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June, 1987 Estimating Trajectories
Yunshyong Chow
Ann. Statist. 15(2): 552-567 (June, 1987). DOI: 10.1214/aos/1176350360

Abstract

Let $f$ be a continuously differentiable function from [0, 1] to the complex plane. Suppose that "at time $n$" we are given the random set $\{f(k/n) + e_{n, k}: 1 \leq k \leq n\}$, where the random errors $e_{n, k}$ are i.i.d. and $(\mathrm{Re} e_{n, 1}, \mathrm{Im} e_{n, 1})$ is $N \big( (0, 0), \sigma^2 \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} \big)$ with $\sigma^2$ known. We do not know which datum belongs to which position $\theta = k/n, k = 1, 2, \cdots, n$. In general, $f$ cannot be determined. In this paper it is shown that a random set $T_n$ can be constructed such that with probability one, $T_n$ converges in the Hausdorff sense to the trajectory $f(\lbrack 0, 1 \rbrack)$.

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Yunshyong Chow. "Estimating Trajectories." Ann. Statist. 15 (2) 552 - 567, June, 1987. https://doi.org/10.1214/aos/1176350360

Information

Published: June, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0629.62042
MathSciNet: MR888425
Digital Object Identifier: 10.1214/aos/1176350360

Subjects:
Primary: 62G05
Secondary: 62J99

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 2 • June, 1987
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