# Bayesian posterior consistency in the functional randomly shifted curves model

Abstract : In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function $f^0$ submitted to a random translation of law $g^0$ in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process $\PP_{f^0,g^0}$ as well as $f^0$ and $g^0$. In this perspective, we adopt a Bayesian point of view and find prior on $f$ and $g$ such that the posterior distribution concentrates around $\PP_{f^0,g^0}$ at a polynomial rate when $n$ goes to $+\infty$. We obtain a logarithmic posterior contraction rate for the shape $f^0$ and the distribution $g^0$. We also derive logarithmic lower bounds for the estimation of $f^0$ and $g^0$ in a frequentist paradigm.
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https://hal-univ-tlse3.archives-ouvertes.fr/hal-00768169
Submitted on : Tuesday, March 12, 2013 - 3:32:22 PM
Last modification on : Wednesday, February 10, 2021 - 9:30:03 AM
Long-term archiving on: : Thursday, June 13, 2013 - 2:37:10 AM

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### Identifiers

• HAL Id : hal-00768169, version 2
• ARXIV : 1212.5429

### Citation

Dominique Bontemps, Sébastien Gadat. Bayesian posterior consistency in the functional randomly shifted curves model. 2013. ⟨hal-00768169v2⟩

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