On the risk of convex-constrained least squares estimators under misspecification

@article{Fang2019OnTR,
  title={On the risk of convex-constrained least squares estimators under misspecification},
  author={Billy Fang and Adityanand Guntuboyina},
  journal={Bernoulli},
  year={2019}
}
We consider the problem of estimating the mean of a noisy vector. When the mean lies in a convex constraint set, the least squares projection of the random vector onto the set is a natural estimator. Properties of the risk of this estimator, such as its asymptotic behavior as the noise tends to zero, have been well studied. We instead study the behavior of this estimator under misspecification, that is, without the assumption that the mean lies in the constraint set. For appropriately defined… Expand

Figures and Tables from this paper

A new computational framework for log-concave density estimation
In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statisticalExpand
Nonparametric Shape-restricted Regression.
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additiveExpand
Distribution-free properties of isotonic regression
It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we proveExpand
of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Seven Number Four November 2021
The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt fürExpand
Local continuity of log-concave projection, with applications to estimation under model misspecification
The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. Prior work by D{u}mbgen et al. (2011) establishes that, with suitableExpand

References

SHOWING 1-10 OF 14 REFERENCES
Sharp oracle inequalities for Least Squares estimators in shape restricted regression
The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the modelExpand
Convergence of linear functionals of the Grenander estimator under misspecification
Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate $n^{1/3}$ if the true density is curved [Sankhy\={a} Ser. A 31 (1969) 23-36]Expand
Regression Shrinkage and Selection via the Lasso
SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than aExpand
Risk bounds in isotonic regression
Nonasymptotic risk bounds are provided for maximum likelihood-type isotonic estimators of an unknown nondecreasing regression function, with general average loss at design points. These bounds areExpand
Penalized isotonic regression
Abstract In isotonic regression, the mean function is assumed to be monotone increasing (or decreasing) but otherwise unspecified. The classical isotonic least-squares estimator is known to beExpand
Living on the edge: phase transitions in convex programs with random data
TLDR
This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems and introduces a summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. Expand
Spiking problem in monotone regression: Penalized residual sum of squares
We consider the estimation of a monotone function at its end-point, where the least square estimate is inconsistent. The least square criterion is penalized to achieve consistency. The limitExpand
Fundamentals of Convex Analysis
Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations ofExpand
Order restricted statistical inference
Isotonic Regression. Tests of Ordered Hypotheses: The Normal Means Case. Tests of Ordered Hypotheses: Generalizations of the Likelihood Ratio Tests and Other Procedures. Inferences about a Set ofExpand
Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective
TLDR
This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones and direct derivations of the general Steiner formula, the conic analogues of the Brianchon–Gram–Euler and the Gauss–Bonnet relations, and the principal kinematic formula are given. Expand
...
1
2
...