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. .     

Über die Berechnung der Leibrente bei Veränderung des Zinsfusses

Abstract

1. In dieser Zeitschrift haben die Herren Steffensen ¹ 1918, Häft. 1–2, S. 82–97. , Meidell ² 1918, Häft. 3–4, S. 180–198. und Palmqvist ³ 1921, Häft. 3, S. 152–178. Näherungsformeln zur Berechnung der Veränderung der Leibrente bei der Veränderung des Zinsfusses veröffentlicht.     

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.     

Kleine Bemerkung zum Zinsfussproblem

Abstract

In den Mitteilungen schweizerischer Versieherungsmathematiker ¹ H. Christen: Das Zinsfussproblem bei der Leibrente (M. d. V. sch. V., Heft 25, 1930). hat H. CHRISTEN eine eingehende Arbeit über das Zinsfussproblem veröffentlieht, zu dem er eine sehr gute Lösung beigetragen hat. Die Hauptergebnisse seiner Untersuehullg hat er aueh in der Skandinavisk Aktuarietidskrift ² H. Christen: Das Zinsfussproblem bei der Leibrente (M. d. V. sch. V., Heft 25, 1930). mitgeteilt.     

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.     

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.     

Une application de la nomographie au système complet de rentes viagères de Blaschke-Gram

Abstract

Blaschke ¹ E. Blaschke: » Ueber eine Anwendung des Sterbegesetzes von Gompertz-Makeham ». Mitteilungen des Verbandes der östeur. und ungar. Versicherungstechniker, 1902. et Gram ² J. P. Gram: »Om Makehams Dødelighedsformel og dens Anvendelse paa ikke normale Liv ». Aktuaren, Heft 1, 1904. ont exprimé la relation entre les rentes viagères de deux ordres de survie quelconques obéissant à la loi de Makeham par les équations fondamentales I_a II_a et III_a ci-après.     

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.      

Note respecting the application of »The reduced number of deaths»

Abstract

This note is merely intended to supplement the remarks on the reduced number of deaths made by Olav Aabakken III his paper in this number of the journal ¹ Olav Aabakken: Appraisement of Under-average Lives in Norway, p. 126. .     

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:     

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     

Book Review

1. Introductory. In the empirical study of demand behaviour, as in other investigations into economic factors and their interrelations, the main statistical tool has been the mean square (m. sq. ¹ Following H. Cramér 1, we adopt this abbreviation for the classical least squares regression method. ) method of regression. ² For a survey, see the standard treatise of H. Schultz 1. It is equally well known, however, that this method has met much criticism. No doubt there are still many questions to discuss in this field.     

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.     

Sur certains mouvements aléatoires discontinus

1. Introduction. Divers problèmes de Physique et de Calcul des Probabilités conduisent à des sehémas de mouvements aléatoires, régis par l'équation de Chapman, dans lesquels un certain point mobile ne se déplaee qu'un nombre fini de fois et reste immobile entre les instants aléatoires auxquels ont lieu les sauts. Dans la théorie des variables aléatoires indépendantes ces schémas donnent lieu à des lois de probabilité généralisant la loi de Poisson (cf. p. e. Khintchine) ¹ Ergebnisse der Mathematik, Asympt. Gesetze, 1933. . Ils se présentent très naturellement dans le cas où le nombre des positions que le point mobile peut occuper est fini, et les distributions de probabilité qu'on rencontre ainsi ont été étudiées par MM. Kolmogoroff ² Math. Ann. t. 104, pp. 415–58, 1931. , Fréchet ³ voir p. e. Recherches Théoriques Modernes, livre I, pp. 231–54, Gauthiers-Villars, 1938. et Hostinský. ⁴ cf. p. e. Journal de Math., t. 16, pp. 267–84, 1937. On comprend très aisément que dans un mouvement où il n'y a qu'un nombre fini de déplacements il n'y a pas lieu d'attacher de l'importance à la structure topologique de l'espace dans lequel a lieu le mouvement et, effectivement, M. Pospísil ⁵ ?asopis Mat. Fys., t. 65, p. 64–76, 1936. a, en généralisant les méthodes de M. B. Hostinsktý, étudié des mouvements ayant lieu dans des espaces abstraits quelconques, mais dans cette étude, on est obligé d'opérer avec des fonctions d'ensembles abstraits, des corps d'ensembles etc., ce qui nuit un peu au caractère intuitif du problème.     

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.     

On Numerical Differentiation of Functions of Several Variables

In a previous paper ¹ This journal (1957), pp. 60-70; in the sequel it will be quoted as N. D. some results concerning numerical differentiation of functions of a single variable were obtained on the basis of the important investigation by W. Barrett ² ‘On the remainders of numerical formulae, with special reference to differentiation formulae’, Journ. Lond. Math. Soc., Vol. 27 (1952). , which involved, inter alia, a considerable simplification of the form of remainder-terms of various formulae for numerical differentiation. The object of the present paper will be an extension of the results obtained for functions of a single variable to functions of several variables in the case of a regular distribution of the points at which the functional values are supposed to be given.     

