Abstract
For $x \in \mathbb{R}$ let $N_\alpha(x) := m\alpha, \operatorname{iff} x \in (\alpha m - \alpha/2, \alpha m + \alpha/2\rbrack$. For a sample $X_1,\ldots, X_n$ we mainly study the asymptotic properties of the estimators $\bar{N}_\alpha := 1/n\sum^n_{i = 1} N_\alpha(X_i)$ and $S^2_\alpha := 1/(n - 1)\sum^n_{i = 1}(N_\alpha(X_i) - \overline{N}_\alpha)^2$ for $\alpha = \alpha_n \rightarrow 0,$ as $n \rightarrow \infty.$ For example, if $E(X^2) < \infty, E(e^{itX}) = o(|t|^{-k}),(|r| \rightarrow\infty)$ for some $k \in \mathbb{N}$ and $\alpha_n = O(n^{-1/(2k + 2)})$ or $X \sim N(\theta, \sigma^2)$ and $\alpha_n \leq 2\pi\sigma(\log n)^{-1/2,}$ we prove that $\sqrt{n}(\overline{N}_{\alpha n} - EX)$ is asymptotically normal. Problems of truncation as well as general maximum likelihood estimation from discrete scale measurements are also considered.
Citation
Wolfgang Stadje. "Estimation Problems for Samples with Measurement Errors." Ann. Statist. 13 (4) 1592 - 1615, December, 1985. https://doi.org/10.1214/aos/1176349757
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