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Sur les Semi-Invariants et Moments Employes dans l'Etude des Distributions Statistiques. by R. Frisch Review by: Arne Fisher Journal of the American Statistical Association, Vol. 22, No. 159 (Sep., 1927), pp. 402-404 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2276818 . Accessed: 15/06/2014 05:22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association. http://www.jstor.org This content downloaded from 185.2.32.58 on Sun, 15 Jun 2014 05:22:47 AM All use subject to JSTOR Terms and Conditions

Sur les Semi-Invariants et Moments Employes dans l'Etude des Distributions Statistiques.by R. Frisch

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Sur les Semi-Invariants et Moments Employes dans l'Etude des Distributions Statistiques. byR. FrischReview by: Arne FisherJournal of the American Statistical Association, Vol. 22, No. 159 (Sep., 1927), pp. 402-404Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2276818 .

Accessed: 15/06/2014 05:22

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

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American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journalof the American Statistical Association.

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402 American Statistical Association [132

Sur les Semi-invariants et Moments Employgs dans l'Etude des Distributions Statis- tiques, par R. Frisch. (From Transactions of the Norwegian Academy of Science) Oslo. 1926. 91 pp. In the analytical determination of the numerical values of the constants or

parameters of a statistical material or a collection of observations three types of symmetrical functions play an important r6le, viz.:

1. The well-known moments mr defined by

mr = 2(x-a)rF(x), or mr = 2xrF(x).

2. The semi-invariants xr defined by

soe r! = SO + r.

3. The factorial moments m[rl defined by x=k

mr] = r! (i) F(x)

{x\ where ( is the usual combinatorial symbol.

The moment concept was already introduced and applied by Laplace and Poisson. It was further developed by Thiele and received what practically amounts to a complete systematic treatment in the hands of Karl Pearson and his disciples. The semi-invariants were first introduced and systematically de- veloped by Thiele in his remarkable Almindelig Iagttagelseslere published in 1889.1 The factorial moments are of a comparatively recent origin, having been intro- duced in mathematical statistics by Professor Steffensen of Copenhagen in 1923. But in spite of the undisputed fact that both the semi-invariants and the facto- rial moments possess certain properties, which often make them preferable to the usual moments, practically all of the English-speaking statisticians have sadly neglected to avail themselves of this fact with the result that many of their dem- onstrations become unduly complicated and unwieldy.

Since both moments, semi-invariants and factorial moments are symmetric functions, it is, of course, possible to express one in terms of the other, as actually has been done by Thiele in the fourth chapter of his Almindelig Iagttagelseslaere, and later on by Steffensen in his derivation of the factorial moments.

One of the achievements of the present memoir by Dr. Frisch is that it treats the three systems or types of symmetrical functions from a common point of

1 My friend, the late Mr. Vigfusson, pointed out to me in 1922 that Thiele's definition of semi-inva- riants in a limited sense already is found in Lacroix's treatise on the calculus. Lacroix did, of course, not treat the relation between the theory of symmetrical functions and the theory of observations (or statistics). Traces of the semi-invariants are, on the other hand, found in the writings of Bessel. But to Thiele belongs the honor to have given the first comprehensive and systematic treatment of that particular form of symmetric functions, and to have applied his theory to observational or statistical data. Lest the casual reader should get confused in the way of nomenclature, it might be well to point out that Thiele's semi-invariants are not the same as the semi-invariants of Sylvester in the algebra of quantics.

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133] Reviews 403

view, taking its root in the mathematical number theory and especially in the type of numbers associated with the name of Bernoulli. In this manner Dr. Frisch reaches in his first chapter some very general formulas from which he on the one hand derives the relations between semi-invariants, moments and facto- rial moments, as established by Thiele and Steffensen, and on the other hand applies in his treatment of incomplete moments and their relations to the in- equalities of Tchebycheff, H6lder, Jensen and Steffensen.

