISBN 0-89871-295-5
• [10] Niederreiter, H.G. (1978). Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84, no. 6, 957–1041
• [11] Wieczorkowski, R. (1995). Algorytmy stochastyczne w optymalizacji dyskretnej przy zaburzonych wartościach funkcji. Praca doktorska. Politechnika Warszawska; Wydział Matematyki i Nauk Informacyjnych.