Sequential Analysis: A Survey

AFTER a short period of relative quiescence, the complex of statistical techniques known as "sequential analysis" has once again begun to attract a steadily increasing amount of interest. These techniques are not only interesting on account of the type of mathematical problems they pose (many of which are not yet solved in a completely satisfactory manner): the results obtained have enabled some statisticians to use new modes of sampling and analysis which can lead to important savings in money and time; and the existence of the techniques has broadened the outlook of many statisticians, even though they have never had occasion to use a formal sequential procedure. The bulk of these techniques have been published and used in the relatively short period of fifteen years, since 1945. A fairly detailed treatment of the subject is therefore possible within the compass of a Journal article. For most other statistical topics, there are longer histories, or more rapid and complicated development, and a correspondingly intensive treatment would not be possible without producing a text more appropriate to a book. It is as well to be clear, from the outset, as to the sense in which the term "sequential" will be understood in this article. It will be understood to apply to any statistical procedure in which the final pattern (including the number) of observations is not determined a priori but depends, in some way or other, on the values observed in the course of the work. This is a widely inclusive definition, but most attention will be concentrated on those techniques in which the sequential element is well in evidence; in particular, on techniques embodying a "stopping rule", so that they include directions on when to cease the investigation. For the sake of completeness a fundamental paper by Box and Wilson (1951) on what may be termed a "sequential approach" to the maximum of a function has been