Geodesic
Estimating a monotone function, being observe in the white noise
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 124-133 
K. L. Zilberburg. Estimating a monotone function, being observe in the white noise. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 124-133. http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a6/
@article{ZNSL_2007_341_a6,
     author = {K. L. Zilberburg},
     title = {Estimating a~monotone function, being observe in the white noise},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {124--133},
     year = {2007},
     volume = {341},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a6/}
}
TY  - JOUR
AU  - K. L. Zilberburg
TI  - Estimating a monotone function, being observe in the white noise
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2007
SP  - 124
EP  - 133
VL  - 341
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a6/
LA  - ru
ID  - ZNSL_2007_341_a6
ER  - 
%0 Journal Article
%A K. L. Zilberburg
%T Estimating a monotone function, being observe in the white noise
%J Zapiski Nauchnykh Seminarov POMI
%D 2007
%P 124-133
%V 341
%U http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a6/
%G ru
%F ZNSL_2007_341_a6

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Monotone functions on the segment is being observed in the Gaussian white noise. The Maximum Likelihood Estimator of such a function is of a piecewise continious sort. A limit theorem on the convergence rate of such an estimator is proved.

[1] I. A. Ibragimov, R. Z. Khasminskii, Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979 | MR

[2] J. Kim, D. Pollard, “Cube root asymptotics”, Ann. Statist., 18:1 (1990), 191–219 | DOI | MR | Zbl

[3] U. Grenander, “On the theory of mortality measurement, II”, Skand. Akt., 39 (1956), 125–153 | MR

[4] B. L. S. Prakasa Rao, “Estimation of a unimodal density”, Sankhaya, Ser. A, 31 (1969), 23–36 | MR | Zbl

[5] P. Groeneboom, “Estimating a monotone density”, Proc. Berkley Conf. in Honor of Jerzy Neyman and Jack Kiefer, Jack Kiefer, Vol. II (Berkeley, Calif., 1983), eds. L. Le Cam and R. A. Olshen, Wadsword, Belmont, Calif., 1985, 539–555 | MR

[6] B. S. Tsirelson, “Geometricheskii podkhod k otsenke maksimalnogo pravdopodobiya dlya beskonechnomernogo gaussovskogo sdviga, I”, Teoriya veroyatnosti i eë primenenie, 27:2 (1982), 388–395 | MR

[7] B. S. Tsirelson, “Geometricheskii podkhod k otsenke maksimalnogo pravdopodobiya dlya beskonechnomernogo gaussovskogo sdviga, II”, Teoriya veroyatnosti i eë primenenie, 30:4 (1985), 772–779 | MR

[8] B. S. Tsirelson, “Geometricheskii podkhod k otsenke maksimalnogo pravdopodobiya dlya beskonechnomernogo gaussovskogo sdviga, III”, Teoriya veroyatnosti i eë primenenie, 31:3, 537–549 | MR

[9] D. Pollard, Empirical processes: theory and applications, Hayward, CA, 1990 | MR

[10] V. N. Sudakov, Geometricheskie problemy teorii beskonechnomernykh veroyatnostnykh raspredelenii, Trudy MIAN, 141, 1976 | MR | Zbl

[11] P. Billingsli, Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR