More about invalidity functions

Abstract

1. In the discussion that followed the reading to the Danish Actuarial Society of the paper quoted below1 it was suggested by Mr N. E. Andersen that the hypothesis T. F. (49), or , employed in the second half of the paper, might with advantage be replaced by x_o being the initial age. In this way it is obtained that and it then follows, by T. F. (6), that      

Some inequalities satisfied by incomplete beta functions

Abstract

Let t (x, n) being defined by Max and .     

On the remainder in the central limit theorem

Abstract

Let X^bv (v = 1,2, ..., n) be independent random variables with the distribution functions F_bvx) and suppose . We define a random variable by where and denote the distribution function of X by F (x.     

On the test of the hypothesis that the probability of an event is contained within given limits

Asbtract

The hypernormal (or Lexian) frequency function can be defined by the integral where θ(p) is the frequency (or density) function of p defined in the interval. We have, of course, that and that .     

Zwei Ungleichungen sur Ergänzung des Tchebycheffschen Lemmas

Abstract

Sei ?(x) eine für ? ∞ < x < + ∞ definierte reelle nichtnegative Funktion und      

On the connection between tests of significance for correlation coefficients and for differences between means

1. Some questions about the connection between statistical tests of significance for simple and multiple correlation coefficients and for differences between sample means (and between sample means and population means) of variables of one or several dimensions are treated in this paper. The distributions of the random variables that are considered in such tests are given, under certain conditions, by frequency functions of the following types ¹ the recently published treatise “Mathematical Methods of Statistics” by Professor Harald Cramér (Uppsala 1945). : where - ∞ < t < ∞, n≧1; where where 0 < t < ∞, k≧1, n≧k; and where .     

Die wahrscheinliche Le bensdauer und die Sterblichkeitsmessung

Im Gegensatz zur durchschnittlichen Lebensdauer hat die wahrscheinliche Lebensdauer in der Versicherungstechnik wohl kaum je eine wirkliche Anwendung gefunden. Der Grund hierfür dürfte hauptsächlich in der recht schwierigen mathematischen Definition dieser Masszahl zu such en sein. Während die durchschnittliche Lebensdauer bekantlich explicite definiert ist, so ist die wahrscheinliche Lebensdauer — wir werden sie mit t^tx bezeichnen — implicite durch die Gleichung definiert.     

On asymptotic properties of an estimate of a functional of a probability density

Abstract

Bhattacharyya & Roussas (1969) proposed to estimate the functional Δ = ∫ _?∞/^∞ f ²(x)dx by , where is a kernel estimate of the probability density f(x). Schuster (1974) proposed, alternatively, to estimate Δ by , where F _n (x) is the sample distribution function, and showed that the two estimates attain the same rate of strong convergence to Δ. In this note, two large sample properties of are presented, first strong convergence of to Δ is established under less assumptions than those of Schuster (1974), and second the asymptotic normality of established.     

A moment inequality with an application to the central limit theorem

Abstract

Consider a sequence of independent random variables (r.v.) X ₁ X ₂, …, X_n , … , with the same distribution function (d.f.) F(x). Let E (X_n ) = 0, _E , E (?(X)) denoting the mean value of the r.v. ? (X). Further, let the r.v. where have the d.f. F _n (x). It was proved by Berry [1] and the present author (Esseen [2], [4]) that Φ(x) being the normal d.f.      

Aktuareksamen 1925 ver Universitetet i Oslo

Abstract

Let be the order statistics in an ordered random sample of size n from the normal population N(±, σ² with mean ± and standard deviation σ. The present paper (1) provides the optimum ranks coefficients and efficiencies of Lloyd's [6] best linear unbiased estimates (BLUE) of ±, σ (when one is known) and (±, σ) based on the k = 1,2, 3, 4 order statistics selected from (Ll) which give the highest efficiencies and (2) demonstrates that efficiencies are only slightly reduced if the BLUE of ± and σ (when one is known) based on the order statistics with ranks are replaced by the BLUE based on the order statistics with ranks , where are the optimum spacings for Ogawa's [8] asymptotic best linear estimates (ABLE) of ± and σ when one parameter is known, or even are replaced directly by the ABLE with their corresponding optimum spacings.     

