Celebratio Mathematica — Blackwell

[1]D. Blackwell: “Conditional expectation and unbiased sequential estimation,” Ann. Math. Stat. 18 : 1 (1947), pp. 105–110. MR 0019903 Zbl 0033.07603 article

It is shown that \[ E\bigl[ f(x) \,E(y \mid x)\bigr] = E(fy) \] whenever \( E(fy) \) is finite, and that \[ \sigma^2 E(y \mid x) \leq \sigma^2 y ,\] where \( E(y \mid x) \) denotes the conditional expectation of \( y \) with respect to \( x \). These results imply that whenever there is a sufficient statistic \( u \) and an unbiased estimate \( t \), not a function of \( u \) only, for a parameter \( \theta \), the function \( E(t \mid u) \), which is a function of \( u \) only, is an unbiased estimate for \( \theta \) with a variance smaller than that of \( t \). A sequential unbiased estimate for a parameter is obtained, such that when the sequential test terminates after \( i \) observations, the estimate is a function of a sufficient statistic for the parameter with respect to these observations. A special case of this estimate is that obtained by Girshick, Mosteller, and Savage [1946] for the parameter of a binomial distribution.

[2]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:

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

[3]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

[4]D. Blackwell: “On the translation parameter problem for discrete variables,” Ann. Math. Stat. 22 : 3 (September 1951), pp. 393–399. MR 0043418 Zbl 0043.13802 article

For any chance variable \( x = (x_1,\dots \), \( x_N) \) having known distribution, the translation parameter estimation problem is to estimate an unknown constant \( h \), having observed \( y = (x_1+h,\dots \), \( x_N+h) \). Extending the work of Pitman [1939], Girshick and Savage [1951] have, for any loss function depending only on the error of estimate, described an estimate whose risk is a constant \( R \) independent of \( h \), and have shown that under certain hypotheses their estimate is minimax. We investigate whether the Girshick–Savage estimate is admissible, i.e., whether it is impossible to find an estimate with risk \( R(h)\leq R \) for all \( h \) and actual inequality for some \( h \). We consider only bounded discrete variables \( x \), and show that, if all values of \( x \) have all integer coordinates and if the loss \( f(d) \) from an error \( d \) is, for instance, strictly convex and assumes its minimum value, the Girshick–Savage estimate is admissible. Two examples in which the Girshick–Savage estimate is not admissible are given.

[5]D. Blackwell: “Comparison of experiments,” pp. 93–102 in Proceedings of the second Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 31 July–12 August 1950). Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1951. MR 0046002 Zbl 0044.14203 inproceedings

Bohnenblust, Shapley, and Sherman [unpublished] have introduced a method of comparing two sampling procedures or experiments; essentially their concept is that one experiment \( \alpha \) is more informative than a second experiment \( \beta \), \( \alpha\supset\beta \), if, for every possible risk function, any risk attainable with \( \beta \) is also attainable with \( \alpha \). If \( \alpha \) is a sufficient statistic for a procedure equivalent to \( \beta \), \( \alpha\succ\beta \), it is shown that \( \alpha\supset\beta \). In the case of dichotomies, the converse is proved. Whether \( \succ \) and \( \supset \) are equivalent in general is not known. Various properties of \( \succ \) and \( \supset \) are obtained, such as the following: if \( \alpha\succ\beta \) and \( \gamma \) is independent of both, then the combination \( (\alpha,\gamma)\succ(\beta,\gamma) \). An application to a problem in \( 2{\times}2 \) tables is discussed.

[6]D. Blackwell: “Equivalent comparisons of experiments,” Ann. Math. Stat. 24 : 2 (1953), pp. 265–272. MR 0056251 Zbl 0050.36004 article

[7]D. Blackwell and J. L. Hodges, Jr.: “Design for the control of selection bias,” Ann. Math. Stat. 28 : 2 (1957), pp. 449–460. MR 0088849 Zbl 0081.36403 article

Suppose an experimenter \( E \) wishes to compare the effectiveness of two treatments, \( A \) and \( B \), on a somewhat vaguely defined population. As individuals arrive, \( E \) decides whether they are in the population, and if he decides that they are, he administers \( A \) or \( B \) and notes the result, until \( nA \)’s and \( nB \)’s have been administered. Plainly, if \( E \) is aware, before deciding whether an individual is in the population, which treatment is to be administered next, he may, not necessarily deliberately, introduce a bias into the experiment. This bias we call selection bias.

We propose to investigate the extent to which a statistician \( S \), by determining the order in which treatments are administered, and not revealing to \( E \) which treatment comes next until after the individual who is to receive it has been selected, can control this selection bias. Thus a design \( d \) is a distribution over the set \( T \) of the \( \binom{2n}{n} \) sequences of length \( 2n \) containing \( nA \)’s and \( nB \)’s.

