ASYMPTOTIC LAWS FOR CHANGE POINT ESTIMATION IN INVERSE REGRESSION

@article{Frick2014ASYMPTOTICLF,
  title={ASYMPTOTIC LAWS FOR CHANGE POINT ESTIMATION IN INVERSE REGRESSION},
  author={Sophie Frick and Thorsten Hohage and Axel Munk},
  journal={Statistica Sinica},
  year={2014}
}
We derive rates of convergence and asymptotic normality for the least squares estimator for a large class of parametric inverse regression models Y = (f )(X) +". Our theory provides a unied asymptotic tretament for estimation of f with discontinuities of certain order, including piecewise polynomials and piece- wise kink functions. Our results cover several classical and new examples, including splines with free knots or the estimation of piecewise linear functions with indirect observations… 
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References

SHOWING 1-10 OF 39 REFERENCES
Consistencies and rates of convergence of jump-penalized least squares estimators
We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence…
Free‐knot polynomial splines with confidence intervals
Summary.  We construct approximate confidence intervals for a nonparametric regression function, using polynomial splines with free‐knot locations. The number of knots is determined by generalized…
Spatially Adaptive Splines for Statistical Linear Inverse Problems
This paper introduces a new nonparametric estimator based on penalized regression splines for linear operator equations when the data are noisy. A local roughness penalty that relies on local support…
Jump estimation in inverse regression
We consider estimation of a step function f from noisy obser- vations of a deconvolution � � f, whereis some bounded L1-function. We use a penalized least squares estimator to reconstruct the signal…
Kink estimation in stochastic regression with dependent errors and predictors
In this article we study the estimation of the location of jump points in the first derivative (referred to as kinks) of a regression function µ in two random design models with different long-range…
Optimal Change‐point Estimation in Inverse Problems
We develop a method of estimating a change‐point of an otherwise smooth function in the case of indirect noisy observations. As two paradigms we consider deconvolution and non‐parametric…
Minimax estimation of sharp change points
We define the sharp change point problem as an extension of earlier problems in change point analysis related to nonparametric regression. As particular cases, these include estimation of jump points…
Optimal change-point estimation from indirect observations
TLDR
The results demonstrate that the best achievable rates of convergence are determined both by smooths of the function away from the change-point and by the degree of ill-posedness of the convolution operator.
Minimax lower bound for kink location estimators in a nonparametric regression model with long-range dependence
A Practical Guide to Splines
  • C. D. Boor
  • Mathematics
    Applied Mathematical Sciences
  • 1978
TLDR
This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
...
...