Showing papers in "Annals of Mathematical Statistics in 1930"

The Sampling Variability of Linear and Curvilinear Regressions: A First Approximation to the Reliability of the Results Secured by the Graphic "Successive Approximation" Method

Abstract: Many statistical problems involve determining the change in one varial!with changes in each of several others, all operating at the same time. I inear multiple correlation provides a method of making this determination, on the assumption that all the relations are linear. In many problems this assumption is not valid. To determine curvilinear relations without making assumptions as to the type of each curve except that it be a continuous function, a method of successive approximations by graphic fitting was presented six years ago; and it was demonstrated empirically that in cases of high correlation this method successfully determined the underlying curves.2 It was also pointed out that multiple regression curves could be fitted by the leastsquares method. if specific parabolae or other first-degree equations were assumed for each variable, following methods previously sugsuggested by Yule.3

Remarks on Regression

Abstract: 1. In a paper published twelve years ago1 I derived a set of formutilae for bivariate regression which were found to give good results on uiilimodal materials of a fairly general nature and which, in the case of mo(lerately skew (listributions, were reduced-to very simple andl easily applicable formls. Two years later I extended the thegry a lso to the case of muitltiple correlations of similar tviles2. TFhese formulae were deduced on the assumption that the correlation surfalce could be expressed by a so-called series of type A3, i. e. that the dev iations from tlhe best fitting normal surface could be eN>l)ressc(l .ix a series, developed according to the derivatives of differenit orders of the Bravais ftunctioni, exp)reSsing that normal surface.

Stieltjes Integrals in Mathematical Statistics

Abstract: Introduction. Stieltjes integrals, introduced into analysis in 1894-51, play an increasingly important role not only in pure mathematics, but also in theoretical physics and in the theory of probability. In mathematical statistics, however, their use, it seems, still remains very limited. And yet, one of the most remarkable features of Stieltjes integrals is that they represent, as the case may be, an integral proper or a sum of an finite or an infinite number of discrete aggregates. Thus the statistician is enabled to treat in a single formula a continuous, as well as a discontinuous distribution. This means far more than a mere simplification of writing. In fact, since Stieltjes integrals have many properties in common with Riemann and Lebesgue definite integrals, we can use all known resources of the theory of definite integrals (mean-value theorem, various inequalities), and therefore readily obtain general results which, otherwise, require special (often complicated) proofs. The advantage of such a treatment is particularly evident in the theory of interpolation, approximation, and mechanical quadratures.