Optimal adaptive estimation of a quadratic functional

Annals of Statistics

Volumn 34, Issue 5, 2006, Pages 2298-2325

  Cai, T Tony ^a,b   Low, Mark G ^a,b  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Adaptation; Block thresholding; Quadratic functional; Wavelets; White noise model

Indexed keywords

EID: 33644888851     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000849     Document Type: Article

References (30)

1. BARAUD, Y. (2000). Model selection for regression on a fixed design. Probab. Theory Related Fields 117 467-493. MR1777129
2. BARAUD, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577-606. MR1935648
3. BARRON, A. R., BIRGÉ, L. and MASSART, P. (1999). Risk bound for model selection via penalization. Probab. Theory Related Fields 113 301-413. MR1679028
4. BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore. MR1245941
5. BICKEL, P. J. and RITOV, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381-393. MR1065550
6. BIRGÉ, L. and MASSART, P. (1997). From model selection to adaptive estimation. In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. Yang, eds.) 55-87. Springer, New York. MR1462939
7. BROWN, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. IMS, Hayward, CA. MR0882001
8. CAI, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898-924. MR1724035
9. CAI, T. and LOW, M. (2005). On adaptive estimation of linear functionals. Ann. Statist. 33 2311-2343. MR2211088
10. CAI, T. and LOW, M. (2005). Nonquadratic estimators of a quadratic functional. Ann. Statist. 33 2930-2956.
11. CAI, T., LOW, M. and ZHAO, L. (2006). Tradeoffs between global and local risks in nonparametric function estimation. Bernoulli. To appear.
12. DONOHO, D. L. and LIU, R. C. (1991). Geometrizing rates of convergence. II, III. Ann. Statist. 19 633-667, 668-701. MR1105839
13. DONOHO, D. L. and NUSSBAUM, M. (1990). Minimax quadratic estimation of a quadratic functional. J. Complexity 6 290-323. MR1081043
14. EFROMOVICH, S. Y. and LOW, M. (1996). On optimal adaptive estimation of a quadratic functional. Ann. Statist. 24 1106-1125. MR1401840
15. FAN, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273-1294. MR1126325
16. FELLER, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York. MR0228020
17. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. MR0270403
18. GAYRAUD, G. and TRIBOULEY, K. (1999). Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law. Statist. Probab. Lett. 44 109-122. MR1706448
19. HALL, P. and HEYDE, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. MR0624435
20. HARDY, G., LITTLEWOOD, J. E. and PÓLYA, G. (1952). Inequalities, 2nd ed. Cambridge Univ. Press. MR0046395
21. JOHNSTONE, I. M. (2001). Chi-square oracle inequalities. In State of the Art in Probability and Statistics. Festschrift for Willem R. van Zwet (M. de Gunst, C. Klaassen and A. van der Waart, eds.) 399-418. IMS, Beachwood, OH. MR1836572
22. JOHNSTONE, I. M. (2002). Function estimation and Gaussian sequence models. Draft of a monograph. Available at stat.stanford.edu/~imj/ baseb.pdf.
23. KLEMELÄ, J. (2006). Sharp adaptive estimation of quadratic functionals. Probab. Theory Related Fields 134 539-564. MR2214904
24. KLEMELÄ, J. and TSYBAKOV, A. B. (2001). Sharp adaptive estimation of linear functionals. Ann. Statist. 29 1567-1600. MR1891739
25. LAURENT, B. and MASSART, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 1302-1338. MR1805785
26. LEPSKI, O. V. (1990). On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454-466. MR1091202
27. LEPSKI, O. V. and SPOKOINY, V. G. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512-2546. MR 1604408
28. MARDIA, K. V. (1972). Statistics of Directional Data. Academic Press, London. MR0336854
29. TRIBOULEY, K. (2000). Adaptive estimation of integrated functionals. Math. Methods Statist. 9 19-38. MR1772223
30. WATSON, G.S. (1983). Statistics on Spheres. Wiley, New York. MR0709262