On the sum or integral of the product of two functions

Abstract

The present paper is a sequel to an earlier paper of mine¹ which will be quoted below as \>2First Paper\>2. Since then, several important contributions to the subject have appeared, in particular papers by RAGNAR FRISCH ², A. W. JOSEPH ³, and G. J. LIDSTONE ⁴ whereby the problem of expanding the product-sum or product-integral in what may be termed Tchebychef and Legendre Moments of the two functions may be considered to have been solved. Also certain valuable papers on orthogonal polynomials by A. C. AITKEN ⁵, G. J. LIDSTONE ⁶, DUNCAN C. FRASER ⁷, and L. TRUKSA ⁸ have a bearing on the subject. — Since this was written a further paper by A. W. JOSEPH has appeared.⁹     

Functions of measurements under general laws of error

Abstract

The first object of this paper is to extend somewhat a theorem of MISES ¹ (1) Fundamentalsätze der Wahrscheinlichkeitsrechnung, (2) Grundlagen der Wahr. Mathematische Zeitschrift (1919) vol. 4, pp. 1–97; vol. 5, pp. 52–99. , who by the use of the STIELTJES integral ² Bliss, Integrals of Lebesgue, Bulletin of the American Mathematical Society, vol. 24 (1917), pp. 1–47. Bliss, A necessary and sufficient condition for the existence of a Stieltjes integral, Proceedings of the National Academy of Sciences vol 3 (1917), pp. 633–637. HILDEBRANDT, On Integrals related to and Extensions of the Lebesgue Integrals, Bulletin of the American Mathematical Society, vol. 24 (1917), pp. 113–144; Ibid., pp. 177–202. CARMICHEAL, Conditions necessary and sufficient for the existence of the STIELTJE'S Integral, Proceedings of the National Academy of Sciences, vol. 5 (1919), pp. 551–555. treats as a single subject continuous (geometric) and discontinuous (arithmetic) probability. Laplace ³ Oeuvres complete de LAPLACE, 3rd Ed., vol. 7, p. 266. , indeed, used a probability function which was in general continuous but had finite jumps, but he left no systematic treatment of the whole field of probability under such general conditions.     

On certain inequalities by Steffensen

Abstract

Extract

In this note we are concerned with the inequalities published by Steffensen in this Journal under the title: ?On a generalization of certain inequalities by Tchebycheff and Jensen?. I will show how these inequalities are represented in one form by the integrals of Stieltjes. These integrals have the advantage of containing both sums and the integrals of Riemann. The proof of this generalized theorem is also more simple, which shows that this theorem appears only in this generalization in its most natural form.     

Edvard Jäderin

Abstract

Im Jahre 1928 hat Frau Dr. H. Pollaczek-Geiringer ² H. Pollaczek-Geiringer: Die Charliersche Entwicklung willkür licher Funktionen, Skandinavisk Aktuarietidskrift 1928, p. 98–111. in dieser Zeitschrift einen Konvergenzbeweis für die Charlier'sche B-Reihe unter der folgenden Voraussetzung bewiesen:     

Sur le problème des moments et le théorème de Parseval correspondant

Abstract

On connaît le rôle que joue le théorème de Parseval. dans la théorie des séries de Fourier et, plus généralement, dans celle des développements suivant des fonctions orthogonales. On sait aussi qu'on peut généraliser la notion d'orthogonalité ainsi que le théorème de Parseval de différentes manierès, p. ex. en se servant de l'intégrale de Stieltjes correspondant à, une distribution de masses positives.     

Zum Markoffschen Lemma

Abstract

Der Sechste Skandinavische Mathematiker-Kongress (Kopenhagen 1925) hat sich unter anderem mit dem Markoffschen Lemma ¹ A. A. Markoff, Wahrscheinlichkeitsrechnung, deutsch von H. Liebmann, 1912, S. 54–56. Vgl. E. Czuber, Wahrscheinlichkeitsrechnung, I, 3. Aufl., 1914, S. 232–233. beschäftigt, und auf diesem Kongress hat Dr. phil. RAGNAR FRISCH im Anschluss an ein Referat von Prof. Alf Guldberg in einem besonderen Referat den Beweis geliefert, dass die in Frage stehende Ungleichung im allgemeinen Fall keine »Einengung» zulasse. ² Ragnar Frisch, Impossibilité de resserrer l'inégalité de Markow dans le cas général. Kongressbericht, S. 203–206. Der Beweis ist schlüssig, aber er kann, wie mir scheint, einfacher erbracht werden.     

