## Asymptotic minimax rates for abstract linear estimators.(English)Zbl 0897.62076

Summary: Abstract linear estimation concerns the estimation of an abstract parameter that depends on the underlying density via a linear transformation. An important subclass is the class of inverse problems where this transformation is naturally described as the inverse of some bounded operator. Suitable preconditioning allows us to restrict ourselves to the inverse of some Hermitian operator, which does not remain restricted to the class of compact operators.
A lower bound to the minimax risk is obtained for the class of all estimators satisfying a natural moment condition and certain submodels. To establish the bound we use the Bayesian van Trees inequality [see H. L. van Trees, Detection estimation and modulation theory. Part 1 (1968; Zbl 0202.18002)] and systems of (pseudo) eigenvectors of the operator involved. We also briefly sketch a general construction method for estimators, based on a regularized inverse of the operator involved, and show that these estimators attain the asymptotic minimax rate in interesting examples.

### MSC:

 62J99 Linear inference, regression 46N30 Applications of functional analysis in probability theory and statistics 62G05 Nonparametric estimation

Zbl 0202.18002
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### References:

 [1] Bertero, M., Linear inverse and ill-posed problems, Adv. Electron. El. Phys., 75, 1-120 (1989) [2] Carroll, R. J.; van Rooij, A. C.M.; Ruymgaart, F. H., Theoretical aspects of ill-posed problems in statistics, Acta Appl. Math., 24, 113-140 (1991) · Zbl 0753.62022 [3] Chauveau, D. E.; van Rooij, A. C.M.; Ruymgaart, F. H., Regularized inversion of noisy Laplace transforms, Adv. in Appl. Math., 15, 186-201 (1993) · Zbl 0806.65136 [4] Devroye, L., A Course in Density Estimation (1987), Birkhäuser: Birkhäuser Boston · Zbl 0617.62043 [5] Dey, A. K.; Mair, B. A.; Ruymgaart, F. H., Cross-validation for parameter selection in inverse estimation problems, Scand. J. Statist. (1996), (to appear) · Zbl 0898.62046 [6] Donoho, D. L.; Liu, R. C.; MacGibbon, K. B., Minimax risk over hyperrectangles, and implications, Ann. Statist., 18, 1416-1437 (1990) · Zbl 0705.62018 [7] Eubank, R. L., Spline Smoothing and Nonparametric Regression (1988), Dekker: Dekker New York · Zbl 0702.62036 [8] Fan, J., Global behavior of deconvolution kernel estimates, Statist. Sin., 1, 541-551 (1991) · Zbl 0823.62032 [9] Gilliam, D. S.; Hall, P.; Ruymgaart, F. H., Rate of convergence of the empirical Radon transform, J. Multivariate Anal., 44, 115-145 (1993) · Zbl 0782.62042 [10] Halmos, P. R., What does the spectral theorem say?, Amer. Math. Monthly, 70, 241-247 (1963) · Zbl 0132.35606 [11] Hoerl, A.; Kennard, R. W., Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12, 55-67 (1970) · Zbl 0202.17205 [12] Hoerl, A.; Kennard, R. W., Ridge regression: applications to nonorthogonal problems, Technometrics, 12, 69-82 (1970) · Zbl 0202.17206 [13] Gill, R. D.; Levit, B. Y., Applications of the van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli, 1 (1995) · Zbl 0830.62035 [14] Johnstone, I. M.; Silverman, B. W., Speed of estimation in positron emission tomography and related inverse problems, Ann. Statist., 18, 251-280 (1990) · Zbl 0699.62043 [15] Johnstone, I. M.; Silverman, B. W., Discretization effects in statistical inverse problems, J. Complexity, 7, 1-34 (1991) · Zbl 0737.62099 [16] Kress, R., Linear Integral Equations (1989), Springer: Springer New York [17] Nussbaum, M., Spline smoothing in regression models and asymptotic efficiency in $$L_2$$, Ann. Statist., 13, 984-997 (1985) · Zbl 0596.62052 [18] Nychka, D. W.; Cox, D. D., Convergence rates for regularized solutions of integral equations from discrete noisy data, Ann. Statist., 17, 556-572 (1989) · Zbl 0672.62054 [19] Reed, M.; Simon, B., Methods of Mathematical Physics. I: Functional Analysis (1972), Academic Press: Academic Press New York [20] Ruymgaart, F. H., A unified approach to inversion problems in statistics, Math. Methods Statist., 2, 130-146 (1993) · Zbl 0798.62010 [21] Speckman, P., Spline smoothing and optimal rates of convergence in nonparametric regression models, Ann. Statist., 13, 970-983 (1985) · Zbl 0585.62074 [22] Tikhonov, A. N.; Arsenin, V. Y., Solutions of Ill-Posed Problems (1977), Wiley: Wiley New York · Zbl 0354.65028 [23] Trees, H. L.van, Detection Estimation and Modulation Theory. Part I (1968), Wiley: Wiley New York · Zbl 0202.18002 [24] Wahba, G., Principal approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal., 14, 651-667 (1977) · Zbl 0402.65032 [25] Zhang, C.-H., Fourier methods for estimating densities and distributions, Ann. Statist., 18, 806-831 (1990) · Zbl 0778.62037
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