Showing papers in "Annals of Mathematical Statistics in 1932"

On the Sampling Distribution of the Multiple Correlation Coefficient

Abstract: The problem of finding the distribution of the, multiple correlation coefficient in samples from a normal population with a non-zero multiple correlation coefficient was solved in 1928 by Fisher' by the application of geometrical methtds. In his derivation he used the facts that the population value p of the multiple correlation cQefficient is invariant under linear transformations.of the independent variates, and that the distribution of the multiple correlation coefficient is independent of all populationi parameters except p . In this paper it will be shown that the distribution of the multiple correlation coefficient can be derived directly from Wishart's2 generalized product moment distribution without making use of geometrical notions and the property bf the invariance of p under linear transformations of the independent variates. Furthermore, it will not be necessary to show that the distributioni will be independent of all population parameters except p . The population value of the multiple correlation coefficient between a variate x and a set of variates x, X x iS the ordinary correlation coefficient between x, and that linear function of the variates x2, x3, . Z.which will make this correlationi a maximum. It can be expressed as p2=J a Al where A is the determinant of the correlations among all of the

The Simultaneous Distribution of Mean and Standard Deviation in Small Samples

Abstract: If, however. the parent population is other than the normal type. thlere appears to be little known regarding the form of FKX, s). In the present paper, we propose to determine the simultaneous frequeney function of the arithmetic mean and standard deviation ;i1 samples of small numbers of items selected at random from a rather arbitrary universe. For convenience, we shall classify frequeiicy distributions according as the range of the independenit variable is (oo ), 0 ' ) or (e,4a)a< We shall further aissume that the total area under the distribution function is unity. 2. The simultaneous distributioin of x and s in samples of ni 2. Let f(x), -wx < oo be the fiequency unction of the variable x Let x, and xz be two independentl erved values of X . write

A Coefficient of Linear Correlation Based on the Method of Least Squares and the Line of Best Fit

Abstract: Given N points in a plane, corresponding to N pairs of values for two variables, X and Y , we find the line of best fit and the line of wtorst fit, by the method of least squares*. Then we derive a cofficient of correlation based on the sum of the squares of the distances of the points from these two lines. The line of best fit is in the line such that the sum of the squares of the distances of the points from it is a minimum. The line of wtporst fit is the one from which the sum of the squares of the distances of the points is a maximum. We shall refer to them, respectively, as the ntinimum and maxinum lines. For convenience we take the origin at the centre of gravity of the points, letting x and y denote deviations of X and Y , respectively, from their arithmetic means. 1. The two lines pass thru the arithmetic means of the X's

Concerning the Limits of a Measure of Skewness

Abstract: Mathematicians from 37 countries to the number of approximately 650 together with 250 members of their families assembled in Zurich, Switzerland, for the Congress held September 4-12. As would be expected, the largest delegation of participants was from Switzerland (140), followed in order by Germany (111), France (68), United States and Canada (68), Italy (63), and Great Britain (38). The total attendance from this country and Canada, including members of the families of the active participants, was 104. Harvard and Brown had the largest delegations, there being four mathematicians from each. Among those who attended were the following 68 mathematicians :

A Study of the Distribution of Means Estimated From Small Samples by the Method of Maximum Likelihood for Pearson's Type II Curve

Abstract: The object of this paper is to study the distribution of estimates of the parameter of location for Pearson's Type II Curve, estimated by the method of maximum likelihood from small samples. R. A. Fisher has assumed,' and Professor Hotelling his proved2 that in large categories of cases the distribution of an optimum statistic approaches normality as the sample size increases. This normality has been assumed to hold for optimum statistics in general whether calculated from large samples or small ones, and it has also been assumed that optimum statistics have minimum variance and always give better fits than do statistics calculated by the method of moments. That this is the case whenever the sample is large and the distribution of optimum statistics normal is made plausible by the reasoning of R. A. Fisher.3 In case the sample is small, however, there may be reason to doubt that the normality of distribution of optimum statistics holds, and that the other conclusions hold. It is with this phase of the subject that we shall be concerned in what follows. Before entering into the topic under discussion it will be con-venient to review some of the more elementary facts regarding the curve with which we are to be concerned. We shall take first the general equation for the curve in the form,