Industrial replacement

The problem of industrial replacement has been attacked on an essentially empirical basis by various writers, more especially and quite lately by E. B. Kurtz ¹ E. B. Kurtz, Life Expectancy of Physical Property, Ronald Press 1930; R. Winfrey and E. B. Kurtz, Life Characteristics of Physical Property, Bulletin 103 of the Iowa Engineering Experiment Station, Iowa State College, 1931. But its mathematical analysis, which is quite within the range of modern technique, has not in prior publications ² A special case has been discussed by H. Bateman, Messenger of Mathematics 1920 Vol. 49, p. 134. A more general discussion has been given by R. Frisch, Statsekonomisk Tidskrift, 1927, pp. 117-152. See also B. Meidell, Skandinavisk Aktuarietidskrift 1926 p. 172; J1. Washington Acad. Sci. 1928 p. 437. been advanced to the point at which it becomes applicable to Kurtz's data. The requisite analysis is, as a matter of fact, readily conducted by methods very similar to those applicable to certain problems of population growth and structure, which indeed closely resemble the problem of industrial replacement. ³ F. R. Sharpe and A. J. Lotka, A Problem in Age Distribution, Phil. Mag. April 1911 p. 435; A. J. Lotka, The Progeny of a Population Element, American Journal of Hygiene, 1928, vol. 8, p. 875; idem The Spread of Generations, Human Biology, 1929, vol. 1, p. 305.      

Correlation once more

Abstract

§ 1. Correlation Generally.

In my thesis for Doctorship of 1919, ¹ This Journal, 1919, p. 1. I have made some critical remarks on the theory of correlation, trying especially to exhibit the rather unaccomplished state of this theory, and the danger of releasing its formulae for general use. In a supplementary note, ² Ibidem, p. 204. I have pushed my criticism a little further. Since that time, more than ten years have passed away, but I find my point of view still sustainable. One reproach, however, I have never been able to reject, viz, of having been exclusively negative. In fact, I find this position rather natural, from a philosophical standpoint. The information value of correlation calculations is indeed, as a rule, very small. And this seems the more regrettable, as the importance of the Στχαστι?? Τ?χυη, even for the most difficult questions of knowledge, ought to be very great. In a paper of 1924, “Quelques questions concernant les principes de la théorie des probabilités” ³ Ibidem, 1924, p. 107. I have tried to explain, how I imagine the development of the theory of correlation in order to be more apt to set about such philosophical questions.     

Eksamen i forsikringsvidenskab og statistik ved københavns universitet. l′ del. den skriftlige prøve

Abstract

Es kann nicht fehlen, dass sich der von Mises in zwei jüngst erschienenen Abhandlungen ¹ “Fundamentalsätze” und. “Grundlagen der Wahrscheinlichkeitsrechnung” in Lichtensteins mathem. Zeitschrift, 4 und 5. Band. angebahnten Neuordnung der Wahrscheinliehkeitsrechnung eine solche der mathematischen Statistik (in der Folge m. St.) anschliesst. Der genannte Forscher hat sie auch bereits angekündigt, ² Ibid. Band 5. S. 52. Wenn ich es trotzdem unternehme, in diesem Zeitpunkte mich über die Grundlagen dieser Wissenschaft zu äussern, so habe ich die Auffassung, dass bei dieser Neuordnung die Bedürfnisse der Praxis nicht übersehen werden sollten. Die Entwicklung, welche die theoretische Statistik auf Grund einer axiomatisch fundierten Wahrscheinlichkeitsrechnung mogölicherweisenehmen könnte, lässt die Befürchtung nicht von der Hand weisen, dass die der Forschung abträgliche Kluft, die dermalen zwischen den mathematisch orientierten und den Verwaltungs-Statistikern besteht, sich noch erweitere. Es ist an sich zu beklagen, dass die grosse schöpferische Arbeit der Ersteren auf die Praxis so gut wie keinen Einfluss genommen hat. In unseren Quellenwerken werden alljährlich grosse Mengen von Absolut- und Relativzahlen ohne jeden Anhaltspunkt fär ihre kritische Wertung veröffentlicht, obgleieh schon in der ersten theoretischen Statistik (von Theodor Wittstein), also schon vor mehr ala 50 Jahren, die bezäglichen Anforderungen klar formuliert und nachmalig des öfteren wiederholt wurden.     

On discontinuous frequency functions of two variables

The theory of discontinuous frequency functions of one variable has been well elaborated, especially due to the important researches of Professor Steffensen ¹ See f. i. Steffensen: Matematisk Iagtagelseslære, p. 32 f. while the theory of discontinuous frequency functions of two and more variables has been almost neglected, in spite of the fact that all statisticians agree as to the importance of this topic.