636 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-19, NO. 5, SEPTEMRER 1973 Estimating a Binomial Parameter with Finite Memory FRANCISCO J. Abstract-This article treats the asymptotic theory of estimating a binomial parameter p with time-invariant finite memory. The approach taken to this problem is as follows. A decision rule is a pair &a) in which t fixes the transition function of a 6nite automaton, and (I is a vector of estimates of p. Attention is restricted to automata whose transition functions allow transitions only between adjacent states. Rules (&a) for which t satisfies this restriction are termed tridiagonal. For the class of prior distributions on [O,l] which have continuous density func- tions, we study the performance of a corresponding class of tridiagonal rules {(t*,a*)} relative to quadratic loss functions. These rules display sensitivity to the shape of the prior, and have the advantage that the Bayes estimate a* (given t*) is easily computed. Within the class of all tridiagonal rules, a particular rule (t *,a*) is shown, for memory size up to 30, to he locally admissible and minimax as well as locally Bayes with respect to the uniform prior. I. INTRODUCTION AND SUMMARY C ONSIDER a statistical problem in which we observe a sequence of random variables X1, * * *,X,. Suppose these observations are treated as inputs for a k-state stochastic automaton. If T(i) represents the state of the automaton after i observations (inputs), then an algorithm of the form T(i) = f(T(i - l),Xi) (1) updates the state of the automaton with each observation. A decision rule based on X1; . *,X,, is said to have finite memory if it dependson X1, - * *,X,, only through the value of T(n). Tests of simple hypotheses with finite memory were treated by Hellman and Cover [6]. In this paper, we propose to study estimation with finite memory. We limit ourselvesto the following special prob- lem. Let {Xi} represent a sequence of independentBernoulli trials with probability p of success. We treat {Xi} as inputs for a finite automaton with state spaceS = {0,1,2,* - -,m}r and stochastic transition function t. We assume that the initial distribution on S is given and that the automaton is time invariant and operates in discretetime. Thus our study differs from that of Roberts and Tooley [7] in which time- varying automata are considered. After n-observations, or Manuscript received June 22, 1972; revised February 20, 1973. This work was supported in part by the National Science Foundation under Grant GP-17868 at the University of California, Los Angeles, Calif., and by the National Science Foundation under Grant GU-2612 at the Florida State University, Tallahassee, Fla. The author is with the Department of Mathematics, University of California, Davis, Calif. 95616. r For the sake of simplifying the formulas and equations that appear in the sequel, we deal with m + 1 state machines rather than m state machines. (This notation differs from that in [2] and [6], for example.) S will eventually play the role of the range for certain discrete random variables, including a binomially distributed variable for which a range of the form (0,. . . ,k} is traditional. SAMANIEGO after “an infinite sample,” a decision rule a (with domain S) is chosen. For each x E S, a, is simply a point 2 in the unit interval [O,l]. We discussin the next section a method of selecting a pair (t,u), and investigate various properties of the chosen pair. Let us first outline the structure of the estimation problem to be studied here. In doing so we will suppress measure-theoretic considerationsand describethis structure in an intuitive fashion. The parameter space P in the binomial problem will be the open or closed unit interval. The action space ~4, or space of estimates,is the closed interval [O,l]. Let T be the set of all possible stochastic transition functions on S (or a subsetof theseto which we might wish to restrict considera- tion). A loss function L is a nonnegative-valued function defined on P x T x d. Suppose we have an S-valued random variable X whose distribution function belongs to a family of distributions indexed by p E P and t E T. Then for any a E [O,l]“” = &, we may define the risk function of the pair (t,u) by RMtdl = i. ~Cp,4df-(~ I P,0 (2) where f is the probability mass function of the random variable X. It is the risk or expected loss function displayed in (2) which will be used to compare different pairs (t,u). The adjectives admissible, equalizer, minimax, and Bayes are used in the sequelin describing a particular choice (t,u), and the reader unfamiliar with theseconceptsor the funda- mental theorems relating to them is referred to [3, ch. 21. The random variable X in which we are interested is the random variable that arises naturally when we summarize a sample of binomial random variables by a (m + 1)-state stochastic automaton. The transition function of such an automaton is completely specified by a pair M,,M, of (m + 1) x (m + 1) stochastic matrices (corresponding to the two possible outcomes 0 and 1). The (i,j) entry in Me, for example, represents the probability of a transition from state i to statej given the observation 0. If c is the initial distribution on S, and A4 = pM1 + (1 - p)MrJ where P(X, = 1) = p, then the probability mass function of X basedon a sampleof size12 is given by the components 2 We choose a to be nonrandomized. That this causes no loss of generality in our results is easily established by the usual decision- theoretic arguments.