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. .     

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.     

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.      

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.     

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.     

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.     

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.     

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.     

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.     

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     

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.     

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. .     

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.      

On the central limit theorem

Abstract

If X and Y are mutually independent random variables whith the d. f. ¹ Distribution function(s) F ₁(χ) and F ₂(χ), it is known ² CRAMÉR (1), p. 35. that the sum X + Y has the d. f. F ₂(χ), defined as the convolution where the integrals are Lebesgue-Stiltjes integrals. One uses the abbreviation More generally the sum X ₁ + X ₂ + … + X _n of n mutually independent random variables with the d. f. ¹ Distribution function(s) F ₁(χ), F ₂(χ) , … , F _n has the d. f.      

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.      

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.     

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.     

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.      

Accounting Education at a Crossroad in 2010 and Challenges Facing Accounting Education in Australia

Of the various reports released in 2010, two purport to examine the state of accounting education in Australia. These are Accounting Education at a Crossroad in 2010 and Challenges Facing Accounting Education in Australia. Both were released as collaborations of the leading academic organisation, the Accounting and Finance Association of Australia and New Zealand (AFAANZ) or professional accounting bodies in Australia including the Institute of Chartered Accountants in Australia (ICAA), the Institute of Public Accountants (IPA),1 Prior to April 2011, the IPA was known as the National Institute of Accountants (or NIA). and Certified Practising Accountants of Australia (CPA Australia). As their titles imply, the main thrust of these reports is to examine the challenges facing accounting education in Australian universities and, as such, they act as the input for this AE Briefing. The main challenges articulated in these reports portray a sector suffering from the combined pressure of a large international student enrolment, high student-to-staff ratios, an inadequate funding model, and an ageing academic staff profile. By way of commentary, we suggest that, if these gloomy circumstances continue to develop unabated, then the future for the sector will play out as a ‘perfect storm’2 Reference to the state of accounting education as moving toward a ‘perfect storm’ is an analogy that has been widely used in the higher education press in Australia and in dialogue at the local level. with the sector suffering on-going troubled development.