Showing papers in "Annals of Mathematical Statistics in 1931"

The Generalization of Student’s Ratio

Abstract: The accuracy of an estimate of a normally distributed quantity is judged by reference to its variance, or rather, to an estimate of the variance based on the available sample. In 1908 “Student” examined the ratio of the mean to the standard deviation of a sample.1 The distribution at which he arrived was obtained in a more rigorous manner in 1925 by R.A. Fisher,2 who at the same time showed how to extend the application of the distribution beyond the problem of the significance of means, which had been its original object, and applied it to examine regression coefficients and other quantities obtained by least squares, testing not only the deviation of a statistic from a hypothetical value but also the difference between two statistics.

On Symmetric Functions and Symmetric Functions of Symmetric Functions

Abstract: The study of symmetric functions is quite an old one. From the time of Girard (1629) even up to the present day this sub. ject has occupied the attention of many eminent mathematicians. The theory of the roots of algebraic equations in one or more variables has furnished the chief incentive for the development of the theory of symmetric functions. Ingenious methods for computing symmetric functions in terms of what are called the elementary symmetric functions have been developed by Hammond, Brioschi, Junker, Dresden and others. Extensive tables of symmetric functions in terms of the elementary symmetric functions may be found in the literature. Symmetric functions play such a pre-eminenit role in the mathenatical theory of statistics and their computation by direct methods or by general formulas, even when assumptions restricting the groupings of the variates about the various means are made, is so excessively tedious that there has seemed to be need

On A Property of the Semi-Invariants of Thiele

Abstract: in which ) W is the r'th semi-invariant of xi. This is perr haps the niost important and useful property of semi-invariants. Each semi-invariant is defined as a certain isobaric function of the moments of weight equal to the order of the semi-invariant. The question to which this note is devoted is whether among such isobaric functions, the property given above belongs uniquely to the semi-invariant. This problem is equivalent to another which

Bayes' Theorem: An Expository Presentation

Abstract: Bayes' theorem made its appearance as the ninth proposition in an essay which occupies pages 370 to 418 of the Philosophical Transactions, Vol. 53, for 1763. An introductory letter written by Richard Price, “Theologian, Statistician, Actuary and Political Writer,”1 begins thus:

The Use of the Relative Residual in the Application of the Method of Least Squares

Abstract: The method of least squares offers a precise method ofc fitting a curve describing the relation between two or more related, measurable variables, but certain criteria must be fulfilled to justify its application. First, the type of equation selected for fitting must be the true mathematical expression of the law governing the relationship of the variables. Secondly, all errore of measuremeiit, made in obtainiing the observed values of the variables when the data were collected, must be distributed according to the wellknown laws of probability.' This paper is concerned with the latter of these two criteria. The fundameintal theory upon which the method of least squares is based can be found in any text-book on the subject and need not be elaborated upon here. However, it may be well to point out a very pertinent, if somewhat elenientary, aspect of the theory which facilitates the ready visualization of the fundamental concepts involved.