The Annals of Statistics

Estimation Problems for Samples with Measurement Errors

Wolfgang Stadje

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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.

Article information

Source
Ann. Statist., Volume 13, Number 4 (1985), 1592-1615.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349757

Digital Object Identifier
doi:10.1214/aos/1176349757

Mathematical Reviews number (MathSciNet)
MR811512

Zentralblatt MATH identifier
0591.62025

JSTOR
links.jstor.org

Subjects
Primary: 62F10: Point estimation
Secondary: 62E20: Asymptotic distribution theory

Keywords
Estimation from discrete scale measurements asymptotic unbiasedness and efficiency maximum likelihood estimation

Citation

Stadje, Wolfgang. Estimation Problems for Samples with Measurement Errors. Ann. Statist. 13 (1985), no. 4, 1592--1615. doi:10.1214/aos/1176349757. https://projecteuclid.org/euclid.aos/1176349757


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