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On convex least squares estimation when the truth is linear.

Yining Chen1, Jon A Wellner1

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Electronic Journal of Statistics
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The convex least squares estimator (LSE) achieves a n^{-1/2} convergence rate in linear regions for density and regression estimation. This method also enables consistent testing for linearity against convex alternatives.

Keywords:
Adaptive estimationconvexitydensity estimationleast squaresregression function estimationshape constraint

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Area of Science:

  • Statistical theory
  • Nonparametric statistics
  • Machine learning

Background:

  • Convexity is a common assumption in statistical modeling.
  • Understanding the convergence rates of estimators is crucial for statistical inference.
  • Existing methods may not fully exploit convexity for optimal estimation.

Purpose of the Study:

  • To establish the pointwise convergence rate of the convex least squares estimator (LSE).
  • To develop a consistent testing procedure for linearity against convex alternatives.
  • To analyze the adaptive properties of the convex LSE at region boundaries.

Main Methods:

  • Pointwise convergence rate analysis for the convex LSE.
  • Characterization of asymptotic distributions using modified invelope processes.
  • Development and analysis of a hypothesis testing procedure for linearity.
  • Investigation of boundary point adaptation properties.

Main Results:

  • The convex LSE achieves a n^{-1/2} pointwise rate of convergence in linear regions.
  • Asymptotic distributions are characterized by a modified invelope process.
  • The convex LSE derivative achieves analogous results for derivative estimation.
  • A consistent test for linearity against convex alternatives is proposed.
  • Optimal rate adaptation (up to a log-log factor) is shown at boundary points.

Conclusions:

  • The convex LSE provides optimal pointwise convergence rates in linear regions for density and regression estimation.
  • The proposed testing procedure offers a reliable method for assessing linearity.
  • The adaptive properties enhance the performance of the convex LSE in practical scenarios.