A note on nonparametric estimation of linear functionals

Abstract

Precise asymptotic descriptions of the minimax affine risks and biasvariance tradeoffs for estimating linear functionals are given for a broad class of moduli. The results are complemented by illustrative examples including one where it is possible to construct an estimator which is fully adaptive over a range of parameter spaces.

Minimax linear estimation at a boundary point

“…This paper complements Zhao [1997] and a line of other researches on the solutions to the minimax estimation problems under various settings 1 , such as Taikov [1968], Legostaeva and Shiryaev [1971], Sacks and Ylvisaker [1978], Sacks and Ylvisaker [1981], Li [1982], Silverman [1984], Wahba [1990], Fan [1993], Korostelev [1994], Cheng, Fan, and Marron [1997], Fan, Gasser, Gijbels, Brockmann, and Engel [1997], Leonov [1999] and Cai and Low [2003] . See Armstrong and Kolesár [2016a,b] for a review of these results along with the construction of honest confidence intervals for nonparametric regression models based on minimax optimal kernels.…”

Optimized Regression Discontinuity Designs

“…Because this estimator is minimax among the class of linear estimators, it is at least as accurate as any local linear regression estimator in a minimax sense over all problems with Var Y i X i = σ 2 i and |µ w (x)| ≤ B. For further discussion of related estimators, see Cai and Low [2003], Donoho [1994], Donoho and Liu [1991], and Juditsky and Nemirovski [2009].…”

“…In early work, Legostaeva and Shiryaev [1971] and Sacks and Ylvisaker [1978] independently studied inference in "almost" linear models that arise from taking a Taylor expansion around a point; see also Cheng et al [1997]. For a broader discussion of minimax linear estimation over non-parametric function classes, see Cai and Low [2003], Donoho [1994], Ibragimov and Khas'minskii [1985], Johnstone [2011], Juditsky and Nemirovski [2009], and references therein. An important result in this literature that, for many problems of interest, minimax linear estimators are within a small explicit constant of being minimax among all estimators [Donoho and Liu, 1991].…”

“…In this paper, motivated by a large literature on minimax linear estimation [Armstrong and , Cai and Low, 2003, Donoho, 1994, Donoho and Liu, 1991, Ibragimov and Khas'minskii, 1985, Johnstone, 2011, Juditsky and Nemirovski, 2009, we study an alternative approach based on directly minimizing finite sample error bounds via numerical optimization, under an assumption that the second derivative of the response surface is bounded away from the boundary of the treatment region. 1 This approach has several advantages relative to local regression.…”

On efficiency of the plug-in principle for estimating smooth integrated functionals of a nonincreasing density

“…Estimation of functionals of data generating distributions is a fundamental research problem in nonparametric statistics. Consequently, substantial effort has gone into understanding estimation of nonparametric functionals (e.g., linear, quadratic, and other "smooth" functionals) both under density and regression (specifically Gaussian white noise models) settings -a comprehensive snapshot of this huge literature can be found in Hall and Marron (1987), Bickel and Ritov (1988), , Donoho and Nussbaum (1990), Fan (1991), Kerkyacharian and Picard (1996), Nemirovski (2000b), Laurent (1996), Cai and Low (2003, 2004, 2005 and the references therein. However, most of the above papers have focused on smoothness restrictions on the underlying (nonparametric) function class.…”