# 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

#### 5 Citations

A new computational framework for log-concave density estimation

- Mathematics
- 2021

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 statistical… Expand

Nonparametric Shape-restricted Regression.

- Mathematics
- 2017

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, additive… Expand

Distribution-free properties of isotonic regression

- Mathematics
- Electronic Journal of Statistics
- 2019

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 prove… Expand

of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Seven Number Four November 2021

- 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 für… Expand

Local continuity of log-concave projection, with applications to estimation under model misspecification

- Mathematics
- 2020

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 suitable… Expand

#### References

SHOWING 1-10 OF 14 REFERENCES

Sharp oracle inequalities for Least Squares estimators in shape restricted regression

- Mathematics
- 2015

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 model… Expand

Convergence of linear functionals of the Grenander estimator under misspecification

- Mathematics
- 2012

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

- Mathematics
- 1996

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 a… Expand

Risk bounds in isotonic regression

- Mathematics
- 2002

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 are… Expand

Penalized isotonic regression

- Mathematics
- 2015

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 be… Expand

Living on the edge: phase transitions in convex programs with random data

- Mathematics, Computer Science
- 2013

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

- Mathematics
- 2008

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 limit… Expand

Fundamentals of Convex Analysis

- Mathematics
- 2004

Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations of… Expand

Order restricted statistical inference

- Mathematics
- 1988

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 of… Expand

Intrinsic Volumes of Polyhedral Cones: A Combinatorial Perspective

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2017

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