×

Rates of convergence for minimum contrast estimators. (English) Zbl 0805.62037

The paper concerns minimum contrast estimators in a nonparameteric setting for independent observations. The main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters. The cases of optimal and suboptimal rates are considered. Several examples illustrate the proved results.
Reviewer: M.Huškova (Praha)

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62J02 General nonlinear regression
Full Text: DOI

References:

[1] Andersen, N.T., Giné, E., Ossiander, M., Zinn, J.: The central limit theorem and the law of iterated logarithm for empirical processes under local conditions. Probab. Theory Relat. Fields77, 271-305 (1988) · Zbl 0618.60022 · doi:10.1007/BF00334041
[2] Bahadur, R.R.: Examples of inconsistency of maximum likelihood estimates. Sankhyã, Ser. A20, 207-210 (1958) · Zbl 0087.34202
[3] Barlow, R.E., Bartholomew, D.J., Bremner, J.M., Brunk, H.D.: Statistical inference under order restrictions. New York: Wiley 1972 · Zbl 0246.62038
[4] Bass, R.F.: Law of the iterated logarithm for set-indexed partial sum processes with finite variance. Z. Warscheinlichkeitstheor. Verw. Geb.70, 591-608 (1985) · Zbl 0575.60034 · doi:10.1007/BF00531869
[5] Birgé, L.: Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 181-237 (1983) · Zbl 0506.62026 · doi:10.1007/BF00532480
[6] Brigé, L.: Stabilité et instabilité du risque minimax pour des variables indépendantes équidistribuées. Ann. Inst. Henri Poincaré, Sect. B20, 201-223 (1984)
[7] Birgé, L.: On estimating a density using Hellinger distance and some other strange facts. Probab. Theory Relat. Fields71, 271-291 (1986) · Zbl 0561.62029 · doi:10.1007/BF00332312
[8] Birgé, L.: The Grenander estimator: a nonasymptotic approach. Ann. Stat.17, 1532-1549 (1989) · Zbl 0703.62042 · doi:10.1214/aos/1176347380
[9] Birgé, L., Massart, P.: TheL 2 entropy with bracketing. Manuscript (1993)
[10] Birman, M.S., Solomjak, M.Z.: Piecewise-polynomial approximation of functions of the classesW p . Mat. Sb.73, 295-317 (1967) · Zbl 0173.16001 · doi:10.1070/SM1967v002n03ABEH002343
[11] Donoho, D.L.: Statistical estimation and optimal recovery. Technical report. University of California. Berkeley (1989) · Zbl 0805.62014
[12] Dudley, R.M.: A course on empirical processes. In: Ecole d’Eté de Probabilités de Saint-Flour XII-1982. Berlin Heidelberg New York: Springer 1984
[13] Efroimovitch, S.Yu., Pinsker, M.S.: Estimation of square-integrable density on the basis of a sequence of observations. Probl. Inf. Transm.17, 182-196 (1981)
[14] Grenander, U.: On the theory of mortality measurement. II. Skand. Aktuarietidskr.39, 125-153 (1956) · Zbl 0077.33715
[15] Grenander, U.: Abstract inference. New-York: Wiley 1980 · Zbl 0505.62069
[16] Groeneboom, P.: Estimating a monotone density. In: LeCam, L.M., Olshen, R.A. (eds.) Proc. of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. 2, pp. 539-555. Monterey, CA: Wadsworth 1985 · Zbl 1373.62144
[17] Huber, P.J.: The behavior of maximum likelihood estimates under non-standard conditions. In: Proc. 5th Berkeley Symp. Math. Stat. Probab., vol. 1, pp. 221-233. Berkeley, CA: University of California Press 1967 · Zbl 0212.21504
[18] Ibragimov, I.A., Has’minskii, R.Z.: On estimate of the density function. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova98, 61-85 (1980)
[19] Ibragimov, I.A., Has’minskii, R.Z.: Estimation of distribution density. J. Sov. Math.21, 40-57 (1983) · Zbl 0507.62041 · doi:10.1007/BF01091455
[20] Kolmogorov, A.N., Tikhomirov, V.M.: ?-entropy and ?-capacity of sets in function spaces. Transl., II. Ser., Am. Math. Soc.17, 277-364 (1961) · Zbl 0133.06703
[21] Le Cam, L.M.: Convergence of estimates under dimensionality restrictions. Ann. Stat.1, 38-53 (1973) · Zbl 0255.62006 · doi:10.1214/aos/1193342380
[22] Le Cam, L.M.: Asymptotic methods in statistical decision theory. Berlin Heidelberg New-York: Springer 1986 · Zbl 0605.62002
[23] Lorentz, G.G.: Approximation of functions. New-York: Holt, Rinehart, Winston 1966 · Zbl 0153.38901
[24] Massart, P.: The tight constant in the D.K.W. inequality. Ann. Probab.18, 1269-1283 (1990) · Zbl 0713.62021 · doi:10.1214/aop/1176990746
[25] Nemirovskii, A.S., Polyak, B.T., Tsybakov, A.B.: Signal processing by the nonparametric maximum-likelihood method. Probl. Inf. Transm.20, 177-192 (1984) · Zbl 0599.62049
[26] Nemirovskii, A.S., Polyak, B.T., Tsybakov, A.B.: Rate of convergence of nonparametric estimates of maximum-likelihood type. Probl. Inf. Transm.21, 258-272 (1985) · Zbl 0616.62048
[27] Ossiander, M.: A central limit theorem under metric entropy withL 2 bracketing. Ann. Probab.15, 897-919 (1987) · Zbl 0665.60036 · doi:10.1214/aop/1176992072
[28] Pfanzagl, J.: On the measurability and consistency of minimum contrast estimates. Metrika14, 249-272 (1969) · Zbl 0181.45501 · doi:10.1007/BF02613654
[29] Pinsker, M.S.: Optimal filtration of square-integrable signals in gaussian noise. Probl. Inf. Transm.16, 120-133 (1980) · Zbl 0452.94003
[30] Prakasa Rao, B.L.S.: Estimation of a unimodal density. Sankhya, Ser. A31, 23-36 (1983) · Zbl 0181.45901
[31] Reiss, R.-D.: Sharp rates of convergence of maximum likelihood estimators in nonparametric models. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 473-482 (1984) · Zbl 0508.62033 · doi:10.1007/BF00533748
[32] Severini, T.A., Wong, W.H.: Convergence rates of maximum likelihood and related estimates in general parameter spaces. (Preprint 1987)
[33] Shorack, G.R., Wellner, J.A.: Empirical processes with applications to statistics. New-York: Wiley 1986 · Zbl 1170.62365
[34] Stone, C.J.: Optimal rates of convergence for nonparametric regression. Ann. Stat.10, 1040-1053 (1982) · Zbl 0511.62048 · doi:10.1214/aos/1176345969
[35] Van de Geer, S.: Estimating a regression function. Ann. Stat.18, 907-924 (1990a) · Zbl 0709.62040 · doi:10.1214/aos/1176347632
[36] Van de Geer, S.: Hellinger-consistency of certain nonparametric maximum likelihood estimates. no 614, Utrecht University (Preprint 1990b) · Zbl 0779.62033
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.