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.     

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     

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.     

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.     

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     

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:     

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. .     

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 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.     

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.     

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.     

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.     

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.     

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.     

Lewis' branching poisson process model from the point of view of the theory of compound poisson processes

Abstract

1. Lewis [1] has used a composite process as a model for computer failure patterns. This model seems to be of a much wider scope of application at least after some modifications, which are discussed in this note. ¹ The present author has read Lewis [1] only in the form of a galley proof with due regard to a few corrections kindly communicated by Lewis. This note was prepared in 1964 before the publication of Lewis' paper.      

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.     

Withdrawal among Norwegian Life Assured 1910–1935

Introduction.

In the series of statistical experiences of the Norwegian life assurance companies report No. IV was issued some time ago. ¹ “Frivillig avgang blant norske livsforsikrede og dödeligheten blant forsikrede med fripolise 1910–1935”. Oslo 1943. The report, which was prepared at the Statistical Bureau of the Norwegian Life Assurance Companies, comprises 2 subjects: 1. Withdrawal among Norwegian life assured

2. Mortality among assured with paid-up policies (free policies) both during the period 1910–1935.