De skandinaviske aktuarforeningers virksomhed i 1934

Abstract

17. Januar 1934. Generalforsamling. Til Medlemmer af Bestyrelsen valgtes: Direktør, cando mag. BJ. Drachmann (Formand), Underdirektør, Beregner, cando act. E. Hude (Næstformand), Aktuar A. Kousgaard Nielsen (Sekretær), Overassistent, cando act. Frk. Margarethe Bjerregaard og Fuldmægtig, cando act. Børge Sørensen.     

A. Palmström

Abstract

1. Mr R. Frisch has recently ¹ Ragnar Frisch: Sur les semi-invariants et moments employés dans l'étude des distributions statistiques, Oslo, 1926 (Det Norske VidenskabsAkademis Skrifter, II N:o 3), p. 26. established the interesting identity     

Mortality experience of life annuitants insured with thirteen Swedish offices during the period 1900–1924.

Abstract

To the Scandinavian Life Insurance Congress in Oslo 1926 an investigation into the mortality of annnitants was presented by thirteen Swedish Life Insurance Companies1 The investigation was executed by a committee consisting of Harald Cramér, Reinh. Palmqvist and Iwar Sjögren.      

Measures of departure from symmetry

Abstract

1. Many different measures of skewness have been proposed. The textbook of Albert Waugh ¹ Elements of Statistical Method. New York and London 1938. for instance presents six different measures of skewness. Other measures of skewness have been proposed by Lindeberg ² Skandinavisk Aktuarietidsskrift. Vol. 7. (1925) p. 106. and Lenz ³ Erna Weber: Einführung in die Variations- nnd Erblichkeitsstatistik. München 1935. p. 92. . All these measures are zero when the distribution is symmetric, but any of them may be zero also when the distribution is unsymmetric. I shall here present some measures of skewness which are zero when and only when the distribution is symmetric.     

Ein Beitrag zum Zinsfussproblem

Abstract

Zu den nachfolgenden Untersuchungen wurde der Verfasser veranlasst durch wertvolle Anregungen seines Lehrers, Prof. Dr. W. Friedli, sowie durch die schönen, das Zinsfussproblem berührenden Arbeiten der Herren Steffensen ¹ Skandinavisk Aktuarietidskrift: I. 1918. On certain inequalities between mean values, and their application to actuarial problems. , Meidell ² Skandinavisk Aktuarietidskrift: I. 1918. Note sur quelques inégalités et formules d'approximation. , Palmqvist ³ Skandinavisk Aktuarietidskrift: IV. 1921. Sur une méhode d'approximation applicable à certains problèmes actuariels. und Poukka ⁴ Skandinavisk Aktuarietidskrift: IV. 1923. Über die Berechnung der Leibrente bei Änderung des Zinsfusses. .     

Summation-formulas of Lubbock's type

Abstract

If it is required to calculate a sum consisting of a great many terms, it is natural to ask oneself whether an approximation might not be obtained by adding up every mth term and multiplying the result by m. If the approximation obtained in this way is not considered sufficient, certain supplementary terms must be added to the result, and these may either be expressed as differences or as differential coefficients of the function under consideration. In the former case we have formulas of Lubbock's, in the latter of Woolhouse's type. We here confine our attention to the formulas of Lubbock's type.     

Short notes on Charlier's method for expansion of frequency functions in series

Abstract

1. In 1905 Charlier outlined some methods for the expansion of functions in series. ¹ C. V. L. Charlier, Über die Darstellung willkürlicher Funktionen (Meddelanden från Lunds Observatorium, Ser. I, nr 27). Particularly he was dealing with frequency functions, but the method has a more general application. As is well known there were two kinds of developments considered, namely in terms of the differentials and in terms of the differences of a conveniently chosen developing function. The outstanding examples are — respectively — the expansions of the so called types A and B. The difference series has later gained a special attention by its deduction being attached to the theory for generating functions. ² I. V. Uspensky, On Ch. Jordan's Series for Probability (Annals of Mathematics, Vol. 32, 1931). The true pivotal function in this respect seems, however, to be the moment generating function. In the following notes it will be shown that the differential series as well as difference series built up by the advancing and the central differences are obtainable in a similar way. By employing some convenient cumulants the different expansions can be written down compactly in symbolic forms which reveal their mutual formal relations. It will further be observed that Charlier's method of expansion is the inversion of a method indicated by Abel.     

