Celebratio Mathematica

D. Blackwell and M. A. Girshick: “On functions of sequences of independent chance vectors with applications to the problem of the ‘random walk’ in $k$ dimensions,” Ann. Math. Stat. 17 : 3 (September 1946), pp. 310–317. MR 0017898 Zbl 0060.29007 article

Consider a sequence $ \{x_i\} $ of independent chance vectors in $ k $ dimensions with identical distributions, and a sequence of mutually exclusive events $ S_1 $, $ S_2, \dots $, such that $ S_i $ depends only on the first $ i $ vectors and $ \sum P(S_i) = 1 $. Let $ \varphi_i $ be a real or complex function of the first $ i $ vectors in the sequence satisfying conditions:

$ E(\varphi_i) = 0 $ and
$ E(\varphi_j \mid X_1, \dots, X_i) = \varphi_i $ for $ j \geq i $.

Let $ \varphi = \varphi_i $ and $ n = i $ when $ S_i $ occurs. A general theorem is proved which gives the conditions $ \varphi_i $ must satisfy such that $ E\varphi = 0 $. This theorem generalizes some of the important results obtained by Wald for $ k = 1 $. A method is also given for obtaining the distribution of $ \varphi $ and $ n $ in the problem of the ”random walk” in $ k $ dimensions for the case in which the components of the vector take on a finite number of integral values.

D. Blackwell and M. A. Girshick: “A lower bound for the variance of some unbiased sequential estimates,” Ann. Math. Stat. 18 : 2 (June 1947), pp. 277–280. MR 0020765 Zbl 0032.04401 article

Consider a sequence of independent chance variables $ x_1 $, $ x_2,\dots $ with identical distributions determined by an unknown parameter $ \theta $. We assume that $ Ex_i=\theta $ and that \[ W_k = x_1 + \dots + x_k \] is a sufficient statistic for estimating $ \theta $ from $ x_1,\dots $, $ x_k $. A sequential sampling procedure is defined by a sequence of mutually exclusive events $ S_k $ such that $ S_k $ depends only on $ x_1,\dots $, $ x_k $ and $ \sum P(S_k)=1 $. Define $ W=W_k $ and $ n=k $ when $ S_k $ occurs. In a previous paper by one of the authors [Blackwell 1947] it was shown that if \[ S_k = W_k C(S_1 + \dots + S_{k-1}) ,\] (where $ C(A) $ denotes the event that $ A $ does not occur), the function \[ V(W,n)=E(x_1\mid W,n) \] is an unbiased estimate of $ \theta $, and \[ \sigma^2(V)\leq \sigma^2(x_1) .\] It is the purpose of this note to obtain a lower bound for $ \sigma^2(V) $. Our result is:

$ \displaystyle\sigma^2(V)\geq \frac{\sigma^2(x_1)}{E(n)} $

K. J. Arrow, D. Blackwell, and M. A. Girshick: “Bayes and minimax solutions of sequential decision problems,” Econometrica 17 : 3/4 (July–October 1949), pp. 213–244. MR 0032173 Zbl 0034.07504 article

The present paper deals with the general problem of sequential choice among several actions, where at each stage the options available are to stop and take a definite action or to continue sampling for more information. There are costs attached to taking inappropriate action and to sampling. A characterization of the optimum solution is obtained first under very general assumptions as to the distribution of the successive observations and the costs of sampling; then more detailed results are given for the case where the alternative actions are finite in number, the observations are drawn under conditions of random sampling, and the cost depends only on the number of observations. Explicit solutions are given for the case of two actions, random sampling, and linear cost functions.

Kenneth J. Arrow	Related
Meyer Abraham Girshick	Related

D. Blackwell and M. A. Girshick: Theory of games and statistical decisions. John Wiley and Sons (New York), 1954. Republished by Dover in 1979. MR 0070134 Zbl 0056.36303 book

D. Blackwell and A. H. Bowker: “Meyer Abraham Girshick 1908–1955,” Ann. Math. Stat. 26 : 3 (1955), pp. 365–367. MR 0070573 Zbl 0065.24507 article

Meyer Abraham Girshick	Related
Albert H. Bowker	Related

D. Blackwell and M. A. Girshick: Theory of games and statistical decisions, reprint edition. Dover Books on Advanced Mathematics. Dover Publications (New York), 1979. Republication of 1954 original. MR 597146 Zbl 0439.62008 book

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