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The purpose of this note is to indicate a direct and natural way of proving theorems stated in [4] by using an explicit expression for the sequence of normalizing constants belonging to a distribution function attracted to a stable law. This results in a remark concerning a counter example given by Tucker and a slightly sharpened version of his Lemma 5.

In a recent paper [1] the authors began the study of the theory of sequential decision functions for stochastic processes with a continuous time parameter. This paper treated the standard problem of testing hypotheses, and the advantage of being able to stop at an arbitrary time point (not necessarily a multiple of some unit given in advance) was demonstrated in several cases, notably in that of deciding between two Poisson processes. The optimal tests were Wald probability ratio tests and thus truly sequential. In the present paper we treat the problem of estimation, and study in detail the Poisson, Gamma, Normal and Negative Binomial processes. It turns out for these processes that, with a proper weight function, the minimax (sequential) rule reduces to a fixed-time rule. Though we confine ourselves to point-estimation it is clear that similar methods apply to interval estimation. It may also be remarked that the case when the time-parameter is discrete need not be treated separately. For example, as described in Section 6.1, the results of Sections 2 and 3 imply analogous results in the case of discrete time, which in turn imply certain results proved in [3] and (in the nonsequential case) in [2] by other methods. The treatment of some other problems in estimation is discussed in Section 6. This paper may be read independently of [1].

In recent years the study of sequential procedures which are asymptotically optimum in an appropriate sense as the cost $c$ per observation goes to zero has received considerable attention. On the one hand, Schwarz (1962) has recently given an interesting theory of the asymptotic shape, as $c \rightarrow 0$, of the Bayes stopping region relative to an a priori distribution $F$, for testing sequentially between two composite hypotheses $\theta \leqq \theta_1$ and $\theta \geqq \theta_2$ concerning the real parameter $\theta$ of a distribution of exponential (Koopman-Darmois) type, with indifference region the open interval $(\theta_1, \theta_2)$. (An example of Schwarz's considerations is described in connection with Figure 4.) One aim of the present paper is to generalize Schwarz's results to the case where (with or without indifference regions) the distributions have arbitrary form and there can be more than two decisions (Sections 2, 3, 4). In this general setting we obtain, under mild assumptions, a family $\{\delta_c\}$ of procedures whose integrated risk is asymptotically the same as the Bayes risk. (In fact, extending Schwarz's result, a family $\{\delta'_c\}$ can be constructed so as to possess this asymptotic Bayes property relative to all a priori distributions with the same support as $F$, or even with smaller indifference region support than $F$.) Procedures like our $\{\delta_c\}$ have already been suggested by Wald (1947) for use in tests of composite hypotheses (e.g., the sequential $t$-test), but his concern was differently inspired. At the same time, we show how such multiple decision problems can be treated by using simultaneously a number of sequential tests for associated two-decision problems. A second aim is to extend, strengthen, and somewhat simplify the asymptotic sequential design considerations originated by Chernoff (1959) and further developed by Albert (1961) and Bessler (1960) (Section 5). Our point of departure here is a device utilized by Wald (1951) in a simpler estimation setting, and which in the present setting amounts to taking a preliminary sample with predesignated choice of designs and such that, as $c \rightarrow 0$, the size of this preliminary sample tends to infinity, while its ratio to the total expected sample size tends to zero. The preliminary sample can then be used to "guess the true state of nature" and thus to choose the future design pattern once and for all rather than to have to reexamine the choice of design after subsequent observations. (In Wald's setting the only "design" problem was to pick the size of the second sample of his two-sample procedure.) The properties of the resulting procedure can then be inferred from the considerations of Sections 2, 3, and 4, where there is no design problem but where most of the work in this paper is done; using Wald's idea, we thereby obtain procedures for the design problem fairly easily, once we have the (non-design) sequential inference structure to build upon. The family $\{\delta^\ast_c\}$ so obtained has the same asymptotic Bayes property as that described above for the family $\{\delta_c\}$ of the non-design problem. Furthermore, a family $\{\delta^{\ast\ast}_c\}$ can be constructed so that, like $\{\delta'_c\}$ in the non-design problem, it is asymptotically Bayes for all a priori distributions with the same support. The value of the asymptotic Bayes risk of such a family is closely related to the lower bound which was obtained by Chernoff et al for the risk function of certain procedures, and which gives another form for the optimality statement. The role of the sequential procedures considered by Donnelly (1957) and Anderson (1960) for hypothesis testing with an indifference region is indicated at the end of Section 1. Asymptotic solutions to the problem of Kiefer and Weiss (1957) are given. An Appendix contains proofs of certain results on fluctuations of partial sums of independent random variables, which are used in the body of the paper.