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.     

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.     

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.     

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.     

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.     

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.     

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. .     

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.     

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.     

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.     

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     

Some remarks on the effect of selection upon mortality with regard to different causes of death

Abstract

In a previous number of this Journal ¹ Zum Studium der Sterblichkeit minderwertiger Leben. Skandinavisk Aktuarietidskrift 1921, p. 31. , I have given an account of an investigation into the increased mortality among certain groups of sub-standard risks based only on statistics of deaths. The material, collected for that investigation, can also be used as basis for an attempt of investigating the influence of Selection on the mortality among stan,dard risks.     

Technical basis for collective (Group) pensions insurance in Norway — K 1946

I. Introduction.

In 1933 O. Aabakken in this journal ¹ Olav Aabakken: “A New Basis of Calculation for Collective Pensions Insurance in Norway”. 1933. dealt with the collective (group) pensions insurance in Norway and especially with the technical basis — K 1931 — which was adopted in 1931 by the life offices in common for this branch of insurance. This basis was worked out from the experiences since 1917, when collective pensions insurance was introduced in Norway. When the National old-age insurance was introduced in 1936, the life offices adopted some new collective pensions benefits which could be employed in the cases where an adjustment to the National old age insurance was desirable. O. Böe has described the mode of adjustment. ² Olav Böe: “Relations between Collective Pensions Insurance, Employer's Pension Funds and National Insurance in Norway”. Transactions of the Eleventh International Congress of Actuaries, Paris 1937.      

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.     

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.     

Optimum stratification of the logarithmic normal distribution: A comment

Abstract

In a recently published article Block has given the optimum points of stratification when the stratification variable and the estimation variable follow a two-dimensional logarithmic normal distribution. ¹ Eskil Block: “Numerical Considerations for the Stratification of Variables Following a Logarithmic Normal Distribution”, this Journal 1958, pp. 185–200.      

Einige Bemerkungen über die Methode der Hilfszahlen

Abstract

1. Die Methode der Hilfszahlen ist bekanntlich ¹ Siehe z. B. J. F. STEFFENSEN: Forsikringsmatematik, S. 325 ff. eine Gruppierungsmethode zur Berechnung der Prämienreserve für Lebensversicherungen. Die Methode, welche nur Gruppierung nach Geburtsjahren erfordert, ist durch Einführung gewisser Hilfszahlen, welche während der ganzen Dauer der Versicherung unverändert bleiben, charakterisiert.     

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: