Conformal measures and density estimation

• Autorzy: Denker M. , Facca T.
• Języki publikacji: EN

Abstrakty

EN

The notion of conformal measures or densities goes back to Patterson’s work [8]. By a theorem of Milnor and Thurston [7] a piecewise monotone and continuous map of the interval is semiconjugate to one with constant slopes. The semiconjugacies can be defined by distribution functions of conformal measures as shown in [2]. In this note we show that for some transformations the conjugacies are estimable functions and can be used to improve estimation procedures, in particular density estimations.

Słowa kluczowe

EN

resampling; interval map; conjugacy; density estimation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 1
• Strony: 161--174
• Opis fizyczny: Bibliogr. 12 poz., tab., wykr.

Twórcy

autor

Denker M.

• Mathematics Department, Pennsylvania State University, McAllister Building, State College, PA 16802, USA

autor

Facca T.

• Management, Marketing and Logistics, Boler School of Business, #209, John Carroll University, University Heights, OH 44118, USA

Bibliografia

• [1] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. Vol. 470, Springer, 1975.
• [2] M. Denker and M. Stadlbauer, Semiconjugacies for skew products of interval maps, in: Dynamics of Complex Systems, RIMS Kokyuroku 1404 (2004), pp. 12-20.
• [3] M. Denker and M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), pp. 563-587.
• [4] S. Efromovich, Nonparametric Curve Estimation, Springer, New York 1999.
• [5] J. J. Faraway, Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models, Chapman & Hall, Boca Raton-London-New York 2006.
• [6] S. Galatolo, M. Hoyrup and C. Rojas, A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties, Theoret. Comput. Sci. 410 (2009), pp. 2207-2222.
• [7] J. Milnor and W. Thurston, On iterated maps of the interval, in: Dynamical Systems (Maryland 1986-1987), Lecture Notes in Math. Vol. 1342, Springer, 1988, pp. 465-563.
• [8] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), pp. 241-273.
• [9] V. A. Rokhlin, On the fundamental ideas of measure theory, Mat. Sb. 25 (1949), pp. 107-150. Translation: Amer. Math. Soc. Transl. 71 (1952), pp. 1-54.
• [10] M. Stadlbauer, On random topological Markov chains with big images and preimages, Stochastics & Dynamics 10 (2010), pp. 77-95.
• [11] D. Sullivan, Conformal dynamical systems, in: Geometric Dynamics, Lecture Notes in Math. Vol. 1007, Springer, 1983, pp. 725-752.
• [12] L. Waterman, All of Nonparametric Statistics, Springer, New York 2006.