Statistica Sinica 24 (2014), 555-575

ASYMPTOTIC LAWS FOR CHANGE POINT ESTIMATION

IN INVERSE REGRESSION

Sophie Frick, Thorsten Hohage and Axel Munk

Universität Göttingen

Abstract: We derive rates of convergence and asymptotic normality of the
least squares estimator for a large class of parametric inverse regression
models Y = (Φf)(X)+ε. Our theory provides a unified asymptotic tretament
for estimation of f with discontinuities of certain order, including piecewise
polynomials and piecewise 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 under a nonlinear
Hammerstein integral operator. Furthermore, we show that ℓ_{0}-penalisation
leads to a consistent model selection, using techniques from empirical process
theory. The asymptotic normality is used to provide confidence bands for f.
Simulation studies and a data example from rheology illustrate the results.

Key words and phrases: Asymptotic normality, change point analysis, confidence bands, dynamic stress moduli, entropy bounds, Hammerstein integral equations, jump detection, penalized least squares estimator, reproducing kernel Hilbert spaces, sparsity, statistical inverse problems.