Estimation problems for samples with measurement errors (Q1074972)

For \(x\in R\), let \(N_{\alpha}(x)=m\alpha\) iff \(x\in (\alpha m-\alpha /2\), \(\alpha m+\alpha /2]\). Also let \(X_ 1,...,X_ n\) be a random sample from the pdf \(f_ X(x)\), and set \[ \bar N_{\alpha}=(1/n)\sum^{n}_{i=1}N_{\alpha}(X_ i)\quad and\quad S^ 2_{\alpha}=[1/(n-1)]\sum^{n}_{i=1}[N_{\alpha}(X_ i)-\bar N_{\alpha}]^ 2. \] The author mainly studies the asymptotic properties of \(\bar N_{\alpha_ n}\) and \(S^ 2_{\alpha_ n}\), as \(n\to \infty.\) For example, if \(E(X^ 2)<\infty\), \(E(e^{itX})=o(| t|^{-k})\) (as \(| t| \to \infty)\) for some \(k\in {\mathbb{N}}\) and \(\alpha_ n=O(n^{-1/(2k+2)})\), or X is distributed as \(N(\theta,\sigma^ 2)\) and \(\alpha_ n\leq 2\pi \sigma (\log n)^{- 1/2}\), the author shows that \(n^{1/2}[\bar N_{\alpha_ n}-E(X)]\) is asymptotically normal. The author also considers problems of truncation and general maximum likelihood estimation from discrete scale measurements.