The Annals of Mathematical Statistics

Approximation Methods which Converge with Probability one

Julius R. Blum

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Abstract

Let $H(y\mid x)$ be a family of distribution functions depending upon a real parameter $x,$ and let $M(x) = \int^\infty_{-\infty} y dH(y \mid x)$ be the corresponding regression function. It is assumed $M(x)$ is unknown to the experimenter, who is, however, allowed to take observations on $H(y\mid x)$ for any value $x.$ Robbins and Monro [1] give a method for defining successively a sequence $\{x_n\}$ such that $x_n$ converges to $\theta$ in probability, where $\theta$ is a root of the equation $M(x) = \alpha$ and $\alpha$ is a given number. Wolfowitz [2] generalizes these results, and Kiefer and Wolfowitz [3], solve a similar problem in the case when $M(x)$ has a maximum at $x = \theta.$ Using a lemma due to Loeve [4], we show that in both cases $x_n$ converges to $\theta$ with probability one, under weaker conditions than those imposed in [2] and [3]. Further we solve a similar problem in the case when $M(x)$ is the median of $H(y \mid x).$

Article information

Source
Ann. Math. Statist., Volume 25, Number 2 (1954), 382-386.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728794

Digital Object Identifier
doi:10.1214/aoms/1177728794

Mathematical Reviews number (MathSciNet)
MR62399

Zentralblatt MATH identifier
0055.37806

JSTOR
links.jstor.org

Citation

Blum, Julius R. Approximation Methods which Converge with Probability one. Ann. Math. Statist. 25 (1954), no. 2, 382--386. doi:10.1214/aoms/1177728794. https://projecteuclid.org/euclid.aoms/1177728794


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