We shall measure the bias of a design by the maximum expected number of correct guesses which an experimenter can achieve, knowing \( d \), attempting to guess the successive elements of a sequence \( t \in T \) selected by \( d \), and being told after each guess whether or not it is correct. The distribution of the number \( G \) of correct guesses depends both on \( d \) and on the prediction method \( p \) used by the experimenter. We shall consider particularly two designs, the truncated binomial, in which the successive treatments are selected independently with probability \( 1/2 \) each until \( n \) treatments of one kind have occurred, and the sampling design, in which all \( \binom{2n}{n} \) sequences are equally likely.

We shall consider particularly two prediction methods, the convergent prediction, which predicts that treatment which has hitherto occurred less often, and the divergent prediction, which predicts that treatment which has hitherto occurred more often, except that after \( n \) treatments of one kind have been administered, the divergent prediction agrees with the convergent predictions that the other treatment will follow; when both treatments have occurred equally often, either method predicts \( A \) or \( B \) by tossing a fair coin, independently for each case of equality.

We find that among all designs, the truncated binomial minimizes the maximum expected number of correct guesses. For this design, the expected number of correct guesses is independent of the prediction method, and is \[ n + n \binom{2n}{n} \big/ 2^{2n} \sim n + \Bigl(\frac{n}{\pi}\Bigr)^{1/2} .\] With the truncated binomial design, the variance in the number of correct guesses is largest for the divergence strategy and is \[ \frac{3n}{2} - D - \frac{D^2}{4} \sim \frac{(3\pi - 2)n}{2\pi} - 2\Bigl(\frac{n}{\pi}\Bigr)^{1/2}, \] where \( D = n \binom{2n}{n} \big/ 2^{2n - 1} \), and is smallest for the convergence strategy, and is \[ \frac{n}{2} - \frac{D^2}{4} \sim \frac{(\pi - 1)n}{2\pi} .\] For the sampling design, convergent prediction maximizes the expected number of correct guesses; this maximum is \[ n + 2^{2n - 1} \!\big/ \binom{2n}{n} - \frac{1}{2} \sim n + \Bigl(\frac{\pi n}{4}\Bigr)^{1/2}. \] Finally we note that, if treatments are selected independently at random, bias of the kind we discuss disappears, but the treatment numbers can no longer be preassigned. Three such designs are considered: the fixed total design, in which the total number of treatments is a fixed number \( s \), the fixed factor design, in which we continue until \[ \frac{1}{X} + \frac{1}{Y} \leq \frac{2}{n} ,\] where \( X \) is the number of \( A \) treatments and \( Y \) is the number of \( B \) treatments administered, and the fixed minimum design, in which we continue until \( \min (X, Y) = n \). For the fixed total design, we find that, for \( s = 2n + 4 \), \[ \mathrm{Pr}\Bigl(\frac{1}{X} + \frac{1}{Y} \leq \frac{2}{n}\Bigr) \sim 0.955 \] for large \( n \); at the expense of 4 extra observations, we have a bias-free design whose variance factor will with probability \( 0.955 \) be smaller than that in which treatment numbers are preassigned. For the fixed factor design, the additional number of observations required to achieve the given precision has for large \( n \) the distribution of the square of a normal deviate. For the fixed minimum design, in which we guarantee precision for the estimated effect of each treatment, the expected number of additional observations is roughly \( 1.13 (n)^{1/2} \).

[8]C. B. Bell, D. Blackwell, and L. Breiman: “On the completeness of order statistics,” Ann. Math. Stat. 31 : 3 (1960), pp. 794–797. MR 0116427 Zbl 0101.12201 article

Let \( X_1, X_2, \dots, X_n \) be a sample of a one-dimensional random variable \( X \); let the order statistic \( T(X_1 \), \( X_2, \dots \), \( X_n) \) be defined in such a manner that \[ T(x_1, x_2, \dots, x_n) = (x^{(1)}, x^{(2)}, \dots, x^{(n)}) \] where \( x^{(1)} \leq x^{(2)} \leq \dots \leq x^{(n)} \) denote the ordered \( x \)’s; and let \( \Omega \) be a class of one-dimensional cpf’s, i.e., cumulative probability functions. The order statistic, \( T \), is said to be a complete statistic with respect to the class \[ \{P^{(n)} \mid P \in \Omega\} \] of \( n \)-fold power probability distributions if \[ E_p^{(n)}\bigl\{h[ T(X_1, \dots, X_n)]\bigr\} = 0 \] for all \( P \in \Omega \) implies \[ h[ T(x_1, \dots, x_n)] = 0 \quad\text{a.e. }P^{(n)} ,\] for all \( F \in \Omega \). The class \( \Omega \) is said to be symmetrically complete whenever the latter condition holds. Since the completeness of the order statistic plays an essential role in nonparametric estimation and hypothesis testing, e.g., Fraser [1954] and Bell [1960], it is of interest to determine those classes of cpf’s for which the order statistic is complete. Many of the traditionally studied classes of cpf’s on the real line are known to be symmetrically complete, e.g., all continuous cpf’s [Lehmann 1959, pp. 131–134, 152–153]; all cpf’s absolutely continuous with respect to Lebesgue measure [Fraser 1957, pp. 23–31]; and all exponentials of a certain form [Lehmann 1959, pp. 131–134]. The object of this note is to present a different [Lehmann 1959, pp. 131–134, 152–153] demonstration of the symmetric completeness of the class of all continuous cpf’s; and to extend this and other known completeness results to probability spaces other than the real line, e.g., Fraser [1954], and Lehmann and Scheffé [1950; 1955].

