Bias-Variance Tradeoffs in Functional Estimation Problems
Open accessAbstract
It is shown in infinite-dimensional Gaussian problems that affine estimators minimax the variance among all estimators of a linear functional subject to a constraint on the bias. Likewise, affine estimators also minimax the square of the bias among all estimates of a linear functional subject to a constraint on the variance.
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When Θ is convex, the minimization to construct the optimal FLCI can be solved as a nested optimization problem, where both the inner and outer minimizations are convex.
“When is convex, this minimization can be solved as a nested optimization problem, where both the inner and outer minimizations are convex (Low, 1995;Armstrong andKolesár, 2018, 2020a).”
From A More Credible Approach to Parallel Trends (2023) · cites this paper as Low, 1995
The discussion on p. 2 describes finding a hardest one-dimensional subfamily and then finding optimal affine estimators over it; it also discusses fixed-length confidence intervals. But neither that passage nor Theorem 2 (p. 6) says that constructing an optimal FLCI is a nested optimization with both minimizations convex. (LLM, conf=0.76)
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Cites
Geometrizing Rates of Convergence, II
Donoho, D. L., & Liu, R. C. (1991) · The Annals of Statistics · doi:10.1214/aos/1176348114 · Cited by 271 · see citing works
Statistical Estimation and Optimal Recovery
Donoho, D. L. (1994) · The Annals of Statistics · doi:10.1214/aos/1176325367 · Cited by 263 · see citing works
Estimating a Bounded Normal Mean
Casella, G., & Strawderman, W. E. (1981) · The Annals of Statistics · doi:10.1214/aos/1176345527 · Cited by 179 · see citing works
Linear Estimation for Approximately Linear Models
Sacks, J., & Ylvisaker, D. (1978) · The Annals of Statistics · doi:10.1214/aos/1176344315 · Cited by 153 · see citing works
On Nonparametric Estimation of the Value of a Linear Functional in Gaussian White Noise
Ibragimov, I. A., & Khas'minskii, R. Z. (1985) · Theory of Probability and Its Applications · doi:10.1137/1129002 · Cited by 137 · see citing works
Renormalization Exponents and Optimal Pointwise Rates of Convergence
Donoho, D. L., & Low, M. G. (1992) · The Annals of Statistics · doi:10.1214/aos/1176348665 · Cited by 106 · see citing works
Self-citationAsymptotically Optimum Kernels for Density Estimation at a Point
Sacks, J., & Ylvisaker, D. (1981) · The Annals of Statistics · doi:10.1214/aos/1176345399 · Cited by 74 · see citing works
Renormalization and White Noise Approximation for Nonparametric Functional Estimation Problems
Low, M. G. (1992) · The Annals of Statistics · doi:10.1214/aos/1176348538 · Cited by 18 · see citing works
Self-citationIMPROVEMENTS ON LINEAR MINIMAX ESTIMATES
Sacks, J., & Strawderman, W. E. (1982) · Elsevier eBooks · doi:10.1016/b978-0-12-307502-4.50022-2 · Cited by 17 · see citing works
A Lower Bound for the Risk in Estimating the Value of a Probability Density
Brown, L. D., & Farrell, R. H. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10474988 · Cited by 16 · see citing works
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A More Credible Approach to Parallel Trends
Rambachan, A., & Roth, J. (2023) · Review of Economic Studies · doi:10.1093/restud/rdad018 · Cited by 1,559 · see citing works
An adaptation theory for nonparametric confidence intervals
Cai, T. T., & Low, M. G. (2004) · The Annals of Statistics · doi:10.1214/009053604000000049 · Cited by 72 · see citing works
Self-citationNeo-Classical Minimax Problems, Thresholding and Adaptive Function Estimation
Donoho, D. L., & Johnstone, I. M. (1996) · Bernoulli · doi:10.2307/3318568 · Cited by 66 · see citing works
Sensitivity analysis using approximate moment condition models
Armstrong, T. B., & Kolesár, M. (2021) · Quantitative Economics · doi:10.3982/qe1609 · Cited by 60 · see citing works
Combining Multiple Biomarker Models in Logistic Regression
Yuan, Z., & Ghosh, D. (2008) · Biometrics · doi:10.1111/j.1541-0420.2007.00904.x · Cited by 45 · see citing works
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Tony, T., Cai, & Low, M. G. (2004) · Cited by 41 · see citing works
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Adaptive estimates of linear functionals
Efromovich, S., & Low, M. G. (1994) · Probability Theory and Related Fields · doi:10.1007/bf01192517 · Cited by 34 · see citing works
Self-citationMINIMAX ESTIMATION OF LINEAR FUNCTIONALS OVER NONCONVEX PARAMETER SPACES
Cai, T., & Low, M. G. (2004) · Cited by 31 · see citing works
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Cai, T. T., & Low, M. G. (2004) · The Annals of Statistics · doi:10.1214/009053604000000094 · Cited by 29 · see citing works
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Armstrong, T. B., & Kolesár, M. (2019) · SSRN Electronic Journal · doi:10.2139/ssrn.3337748 · Cited by 29 · see citing works
A note on nonparametric estimation of linear functionals
Cai, T., & Low, M. G. (2003) · The Annals of Statistics · doi:10.1214/aos/1059655908 · Cited by 27 · see citing works
Self-citationCHANGE-POINTS VIA WAVELETS FOR INDIRECT DATA
Wang, Y. (1999) · Cited by 16 · see citing works
Confidence Intervals for Nonparametric Empirical Bayes Analysis
Ignatiadis, N., & Wager, S. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.2008403 · Cited by 14 · see citing works
Low, M. G., & Kang, Y.-G. (2002) · Statistics & Probability Letters · doi:10.1016/s0167-7152(01)00179-1 · Cited by 11 · see citing works
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Input layer regularization for magnetic resonance relaxometry biexponential parameter estimation
Rozowski, M., Palumbo, J., Bisen, J., Bi, C., Bouhrara, M., Czaja, W., & Spencer, R. G. S. (2022) · Magnetic Resonance in Chemistry · doi:10.1002/mrc.5289 · Cited by 7 · see citing works
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On lower bounds for the bias-variance trade-off
Derumigny, A., & Schmidt-Hieber, J. (2023) · The Annals of Statistics · doi:10.1214/23-aos2279 · Cited by 5 · see citing works
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Notes on the bias-variance trade-off phenomenon
Chen, J. (2004) · Lecture notes-monograph series · doi:10.1214/lnms/1196285391 · Cited by 3 · see citing works
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Armstrong, T. B., & Kolesar, M. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.3087420 · Cited by 1 · see citing works
Neo-classical minimax problems, thresholding and adaptive function
Donoho, D. L., & Johnstone, I. M. (1996)
- Authors
- Low, M. G.
- Corresponding author
- Low, Mark G.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1995-06-01
- Volume
- 23
- Issue
- 3
- Pages
- 824-835
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2028000850
- Cited by
- 23
- References
- 10
- Citations by year
- 2023: 2 · 2022: 1 · 2021: 3 · 2020: 1 · 2019: 1 · 2018: 2 · 2017: 1
- FWCI
- 0.94
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Keywords
- bias-variance trade-off, functional estimation, linear functionals, minimax estimation, constrained estimation
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