Abstract: The foundations of a general theory of statistical decision functions, including the classical non-sequential case as well as the sequential case, was discussed by the author in a previous publication [3]. Several assumptions made in [3] appear, however, to be unnecessarily restrictive (see conditions 1-7, pp. 297 in [3]). These assumptions, moreover, are not always fulfilled for statistical problems in their conventional form. In this paper the main results of [3], as well as several new results, are obtained from a considerably weaker set of conditions which are fulfilled for most of the statistical problems treated in the literature. It seemed necessary to abandon most of the methods of proofs used in [3] (particularly those in section 4 of [3]) and to develop the theory from the beginning. To make the present paper self-contained, the basic definitions already given in [3] are briefly restated in section 2.1.

Abstract: Professor William Hovanitz called my attention to the following problem: An observer catches butterflies, marks them, and immediately releases them. It is assumed that a butterfly, no matter how many times it has been caught before, has the same susceptibility to capture as any other butterfly in the population which is supposed stable while the captures are being made. Records are kept of f, the frequency of cases in which the same butterfly is caught x times, x = 1, 2, ..., until a total of s captures of r different butterflies have been made. (Efx = r; Zxfx = s.) The number, fo, of butterflies which escape is not observed; the problem is to estimate from the values offx the total population n of butterflies on the area assumed well defined. The estimation of biological populations by means of capture-recapture data is by no means a new problem, though papers dealing with it from the mathematical-statistical point of view are largely quite recent. (In particular, see the papers by Leslie & Chitty (1951), Bailey (1951), Moran (1951, 1952), and the bibliographies quoted by them.) However, the experimental conditions and the mathematical models for the present study arpear to differ in essential ways from those previously considered. The important point of departure is that each butterfly on being netted is immediately marked (with a spot of nail polish) and released. The butterflies (Colias eurytheme) were caught in one of two isolated alfalfa fields, which they inhabit, in southern California. Each catch was made during the same day at times when the butterflies were freely flying. Thus, it seemed reasonable to assume that the population was stable during a catch. The experimenter, Prof. Hovanitz, endeavoured to give each butterfly an equal chance of capture, walking in straight lines across the field and deviating in direction before reaching a boundary only when he noticed that a butterfly just caught tended to fly down his path. One check of the suitability of a mathematical model is to test the agreement of the experimental results with respect to the number of butterflies caught once, twice, etc., with those predicted from the model. I will return to this point at the end of the paper. Two mathematical models seem appropriate to serve as a basis for discussion of this estimation problem. It is of some interest to see that both lead to approximately the same estimates with little difference in their precision for large samples. It may be of more interest that for both models in which the population size is regarded as a parameter, though maximum likelihood estimates exist and agree substantially with moment estimates in all sixteen of the actual field experiments for which I have data, nevertheless with increasing sample size meaningful solutions of the likelihood equation do not exist.

Abstract: A two-lane road is assumed to be occupied mainly by platoons of cars travelling along the two opposite directions at constant speeds A single car is assumed to try to move at a speed higher than that of the other cars moving in its direction It can pass only when there is enough room for it without breaking up a platoon in either direction The sizes of the platoons are supposed to be given by a borel distribution (e Borel, Comptes Rendus Acad Sci Paris, vol 214, 1942, pp 452-456) An interesting result is that when the traffic flows increase beyond a certain level, appreciably below the theoretical capacity of the road, the "fast" vehicle cannot move any faster than the other vehicles It spends most of its time waiting for an opportunity to pass, and the mean waiting time until passing goes to infinity The average speed was tabulated after numerical computations corresponding to a wide range of the constants of the model These numerical results lead to two general conclusions: First, that increasing the acceleration capability even at the expense of reducing the maximum speed, (eG, by lowering the top gear ratio), would result in increased journey speeds for most modern cars under typical traffic conditions And second, that reducing the safety margins in passing would not normally provide any worthwhile increase in mean journey speed Language: en

Abstract: i=1 In certain technological applications also it is found that individual errors compound into a total error obtained by the addition of numerical values of individual errors. We may mention the following example: In the manufacture of metal rods it is found that the lengths of individual rods differ from specification by a normal error variable. Pairs of rods with lengths xl and x2 are assembled in a manner requiring the longer of the pair to be filed down to the length of the shorter one. If the cross-section of the rods is constant, the total loss of weight through filing is proportional to E X -x2 1, which is a sum of independent modnormal variates. Occasions also arise when special mechanisms give rise to a total error which is a sum of mod-normal errors or a sum of normal and mod-normal errors, although in such situations there are only two or three error components. For instance, in the manufacture of screw gauges two main errors arise, an error x in the diameter of the gauge and an error y in its pitch. These two result in a total vertical displacement of the points of contact of screw and gauge which is proportional to x + c: y l, where c depends on the pitch angle. While dealing with some special cases of correlated error variables, the present author was concerned with the evaluation of the moments of T and was thereby naturally led to the calculation of the 'absolute moments' of the multivariate normal distribution, namely,

Abstract: Finney (1948) has given a table which may be used to test the significance of the deviation from proportionality in any 2 x 2 contingency table having both the frequencies in one of its margins less than or equal to 16. The table printed below extends the range of Finney's table up to marginal frequencies of 20. As the interpretation and uses of the new table are exactly similar to those of the 1948 table, only a brief introductory statement is required.* Using Finney's notation, the contingency table should be arranged in the form