In Chapter II Dr. Frisch discusses the parameters of the point binomial from both the standpoints of semi-invariants and moments. The most interesting part of this chapter appears to be the decidedly elegant manner in which he de- termines the exact expression for the incomplete moment of the point binomial in the form

(x - ($Sp) P =tqPt =qs tq 8-t

x=t t pq

The reader, comparing Dr. Frisch's treatment of the incomplete binomial with the corresponding treatment by Pearson and his scholars, cannot fail to be im- pressed by the more elegant and certainly less laborious demonstration by the Norwegian scholar, compared with which some of the demonstrations in Bio- metrika appear almost clumsy. In this connection it might be well to mention that Dr. Frisch points out that the formula given in Biometrika (Vol. XV, 1924, p. 202) as

A0= z Pr = t( g I (1 -x)8-tdx

already was given by Laplace in a slightly different form. Incidentally we might also point out that the formula in the same notation as that of Pearson's is found in Meyer's treatise on probabilities, published in Bruxelles in 1874. Moreover, the eminent American telephone engineer, Mr. E. C. Molina, used Laplace's formula as a starting point for important extensions in relation to the Poisson exponential in an interesting article which he wrote for the June, 1913, number of the American Mathematical Monthly. It appears, therefore, that much of the present feverish activity among the students of the Pearsonian school along these lines contains comparatively little in the way of novelty or originality.

Dr. Frisch has also in his memoir given us an important contribution in a series of approximate formulas which enables us to compute with very little labor or trouble approximate values for the incomplete binomial. As an example he gives the calculation of the upper tail end of the point binomial

(i4+1)2 from t=7.

Apart from the denominator, 531,441, the exact value of this expression is 35,313, while the three approximations from Dr. Frisch's formulas are 39,424, 38,614 and 35,953 respectively, indicating an error of less than 2 per cent in case of the third formula. In view of the fact that the older methods, as well as the

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404 American Stattstical Association [134

Pearsonian methods, are rather lengthy and time-consuming, a close study of Dr. Frisch's method might very well prove profitable to those statisticians who have occasion to work with incomplete binomial functions. (For higher exponents the approximations are even closer than the above example.)

The third chapter deals with the parameters of the hypergeometric series and their relation to the binomial series, and introduces the Laplacean concept of generating functions. As shown by Steffensen, the introduction of factorial mo- ments often lessens the analysis of the hypergeometric series. This chapter is perhaps the most difficult of Dr. Frisch's brochure, and the present reviewer hesitates to offer any critical comments thereon.

In conclusion it should be pointed out that the statistician who is not thor- oughly grounded in higher mathematical analysis and the theory of numbers will find little cheer in the Frisch memoir, and our Norwegian scholar's mathematical treatment is perhaps too formidable and goes over the head of the average student of statistics. These difficulties should, on the other hand, not alarm, but rather invite the mature student to tackle this brochure. A careful reading of Frisch will not alone prove stimulating, but yield practical benefits as well to those who have the time and mental energy to do a little digging on their own account. For this reason the present reviewer considered it not alone a privilege, but also a pleasure to prepare the above comments.

ARNE FISHER

The National Income, 1924. A Comparative study of the income of the United Kingdom in 1911 and 1924, by Arthur L. Bowley and Sir Josiah Stamp. New York: Oxford University Press, American Branch. 1927. 59 pp. In this book the authors bring down to date, as of 1924, a previous estimate

of national income. It is, as well might be expected, an excellent example of skillful statistical work. The task calls for ingenuity in finding and filling de- ficiencies and gaps in quantitative data by estimates that are well-grounded and well-reasoned guesses, rather than applications of formally mathematical ideas. Skillful work in such matters supposes not only familiarity with available data, but also a high degree of conversance with the underlying concrete facts, the latter requirement being especially exacting, even when thus qualified for degree. An acceptable result also supposes rare powers of statistical analysis.

At this distance it would, of course, not be easy, if there were occasion, to check up the results. Some conclusions are of special interest.

The authors find that, whereas aggregate income in Great Britain and North Ireland increased 105 per cent in the thirteen years, social income, reached by subtracting transfers (pensions, relief, education, etc.), increased only 90 per cent. Wages have held their own, in terms of proportion of the total, despite a reduc- tion of the working week by about 10 per cent and an increase of unemployment by one-twentieth of the normally occupied population. The proportion of earned income other than wages has slightly increased. Taxes have increased, in terms of proportion, from 11 per cent to 20.2 per cent of taxable income, most of the

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