Minimax estimation of linear functionals over nonconvex parameter spaces. (English) Zbl 1048.62054

Cai, T. Tony; Low, Mark G.

Ann. Stat. 32, No. 2, 552-576 (2004).

Summary: The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However, in contrast to the theory for convex parameter spaces, rate optimal procedures are often required to be nonlinear. A construction of such nonlinear procedures is given. The results developed in this paper have important applications to the theory of adaptation.

MSC:

62G99	Nonparametric inference
62C20	Minimax procedures in statistical decision theory
62F12	Asymptotic properties of parametric estimators
62M99	Inference from stochastic processes

Keywords:

constrained risk inequality; linear functionals; minimax estimation; modulus of continuity; nonparametric functional estimation; white noise model

References:

[1]	Abramovich, F., Benjamini, Y., Donoho, D. and Johnstone, I. (2000). Adapting to unknown sparsity by controlling the false discovery rate. Technical report 2000-19, Dept. Statistics, Stanford Univ. · Zbl 1092.62005
[2]	Baraud, Y. (2000). Nonasymptotic minimax rates of testing in signal detection. Technical report, Ecole Normale Supérieure. · Zbl 1007.62042
[3]	Brown, L. D. and Low, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524–2535. · Zbl 0867.62023 · doi:10.1214/aos/1032181166
[4]	Cai, T. and Low, M. (2002). On modulus of continuity and adaptability in nonparametric functional estimation. Technical report, Dept. Statistics, Univ. Pennsylvania.
[5]	Donoho, D. L. (1994). Statistical estimation and optimal recovery. Ann. Statist. 22 238–270. JSTOR: · Zbl 0805.62014 · doi:10.1214/aos/1176325367
[6]	Donoho, D. L., Johnstone, I. M., Hoch, J. C. and Stern, A. S. (1992). Maximum entropy and the nearly black object (with discussion). J. Roy. Statist. Soc. Ser. B 54 41–81. · Zbl 0788.62103
[7]	Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57 301–369. · Zbl 0827.62035
[8]	Donoho, D. L. and Liu, R. C. (1991a). Geometrizing rates of convergence. II. Ann. Statist. 19 633–667. · Zbl 0754.62028 · doi:10.1214/aos/1176348114
[9]	Donoho, D. L. and Liu, R. G. (1991b). Geometrizing rates of convergence. III. Ann. Statist. 19 668–701. · Zbl 0754.62029 · doi:10.1214/aos/1176348115
[10]	Efromovich, S. and Low, M. G. (1994). Adaptive estimates of linear functionals. Probab. Theory Related Fields 98 261–275. · Zbl 0796.62037 · doi:10.1007/BF01192517
[11]	Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1 , 3rd ed. Wiley, New York. · Zbl 0155.23101
[12]	Ibragimov, I. A. and Has’minskii, R. Z. (1984). Nonparametric estimation of the value of a linear functional in Gaussian white noise. Theory Probab. Appl. 29 18–32. · Zbl 0575.62076 · doi:10.1137/1129002
[13]	Lepski, O. V. (1990). On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454–466. · Zbl 0725.62075
[14]	Low, M. G. (1995). Bias-variance tradeoffs in functional estimation problems. Ann. Statist. 23 824–835. JSTOR: · Zbl 0838.62006 · doi:10.1214/aos/1176324624