Estimation of the mean with time-varying finite memory (Corresp.)

Author

Wagner, Terry J.

Volume

Issue

fYear

1972

fDate

7/1/1972 12:00:00 AM

Firstpage

523

Lastpage

525

Abstract

Let $X_1,X_2,\\cdots$ be a sequence of independent identically distributed observations with a common mean $\\mu$ . Assume that $0 \\leq X_i \\leq 1$ with probability 1. We show that for each $\\varepsilon > 0$ there exists an integer $m$ , a finite-valued statistic $T_n = T_n(X_1, \\cdots , X_n) \\in {t_1,\\cdots ,t_m}$ and a real-valued function $d$ defined on ${t_1,\\cdots ,t_m}$ such that $i$ ) $T_{n+1} = f_n(T_n,X_{n+1})$ ; ii) $P[\\lim \\sup \\mid d(T_n) - \\mu \\mid \\leq \\varepsilon ] = 1$ . Thus we have a recursive-like estimate of $\\mu$ , for which the data are summarized for each $n$ by one of $m$ states and which converges to within $\\varepsilon$ of $\\mu$ with probability 1. The constraint on memory here is time varying as contrasted to the time-invariant constraint that would have $T_{n+1} = f(T_n, X_{n+1})$ for all $n$ .

Keywords

Estimation; Finite-memory methods; Band pass filters; Bandwidth; Binary codes; Electrons; Gaussian processes; Pulse modulation; Quantization; Sampling methods; Solids; Telecommunications;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1972.1054853

Filename

1054853

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=917874