C. B. Bell	Related
Leo Breiman	Related

[9]P. J. Bickel and D. Blackwell: “A note on Bayes estimates,” Ann. Math. Stat. 38 : 6 (1967), pp. 1907–1911. MR 0219175 Zbl 0155.26103 article

Throughout this paper we are concerned with the problem of estimating a real parameter when the loss function is such that the Bayes estimate exists, is unique, and satisfies a simple Equation, (1.5). If the estimate is unbiased (in the general sense of [Lehmann 1951]) we show under weak conditions that it must satisfy another Equation, (1.14). The main result of Section 1, Theorem 1.3, shows that, in general, these two equations are incompatible unless the Bayes risk is 0. This extends Theorem 11.24 of [Blackwell and Girshick 1954] which states that in estimation with quadratic loss, unbiased Bayes estimates have Bayes risk 0. Some counter-examples at the end of the section indicate the limits of this incompatibility result.

[10]D. Blackwell and R. V. Ramamoorthi: “A Bayes but not classically sufficient statistic,” Ann. Statist. 10 : 3 (1982), pp. 1025–1026. MR 663456 Zbl 0485.62004 article

[11]D. Blackwell: “A hypothesis-testing game without a value,” pp. 79–82 in A Festschrift for Erich L. Lehmann. Edited by P. J. Bickel, K. A. Doksum, and J. L. Hodges. Wadsworth Statistics/Probability Series. Wadsworth (Belmont, CA), 1983. MR 689739 Zbl 0525.62006 incollection

Kjell Andreas Doksum	Related
Joseph Lawson Hodges, Jr.	Related
Erich Leo Lehmann	Related
Peter J. Bickel	Related

[12]M. Proschan: “A note on D. H. Blackwell and J. L. Hodges, Jr.: ‘Design for the control of selection bias’, and P. Diaconis and R. L. Graham: ‘The analysis of sequential experiments with feedback to subjects’,” Ann. Statist. 19 : 2 (1991), pp. 1106–1108. The paper of Diaconis and Graham is Ann. Statist. 9:1 (1981) pp. 3–23. Commentary on Ann. Math. Stat. 28:2 (1957). MR 1105868 Zbl 0747.62021 article

Ronald Lewis Graham	Related
Joseph Lawson Hodges, Jr.	Related
Michael Proschan	Related
Persi Diaconis	Related

[13]D. Blackwell: “Minimax vs. Bayes prediction,” Probab. Engrg. Inform. Sci. 9 : 1 (1995), pp. 53–58. Available open access here. MR 1336801 article

[14]K. W. Wachter, D. Blackwell, and E. A. Hammel: “Testing the validity of kinship microsimulation,” Math. Comput. Modelling 26 : 6 (1997), pp. 89–104. Zbl 0884.92034 article

Computer microsimulation is capable of generating detailed reconstructions and forecasts of the numbers of living kin that members of a population have. The available detail, specificity, and consistency of vital rates for the past and the accuracy of vital rates projected for the future necessarily limit the accuracy of the kinship estimates that can be produced.

This study is an external validity test of Reeves’ 1982 reconstruction and forecast of U.S. kin counts with the SOCSIM microsimulation program [Reeves 1982]. The external standard is provided by estimates from the 1987-1988 wave of the National Survey of Families and Households, the first detailed information on numbers and ages of kin for the United States. We compare forecasts with estimates for average numbers of living grandchildren, living full-siblings, and living half-siblings as functions of age.

We find remarkably close agreement for some of the predictions, along with several instances of systematic discrepancies. The discrepancies most likely stem from errors in the forecasts rather than in the survey estimates. Interacting rather than isolable errors appear to be responsible. Our validity tests provide a basis for qualified faith in the effectiveness of kinship microsimulation.

E. A. Hammel	Related
Kenneth Willcox Wachter	Related