Three examples will illustrate these points: (1) An investigator may wish to estimate to within 10 per cent the mean weekly expenditure on tobacco per household. Alternatively, he may have no intrinsic interest in the intermediate results but may be able to achieve economy in sample size by taking them into account. The investigator might wish to have an up-to-date record at any stage, either for general information or because the appropriate sample size depends on quantities that he can estimate only from the data themselves. A sequential procedure might be desirable for various reasons. In a sequential investigation observations must be examined either one by one as they are collected or at certain stages during collection. For example, preliminary experience in an experiment may suggest changes in the treatments being compared in a social survey a small pilot survey may lead to modifications in the design of the main investigation. 'The term 'sequential' is occasionally extended to cover also investigations in which various aspects of the design may be changed according to the observations made. The motivation for most sequential investigations is that when the ends achieved are measured against the costs incurred (including the cost of making observations), sequential designs are typically more efficient than non-sequential designs. 'Sequential analysis is the branch of statistics concerned with investigations in which the decision whether or not to stop at any stage depends on the observations previously made. All this, plus many special problems, were gathered together in Sequential Analysis (1947), a book surprisingly easy to read, less formal and more elementary in structure than his work on decision functions' (DSB). He found operating characteristic and average sample number functions he introduced, if he did not completely solve, the problem of sequential tests of composite hypotheses (utilizing weight functions) and he began vital discussions of such basic topics as multivalued decisions and optimal sequential estimation. He introduced the particular method of the sequential probability ratio test and, with Wolfowitz (1948), showed its optimal properties. But again it was Wald, in 1943, who first formulated mathematically and solved quite generally the problem of sequential tests of statistical hypotheses. This notion and the partial mathematical formulation of it were to be found in the statistical literature among those who dealt with it before Wald was Walter Bartky of Chicago, and among Wald's contemporaries, George Barnard, working in England. Intuitive support for this notion is immediate if the evidence shown in sequentially unfolding data is sharply one-sided, it seems reasonable to believe that the inquiry can be terminated early, with lengthier inquiries reserved for those situations in which the issue at hand appears, via the sequentially unfolding data, to be in greater doubt. The notion that in some sense it is economical to observe and analyze data sequentially, rather than to observe and analyze a single sample of predetermined fixed size, was not a new one. 'Wald's second major achievement in mathematical statistics is sequential analysis. It is here offered in the original 'restricted' reports it was published four years later in his well-known book of the same title. First edition, very rare, of Wald's seminal invention of 'Sequential analysis', developed while he was in charge of Columbia University's Statistical Research Group in response to the demand for more efficient methods of industrial quality control during World War II. THE INVENTION OF SEQUENTIAL ANALYSIS - THE ORIGINAL RESTRICTED REPORTS.

Sequential analysis statistics