Donoho, David L.; Nussbaum, Michael Minimax quadratic estimation of a quadratic functional. (English) Zbl 0724.62039 J. Complexity 6, No. 3, 290-323 (1990). Let \(f\) be a real function and consider its noisy sample observations \(v_ i=f(t_ i)+z_ i\), \(i=1,2,...,n\), where \(t_ i=-\pi +2\pi (i/n)\) are equispaced on \([-\pi,\pi]\) and the noise terms \(z_ i\) are iid \(N(0,\sigma^ 2)\). Let \(f^{(k)}(t)\) be the \(k\)th derivative of \(f\). The authors consider the estimation of the quadratic functional \(Q(f)=\int^{\pi}_{-\pi}(f^{(k)}(t))^ 2\,dt\) and derive a simple quadratic estimate which is asymptotically minimax among quadratic estimates. The given estimate is also shown to be rate optimal. Reviewer: Rasul A. Khan (Cleveland) Cited in 1 ReviewCited in 49 Documents MSC: 62G07 Density estimation 62G99 Nonparametric inference Keywords:integrated squared derivative; periodic function; incomplete data; noisy sample observations; asymptotically minimax among quadratic estimates PDFBibTeX XMLCite \textit{D. L. Donoho} and \textit{M. Nussbaum}, J. Complexity 6, No. 3, 290--323 (1990; Zbl 0724.62039) Full Text: DOI References: [1] Donoho, D. L., Statistical Estimation and Optimal Recovery, (Techical Report 214 (1989), Department of Statistics: Department of Statistics U. C. Berkeley) · Zbl 0802.62007 [2] Donoho, D. L.; Liu, R. C., Geometrizing Rates of Convergence, II, (Technical Report No. 120 (1988), Department of Statistics: Department of Statistics U. C. Berkeley) · Zbl 0754.62028 [3] Donoho, D. L.; Low, M., White Noise Approximation and Minimax Risk, (Technical Report (1990), Department of Statistics: Department of Statistics U. C. Berkeley) [4] Fan, J., Nonparametric Estimation of Quadratic Functionals in Gaussian White Noise, (Technical Report 166 (1988), Department of Statistics: Department of Statistics U. C. Berkeley), Ann. Statistics, to appear [5] Hall, P.; Marron, J. S., Estimation of integrated squuared density derivatives, Statist. Probab. Lett., 6, 109-115 (1987) · Zbl 0628.62029 [6] Ibragimov, I. A.; Nemirovskii, A. S.; Has’minskii, R. Z., Some problems on nonparametric estimation in Gaussian white noise, Teor. Veroyatnost. i Primenen., 32 (1987), [in Russian] · Zbl 0623.62028 [7] Le Cam, L., Asymptotic Methods in Statistical Decision Theory (1986), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0605.62002 [8] Levit, B. Y., Asymptotically efficient estimation of nonlinear functionals, Problemy Peredachi Informatsii, 14, 65-72 (1978) · Zbl 0422.62034 [9] Low, M. G., Towards a Unified Theory of Asymptotic Minimax Estimation, (Ph. D. thesis (1988), Cornell Univ) [10] Micchelli, C. A., Optimal Estimation of Linear Functionals, IBM Research Report 5729 (1975) · Zbl 0294.41032 [11] Micchelli, C. A.; Rivlin, T. J., A survey of optimal recovery, (Micchelli; Rivlin, Optimal Estimation in Approximation Theory (1977), Plenum: Plenum New York), 1-54 · Zbl 0386.93045 [12] Nussbaum, M., Spline smoothing and asymptotic efficiency in \(L_2\), Ann. Statist., 13, 984-987 (1985) [13] Pinsker, M. S., Optimal filtering of square integrable signals in Gaussian white noise, Problems of Inform. Transmission, 16, 2, 52-68 (1980) · Zbl 0452.94003 [14] Ritov, Y.; Bickel, P. J., Estimating integrated squared density derivatives: Sharp best order of convergence, Sankhya, 50, 381-393 (1988) · Zbl 0676.62037 [15] Traub, J. F.; Wasilkowski, G. W.; Woźniakowski, H., Information, Uncertainty, Complexity (1983), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0522.68041 [16] Traub, J. F.; Wasilkowski, G. W.; Woźniakowski, H., Information-Based Complexity (1988), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0674.68039 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.