The Annals of Mathematical Statistics

The Efficiency of Sequential Estimates and Wald's Equation for Sequential Processes

J. Wolfowitz

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Abstract

Let $n$ successive independent observations be made on the same chance variable whose distribution function $f(x, \theta)$ depends on a single parameter $\theta$. The number $n$ is a chance variable which depends upon the outcomes of successive observations; it is precisely defined in the text below. Let $\theta^\ast(x_1, \cdots, x_n)$ be an estimate of $\theta$ whose bias is $b(\theta)$. Subject to certain regularity conditions stated below, it is proved that $\sigma^2(\theta^\ast) \geq \big(1 + \frac{db}{d\theta}\big)^2\big\lbrack EnE\big(\frac{\partial\log f}{\partial\theta}\big)^2\big\rbrack^{-1}.$ When $f(x, \theta)$ is the binomial distribution and $\theta^\ast$ is unbiased the lower bound given here specializes to one first announced by Girshick [3], obtained under no doubt different conditions of regularity. When the chance variable $n$ is a constant the lower bound given above is the same as that obtained in [2], page 480, under different conditions of regularity. Let the parameter $\theta$ consist of $l$ components $\theta_1, \cdots, \theta_l$ for which there are given the respective unbiased estimates $\theta^\ast_1(x_1, \cdots, x_n), \cdots, \theta^\ast_1(x_1, \cdots, x_n)$. Let $\|\lambda_{ij}\|$ be the non-singular covariance matrix of the latter, and $\|\lambda^{ij}\|$ its inverse. The concentration ellipsoid in the space of $(k_1, \cdots, k_l)$ is defined as $\sum_{i,j} \lambda^{ij}(k_i - \theta_i)(k_j - \theta_i) = l + 2.$ (This valuable concept is due to Cramer). If a unit mass be uniformly distributed over the concentration ellipsoid, the matrix of its products of inertia will coincide with the covariance matrix $\|\lambda)_{ij}\|$. In [4] Cramer proves that no matter what the unbiased estimates $\theta^\ast_1, \cdots, \theta^\ast_l$, (provided that certain regularity conditions are fulfilled), when $n$ is constant their concentration ellipsoid always contains within itself the ellipsoid $\sum_{i,j} \mu_{ij}(k_i - \theta_i)(k_j - \theta_j) = l + 2$ where $\mu_{ij} = nE\big(\frac{\partial\log f}{\partial\theta_i}\frac{\partial\log f}{\partial\theta_i}\big).$ Consider now the sequential procedure of this paper. Let $\theta^\ast_1, \cdots, \theta^\ast_l$ be, as before, unbiased estimates of $\theta_1, \cdots, \theta_l$, respectively, recalling, however, that the number of $n$ of observations is a chance variable. It is proved that the concentration ellipsoid of $\theta^\ast_1, \cdots, \theta^\ast_l$ always contains within itself the ellipsoid $\sum_{i,j} \mu'_{ij}(k_i - \theta_i)(k_j - \theta_j) = l + 2$ where $\mu'_{ij} = EnE\big(\frac{\partial\log f}{\partial\theta_i}\frac{\partial\log f}{\partial\theta_j}\big).$ When $n$ is a constant this becomes Cramer's result (under different conditions of regularity). In section 7 is presented a number of results related to the equation $EZ_n = EnEX$, which is due to Wald [6] and is fundamental for sequential analysis.

Article information

Source
Ann. Math. Statist., Volume 18, Number 2 (1947), 215-230.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177730439

Digital Object Identifier
doi:10.1214/aoms/1177730439

Mathematical Reviews number (MathSciNet)
MR21288

Zentralblatt MATH identifier
0032.04203

JSTOR
links.jstor.org

Citation

Wolfowitz, J. The Efficiency of Sequential Estimates and Wald's Equation for Sequential Processes. Ann. Math. Statist. 18 (1947), no. 2, 215--230. doi:10.1214/aoms/1177730439. https://projecteuclid.org/euclid.aoms/1177730439


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