Estimation in autoregressive model with measurement error. (3rd October 2014)

• Record Type: Journal Article
• Authors: Dedecker, Jérôme; Samson, Adeline; Taupin, Marie-Luce
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 277
• Page End: 307
• Publication Date: 2014-10-03
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013037
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>Consider an autoregressive model with measurement error: we observe <italic>Z</italic><sub><italic>i</italic></sub> = <italic>X</italic><sub><italic>i</italic></sub> + <italic>ε</italic><sub><italic>i</italic></sub>, where the unobserved <italic>X</italic><sub><italic>i</italic></sub> is a stationary solution of the autoregressive equation <italic>X</italic><sub><italic>i</italic></sub> = <italic>g</italic><sub><italic>θ</italic><sup>0</sup></sub>(<italic>X</italic><sub><italic>i</italic> âˆ' 1</sub>) + <italic>ξ</italic><sub><italic>i</italic></sub>. The regression function <italic>g</italic><sub><italic>θ</italic><sup>0</sup></sub> is known up to a finite dimensional parameter <italic>θ</italic><sup>0</sup> to be estimated. The distributions of <italic>ξ</italic><sub>1</sub> and <italic>X</italic><sub>0</sub> are unknown and <italic>g</italic><sub><italic>θ</italic></sub> belongs to a large class of parametric regression functions. The distribution of <italic>ε</italic><sub>0</sub> is completely known. We propose an estimation procedure with a new criterion computed as the Fourier transform of a weighted least square contrast. This procedure provides an asymptotically normal estimator <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\hat \theta$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgr3rhj" xmlns:xlink="http://www.w3.org/1999/xlink" /><textual-form><italic>θ̂</italic></textual-form></alternatives></inline-formula> of <italic>θ</italic><sup>0</sup>, for a large class of regression functions and various noise distributions.</p> </abstract>