Estimation Problems for Samples with Measurement Errors

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.