Geodesic
Bias of the nonpametric entropy estimator for Markov measures
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 16, Tome 384 (2010), pp. 267-290 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider the problem of the nonparametric entropy (entropy rate) estimation. We investigate the technique of the nonparametric entropy estimation based on so-called “nearest neighbor distances” and obtain a closed-form expression of the bias for Markov measures. This bias is a discontinuous function of transition probabilities. Bibl. 20 titles. 
@article{ZNSL_2010_384_a13,
     author = {E. A. Timofeev},
     title = {Bias of the nonpametric entropy estimator for {Markov} measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {267--290},
     year = {2010},
     volume = {384},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_384_a13/}
}
TY  - JOUR
AU  - E. A. Timofeev
TI  - Bias of the nonpametric entropy estimator for Markov measures
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2010
SP  - 267
EP  - 290
VL  - 384
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2010_384_a13/
LA  - ru
ID  - ZNSL_2010_384_a13
ER  - 
%0 Journal Article
%A E. A. Timofeev
%T Bias of the nonpametric entropy estimator for Markov measures
%J Zapiski Nauchnykh Seminarov POMI
%D 2010
%P 267-290
%V 384
%U http://geodesic.mathdoc.fr/item/ZNSL_2010_384_a13/
%G ru
%F ZNSL_2010_384_a13
E. A. Timofeev. Bias of the nonpametric entropy estimator for Markov measures. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 16, Tome 384 (2010), pp. 267-290. http://geodesic.mathdoc.fr/item/ZNSL_2010_384_a13/

[1] N. K. Bari, Trigonometricheskie ryady, GIFML, M., 1961 | MR

[2] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[3] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Nauka, M., 1971

[4] R. L. Dobrushin, “Uproschennyi metod eksperimentalnoi otsenki entropii statsionarnoi posledovatelnosti”, Teoriya veroyatn. i ee primen., 3:4 (1958), 462–464 | MR | Zbl

[5] A. Zigmund, Trigonometricheskie ryady, Mir, M., 1965 | MR

[6] D. Knut, Iskusstvo programmirovaniya dlya EVM, v. 3, Mir, M., 1978 | MR | Zbl

[7] N. Martin, Dzh. Inglend, Matematicheskaya teoriya entropii, Mir, M., 1988 | MR | Zbl

[8] E. A. Timofeev, “Statisticheski otsenivaemye invarianty mer”, Algebra i analiz, 17:3 (2005), 204–236 | MR | Zbl

[9] P. Erdős, “On the smoothness properties of a family of Bernoulli convolutions”, Amer. J. Math., 62:1 (1940), 180–186 | DOI | MR | Zbl

[10] P. Erdős, “On a family of symmetric Bernoulli convolutions”, Amer. J. Math., 61:4 (1939), 974–976 | DOI | MR | Zbl

[11] P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2008 | MR | Zbl

[12] A. M. Garsia, “Arithmetic properties of Bernoulli convolutions”, Trans. Amer. Math. Soc., 102:3 (1962), 409–432 | DOI | MR | Zbl

[13] A. Kaltchenko, N. Timofeeva, “Entropy estimators with almost sure convergence and an $O(n^{-1})$ variance”, Adv. Math. Comm., 2:1 (2008), 1–13 | DOI | MR | Zbl

[14] A. Kaltchenko, N. Timofeeva, “Rate of convergence of the nearest neighbor entropy estimator”, AEU – Intern. J. Electr. Comm., 64:1 (2010), 75–79 | DOI

[15] I. Kontoyiannis, Yu. M. Suhov, “Prefixes and the entropy rate for long-range sources”, Probability Statistics and Optimization, ed. F. P. Kelly, Wiley, New York, 1994 | MR | Zbl

[16] Y. Peres, W. Schlag, B. Solomyak, “Sixty years of Bernoulli convolutions”, Fractals and Stochastics, v. II, Progress in Probability, 46, eds. C. Bandt, S. Graf, M. Zähle, Birkhauser, 2000, 39–65 | MR | Zbl

[17] R. Salem, “Sets of uniqueness and sets of multiplicity”, Trans. Amer. Math. Soc., 54:2 (1943), 218–228 | DOI | MR

[18] B. Solomyak, “On the random series $\sum\pm\lambda^n$ (an Erdős problem)”, Ann. Math. 2nd Ser., 142:3 (1995), 611–625 | DOI | MR | Zbl

[19] W. Szpankowski, Average Case Analysis of Algorithms on Sequences, John Wiley Sons, New York, 2001 | MR

[20] A. Wintner, “On convergent Poisson convolutions”, Amer. J. Math., 57:4 (1935), 827–838 | DOI | MR | Zbl