Birge, Lucien Approximation dans les espaces métriques et théorie de l’estimation. (French) Zbl 0506.62026 Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 181-237 (1983). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 64 Documents MSC: 62G05 Nonparametric estimation 41A45 Approximation by arbitrary linear expressions Keywords:Hellinger distance; minimax risk; speed of estimation; metric dimension; density estimation; spectral density estimation; estimation of Markov transitions × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Assouad, P.: Espaces métriques, plongements, facteurs. Thèse d’Etat, Orsay (1976) · Zbl 0396.46035 [2] Assouad, P., Classes de Vapnik-Červonenkis et vitesse d’estimation (1982), Orsay: Prépublication, Orsay [3] Bahadur, R. R., Examples of inconsistency of maximum likelihood estimates, Sankhya, 20, 207-210 (1958) · Zbl 0087.34202 [4] Beckenbach, E. F.; Bellman, R., Inequalities (1961), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0097.26502 · doi:10.1007/978-3-642-64971-4 [5] Birgé, L., Vitesses optimales de convergence des estimateurs, Astérisque, 68, 171-185 (1979) · Zbl 0436.62030 [6] Birgé, L.: Sur un théorème de minimax et son application aux tests. Probab. Math. Statist. III, 2 [To appear 1983] · Zbl 0571.62036 [7] Birgé, L., Robust testing for independent non-identically distributed variables and Markov chains, Dans: Specifying statistical models, 134-162 (1983), New York-Heidelberg-Berlin: Springer, New York-Heidelberg-Berlin · Zbl 0509.62036 · doi:10.1007/978-1-4612-5503-1_9 [8] Bretagnolle, J.; Huber, C., Estimation des densités: risque minimax, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47, 119-137 (1979) · Zbl 0413.62024 · doi:10.1007/BF00535278 [9] Čencov, N. N., Statistical decision rule and optimal inference (1972), En russe. Moscou: Nauka, En russe. Moscou [10] Dacunha-Castelle, D., Ecole d’Eté de Probabilités de Saint-Flour VII (1977), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York [11] Donsker, M. D.; Varadhan, S. R.S., Asymptotic evaluation of certain Markov process expectations for large time, I, Comm. Pure Appl. Math., 28, 1-47 (1975) · Zbl 0323.60069 · doi:10.1002/cpa.3160280102 [12] Farrell, R. H., On the best obtainable asymptotic rates of convergence in estimation of a density function at a point, Ann. Math. Statist., 43, 170-180 (1972) · Zbl 0238.62049 · doi:10.1214/aoms/1177692711 [13] Farrell, R. H., Asymptotic lower bounds for the risk of estimators of the value of a spectral density function, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 49, 221-234 (1979) · Zbl 0425.62075 · doi:10.1007/BF00534259 [14] Huber, C: Thèse d’Etat: Orsay 1979 [15] Huber, P. J., A robust version of the probability ratio test, Ann. Math. Statist., 36, 1753-1758 (1965) · Zbl 0137.12702 · doi:10.1214/aoms/1177699803 [16] Ibragimov, I. A.; Khas’minskii, R. Z., Statistical estimation, Asymptotic theory (1981), New York-Heidelberg-Berlin: Springer, New York-Heidelberg-Berlin · Zbl 0467.62026 [17] Ibragimov, I. A.; Khas’minskii, R. Z., On estimate of the density function, En russe. Zap. Nauchn. Semin. LOMI, 98, 61-85 (1980) · Zbl 0482.62025 [18] Ibragimov, I. A.; Khas’minskii, R. Z., On the non-parametric density estimates, En russe. Zap. Nauchn. Semin. LOMI, 108, 73-89 (1981) [19] Khas’minskii, R. Z., A lower bound on the risks of non-parametric estimates of densities in the uniform metric, Theory Probability Appl., 23, 794-796 (1978) · Zbl 0449.62032 · doi:10.1137/1123095 [20] Kolmogorov, A. N.; Tihomirov, V. M., ε-entropy and ε-capacity of sets in function spaces, Amer. Math. Soc. Transl., 17, 2, 277-364 (1961) · Zbl 0133.06703 [21] Kraft, C., Some conditions for consistency and uniform consistency of statistical procedures, Univ. of Calif. Publ. in Statistics, 1, 125-142 (1955) · Zbl 0066.12202 [22] Le Cam, L.: Théorie asymptotique de la décision statistique. Presses de l’Université de Montréal: 1968 · Zbl 0203.51601 [23] Le Cam, L., Convergence of estimates under dimensionality restrictions, Ann. Statist., 1, 38-53 (1973) · Zbl 0255.62006 · doi:10.1214/aos/1193342380 [24] Le Cam, L., On local and global properties in the theory of asymptotic normality of experiments, Stochastic processes and related topics 1, 13-54 (1975), New York-San Francisco-London: Academic Press, New York-San Francisco-London · Zbl 0389.62011 [25] Le Cam, L.: An inequality concerning Bayes estimates. Prépublication (1979) [26] Le Cam, L.: Asymptotic methods in statistical decision theory. Livre à paraître · Zbl 0605.62002 [27] Lorentz, G. G., Metric entropy and approximation, Bull. Amer. Math. Society, 72, 903-937 (1966) · Zbl 0158.13603 · doi:10.1090/S0002-9904-1966-11586-0 [28] Lorentz, G. G., Approximation of functions (1966), New York: Holt, Rinehart, Winston, New York · Zbl 0153.38901 [29] Okamoto, M., Some inequalities relating to the partial sum of binomial probabilities, Ann. Inst. Statis. Math., 10, 29-35 (1958) · Zbl 0084.14001 · doi:10.1007/BF02883985 [30] Pinsker, M. S., Optimal filtration of square-integrable signals in gaussian noise, Problems of Inform. Transmission, 16, 120-133 (1980) · Zbl 0452.94003 [31] Strasser, H., Convergence of estimates: parts I and II, J. Multivariate Anal., 11, 127-172 (1981) · Zbl 0487.62032 · doi:10.1016/0047-259X(81)90105-6 [32] Vitushkin, A. G., Theory of transmission and processing of information (1961), New York: Pergamon Press, New York · Zbl 0122.12001 [33] Wahba, G., Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation, Ann. Statist., 3, 15-29 (1975) · Zbl 0305.62021 · doi:10.1214/aos/1176342997 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.