Consistent estimators of the parameters of a linear structural relation

Abstract

1. Summary of results. Let E and Eo be chance variables at least one of which is not normally distributed (throughout the present paper a chance variable which is constant with probability one will be considered to be normally distributed with variance zero), and whose distribution is otherwise unknown, except that it is known that with probability one, where 0 and p are unknown constants, . Let (u; v) be jointly normally distributed chance variables with unknown covariance matrix, distributed independently of (ε, ε₀). Without loss of generality we assume that the expected values E u and E v, of u and v respectively, are both zero. Define      

Das Restglied der Taylorschen Reihe des Rentenbarwertes und einige Formeln des Zinsfussproblems für grosse Zinsspannungen

Abstract

Im Zinsfussproblem spielen eine wichtige Rolle drei Hilfsfunktionen der Summen der diskontierten Zahlen, die wir vorweg kurz erwahnen wollen. Unter der nten Summe der diskontierten Zahlen D_x verstehen wir SpezieU haben wir:      

On an inequality for gamma functions

Abstract

For all real values of α and λ satisfying the following inequality holds. When compared with a similar inequality due to Gurland [3] this is seen to be stronger for a certain range of α.     

Sur un point de la théorie des erreurs accidentelles

Abstract

Dans son traité de la théorie des erreurs ¹ Theorie der Beobachtungsfehler, Leipzig 1891. M. E. Czuber s'occupe page 202–204 du calcul de la valeur probable de la plus petite erreur dans une série d'observations. En admettant la loi de Gauss il en trouve l'expression , où n est le nombre des observations et . Afin d'évaluer l'intégrale qui représente σ, M. Czuber la remplace par , eu remarquant que la valeur de est très petite, dès que k est en quelque façon considérable. Pour les petites valeurs de x la fonction θ(x) ne diffère que légèrement de hx, dit-il ensuite, de sorte que nous aurons comme valeur approchée de σ . En supposant le nombre n très grand, il est clair, dit M. Czuber, qu'on peut choisir k de manière que (1-hk)ⁿ⁺¹ soit négligeable; il parvient ainsi à la valeur definitive .     

On the ω test of dependence between statistical variables

Abstract

1. In an earlier Note¹ I have suggested to measure the dependence between statistical variables by the expression where p_ij is the probability that x assumes the value x_i and y the value y_j , while By is meant summation with respect to all i and j for which p_ij > p_i* p_*j .     

Über den Einfluss einer Änderung der Sterblichkeit auf die Prämienreserve

Abstract

In einer Note über die Theorie des Deekungskapitales habe ieh für das reduzierte Kapital der gemisehten Versicherung auf die Beträge At den Ausdruck gebraueht (1) wobei als Deckungsintensität bezeichnet wurde.     

Aktuareksamen og Eksamen i Statistik ved Københavns Universitet

Abstract

Extract

d1. Vis, at man for n ? 2 har når x ikke antager nogen af værdierne 0, ?1, ..., ?n+1, og når x ikke antager nogen af værdierne 0, 1, ..., n+1.     

Notes on collective risk theory

Abstract

A complete proof of existence of a probability measure m the space Ω of all sample functions was given by Cramér [4]. For a finitc period, a simplified proof was given in my paper [2]. The latter proof could be restricted to the space of sample functions having only a finite number of jumps, as the probability of an infinite number of jumps is zero in this case. In fact, dividing the space Ω into disjunct subspaces Ω_n containing exactly n jumps we have: The measure of Ω_n m the case of a finite period of length x is: Thus and consequently P (Ω_∞) = 0. Therefore the set Ω_∞ and all its subsets can be neglected.     

The Problem of Optimum Stratification

Abstract

Although most applications of stratified sampling represent sampling from a finite population, π(N), consisting of k mutually exclusive sub-populations or strata, n, (N,), it is for purposes of theoretical investigations convenient to deal with a hypothetical population n, represented by a distribution function f(y), a < y < b. This hypothetical population likewise consists of k mutually exclusive strata, π_i , i = 1,.2 ... k. The mean of this population is µ_i being the mean of ni. By means of a random sample of n observations, n_i of which are selected from π_i , µ, is estimated by: being the estimate of µ_i .     

Analytical studies in stop-loss reinsurance. II

Abstract

I

In an earlier paper [5] we discussed the problem of finding an unbiased estimator of where p (x, 0) is a given frequency density and 0 is a (set of) parameter(s). In general, will not be an unbiased estimator of (1), when Ô is an unbiased estimate of O. In [5] it was shown that is an unbiased estimator of (1), if we define y_i , as the larger of 0 and X _j - c. It was emphasized that the resulting estimate may very well be zero, even when it is unreasonable to assume that the premium for a stop.loss reinsurance. defined by a frequency p (x, 0) of claims x and a critical limit c, should be zero when the critical limit has not been exceeded during the n years considered for the determination of the premium.