A median-unbiased estimator of the \(AR(1)\) coefficient (Q2703248)

The paper is concerned with the median-unbiased estimation of the stationary first-order autoregressive process NEWLINE\[NEWLINE X_{t}=\alpha X_{t-1}+Y_{t}, NEWLINE\]NEWLINE with independent innovations \(Y_{t}\) and \(t=\dots,-1,0,1,\dots\) The problem is to estimate \(\alpha\) from an observed segment \(X_{1}, X_{2},\dots, X_{n}\) of the process \(X_{t},\) where \(n\) is fixed (non-asymptotic approach). There exist only two papers on the subject which contain some constructive results, namely that of \textit{L. Hurwicz} [T.C. Koopmans (ed.), Statistical inference in dynamic economic models, 365-383 (1950); see also ibid., 329-344] and that of \textit{D.W.K. Andrews} [Econometrica 61, No. 1, 139-165 (1993; Zbl 0772.62064)]. Both are concerned with the case when the \(Y_{t}\) are independent and identically distributed normal \(N(0,1)\) r.v.s.NEWLINENEWLINENEWLINEThe aim of this note is to prove that the Hurwicz conjecture concerning median-unbiasedness is true whenever the medians of independent (not necessarily identically distributed) innovations \(Y_{1}, Y_{2},\dots,Y_{n}\) are equal to zero. It follows that the Hurwicz estimator is median-bias robust against heavy tails of innovations as well as against \(\epsilon\)-contamination with contaminants symmetric around zero.