Recent Taxes on Mathematical Statistics

Abstract

STATISTICS books do not fall from heaven at random. They come in runs. For a shockingly long period no adequate textbooks on modern statistics were available, so that the interested student had to refer directly to the technical journals and to pursue each topic in many loosely connected installments. All that is changed now; a number of first-rate books have appeared and more are promised every day. Postwar periods seem to offer a favorable time for the consolidation of knowledge. In the I920's appeared such excellent works as Whittaker and Robinson, The Calculus of Observations, the Rietz symposium, Handbook of Mathematical Statistics, Arne Fisher, Mathematical Theory of Probability, and a number of other works. Good as these were in their day and valuable as parts of them still remain, it is nonetheless true that they do not present a fair picture of modern statistical theory. R. A. Fisher's Statistical Methods for Research Workers-whose first edition goes back to this period -remains an indispensable book; but there has always been something anomalous about the situation in which students and workers had to depend upon a cook-book, however useful the recipes contained therein; and no student can do justice to the truly fundamental contribution of R. A. Fisher without going back to his numerous research memoirs. Textbooks are important to the wives and children of their authors, but they have an importance that goes beyond that. Nowhere is this more important than in the field of statistics, which represents a house divided against itself. The bone of contention is of course nathematics. This subject rears its ugly head even in a field like economics. These days a young student can aspire to become a first-rate economic theorist without knowing any mathematics, but this is doing the job the hard way and he must be just that much more brilliant to make his mark. Still he can do it. In the field of statistical theory, this is simply not possible. Statistical theory is a branch of applied mathematics and no one can get anywhere in the subject without recognizing this basic fact. Moreover, the amount of mathematics needed in statistics is immeasurably greater than that required for even mathematical economics: the statistician must know the properties of many basic mathematical functions (exponential, trigonometric, Fourier transforms, and worse), not just be able to talk about f (x). The economist can go most of the way on the mathematics in R. G. D. Allen, Mathematical Analysis for Economists, plus a little matrix algebra and difference equations; but not so the statisticians. All this is realized by the members of the Institute of Mathematical Statistics, who in recent years have set up committees to remedy the present deplorable state of the teaching of college statistics. At many institutions this is assigned to young men, who teach statistics not because they like to do so or are prepared to teach the subject, but simply because they are young and at the bottom of the ladder and unable to avoid the task. There is no shortage of elementary textbooks on descriptive statistics to make life tolerable for the student and barely tolerable for the instructor; indeed the latter may choose between psychological, economic, business, educational, and biological statistics texts. It would not be true to say that these texts contain no mathematics; invariably they have a table of logarithms and squares, and a number of error formulas, of which many are usually wrong. As a genus, these works fail to distinguish between sample statistics (e.g., sample standard deviation) and population parameter (e.g., standard deviation of the universe sampled); they are shaky in the concept of degrees of freedom, and where they introduce Student's distribution, chi square, and other so-called small sampling theory, it is often too obvious where this discussion has