Die sterblichkeit der versicherten in schwedischen lebensversicherungsgesellschaften während der Zeit vom 1. Juli 1918–30. Juni 1919

Abstract

Die grossen Influenzaepidemien von 1918 und den folgenden Jahren haben aus natürlichen Gründen das Interesse der Lebensversicherungsstatistiker der ganzen Welt auf sich gelenkt. Statistische Ergebnisse über den Einfluss dieser Epidemien auf die Sterblichkeit von Lebensversicherten findet man so z. B. in folgenden Abhandlungen: Dr. A. WINTER, De sterfte in de laatste jaren. Het Verzekerings-Archief, Jaargang II, No. 2; JAMES D. CRAIG and LOUIS I. DUBLIN, The influenza Epidemic of 1918. Transactions of the Actuarial Society of America, vol. 20, No. 61; R. J. BACKLUND, Dödligheten i spanska sjukan i Kaleva 1918–1920. Nordisk Försäkringstidskrift, Årg. I, n:r 4.     

Formulae of recurrence for the semiinvariants at a class of frequency functions of more variables

Abstract

In a paper in this journal¹ Mr. Paul Qvale has proved following theorem:     

De skandinaviske Aktuarforeningers Virksomhed i 1939

Abstract

Der er afholdt 7 Møder. 11. Januar. Generalforsamling. Aktuar W. Simonsen valgtes til Sekretær og Aktuar F. Howitz og Fuldmægtig H. P. Rasmussen til Medlemmer af Bestyrelsen.     

On Charlier's generalized frequency-function

Abstract

It is well known, that Charlier has suggested to develop a frequency-function f(x) in a series of the form where ?(x) stands for a particular, given frequency-function, while the symbol ? denotes the ascending difference, that is ??(x)=?(x)-?(x-1). In the form proposed by Charlier this method is open to objections of which he is partly himself aware; the chief objecti.on being, that no account is taken of questions of convergence. It seems, therefore, of interest to examine what becomes of the method, if it is not carried beyond legitimate bounds. In doing so, I shall try to simplify the determination of the constants, a problem which has been attacked by Charlier ¹ C. V. L. Charlier: Über die Darstellung willkürlicher Funktionen (Arkiv för matematik, astronomi och fysik, Bd 2, N:r 20). himself, and by N. R. Jørgensen ² N. R. Jørgensen: Note sur la fonction de répartition de type B de M. Charlier (ibid., Bd 10, N:r 15); Undersøgelser over Frequensflader og Korrelation (Copenhagen, 1916), S. 5–15. in a special case. For this purpose I avail myself of a class of symmetrical functions of the observations for which I have proposed the name of “factorial moments” and the systematical use of which I recommended in my paper “Factorial Moments and Discontinuous Frequency-Functions”. ³ This Journal, 1923, p. 73. See also the author's book “Matematisk Iagttagelseslære” (Copenhagen, 1923), I, § 2. I shall assume, that the reader is familiar with the notation employed in that paper which differs in some respects from the usual notation of moments.     

Zur dispersionstheorie der statistischen reihen

Abstract

In dermathematischen Statistik nimmt die von DORMOY und LEXIS begründete Dispersionstheorie eine zentrale Stelle ein.     

A remark on the Pearsonian frequency curves

Abstract

In the well-known volume:»Frequency curves and correlation » Mr. W. PALIN ELDERTON gives an exposition of the Pearsonian frequency curves.     

A correlation-formula

Abstract

The late Professor T. N. THIELE has pointed out, ¹ Bulletins de l'Académie Royale de Danemark, 1906 N:o 3 p. 149–152. that a given correlation may sometimes be brought to vanish by a suitable linear transformation of the coordinates. Unfortunately his indications in this respect are very brief; and as the subject is not treated by means of frequency-surfaces, but only by a consideration of the first few moments (or rather “half-invariants ” ² A kind of rational functions of the moments, see THIELE: Theory of Ohservations, London 1903, p. 24. — The same incomplete method has heen applied by C. Burrau (Meddelelser fra den antropologiske Komite p. 243–260). ) in a particular numerical case, his efforts have not resulted in establishing a correlation-formula which alone, by comparison with the observations, could prove his assertion right or wrong. I therefore propose to resume the subject, beginning with a few remarks on frequency-distributions with one single variable, and repeating, for the sake of completeness, a certain amount of known matter.