Minimax and minimax adaptive estimation in multiplicative regression: Locally bayesian approach

Probability Theory and Related Fields

Volumn 153, Issue 3-4, 2012, Pages 543-586

  Chichignoud, M ^a  

^a AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Adaptive bandwidth selector; Lepski's method; Local bayesian fitting; Multiplicative regression; Optimality criterion

Indexed keywords

EID: 84864385532     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-011-0354-7     Document Type: Article

References (21)

1. Allon G., Beenstock M., Hackman S., Passy U., Shapiro A.: Nonparametric estimation of concave production technologies by entropic methods. J. Appl. Econom. 22(4), 795-816 (2007).
2. Barron A., Birgé L., Massart P.: Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113(3), 301-413 (1999).
3. Goldenshluger A., Lepski O.: Universal pointwise selection rule in multivariate function estimation. Bernoulli 14(3), 1150-1190 (2008).
4. Goldenshluger A., Lepski O.: Structural adaptation via lp-norm oracle inequalities. Probab. Theory Relat. Fields 143, 41-71 (2009).
5. Has'minskii R. Z., Ibragimov I. A.: Statistical Estimation, Asymptotic Theory. Applications of Mathematics. Springer, Berlin (1981).
6. Härlde W., Hart J., Marron J. S., Tsybakov A.: Bandwidth choice for average derivative estimation. J. Am. Stat. Assoc. 87, 417 (1992).
7. Juditsky A., Lepski O., Tsybakov A.: Nonparametric estimation of composite functions. Ann. Stat. 37(3), 1360-1404 (2009).
8. Katkovnik V., Spokoiny V.: Spatially adaptive estimation via fitted local likelihood techniques. IEEE Trans. Imag. Process. 56(3), 873-886 (2008).
9. Klutchnikoff, N.: On the adaptive estimation of anisotropic functions. PhD thesis, Aix-Masrseille 1, (2005).
10. Korostelëv, A. P., Tsybakov, A.: Minimax theory of image reconstruction. Lecture Notes in Statistics, vol. 82. Springer, New York. ISBN 0-387-94028-6 (1993).
11. Lepski O.: On a problem of adaptive estimation in gaussian white noise. Theory Probab. Appl. 35(3), 454-466 (1990).
12. Lepski O.: Asymptotically minimax adaptive estimation i. upper bounds. optimally adaptive estimates. Theory Probab. Appl. 36, 682-697 (1991).
13. Lepski O.: Asymptotically minimax adaptive estimation ii. statistical models without optimal adaptation. adaptive estimators. Theory Probab. Appl. 37, 433-468 (1992).
14. Lepski O., Mammen E., Spokoiny V.: Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Stat. 25(3), 929-947 (1997) ISSN 0090-5364.
15. Lepski O., Spokoiny V.: Optimal pointwise adaptive methods in nonparametric estimation. Ann. stat. 25(6), 2512-2546 (1997).
16. Nemirovski A.: Topics in non-parametric statistics. Lecture Notes in Mathematics, vol. 1738. Springer, Berlin (2000).
17. Polzehl J., Spokoiny V.: Propagation-separation approach for local likelihood estimation. Probab. Theory Relat. Fields 135(3), 335-362 (2006).
18. Simar L., Wilson P.: Statistical inference in nonparametric frontier models: The state of the art. J. Prod. Anal. 13, 49-78 (2000).
19. Spokoiny V.: Adaptive hypothesis testing using wavelets. Ann. Stat. 24(6), 2477-2498 (1996).
20. Tsybakov A.: Pointwise and sup-norm sharp adaptive estimation of function on the sobolev classes. Ann. Stat. 26(6), 2420-2469 (1998).
21. Tsybakov A.: Introduction to Nonparametric Estimation. Springer, Berlin (2008).