Showing papers in "Scandinavian Actuarial Journal in 1971"
TL;DR: In this paper, the authors further generalized the Cramer treatment of the classical Poisson case to the more general case with a negative gross risk premium and showed that the identity shown by him in an earlier paper is also capable of being generalized.
Abstract: In a paper [13] in this journal the author has further developed the ruin theory—first presented by E. Sparre Andersen [1] to the XVth International Congress of Actuaries, New York, 1957—in case the epochs of claims form a renewal process and the gross risk premium is positive. It turned out that much of the Cramer treatment [6] of the classical ruin theory (Poisson case) could be generalized to the more general situation. In the present paper the author points out that the identity shown by him in an earlier paper [12] for the Poisson case is also capable of being generalized. Some comments concerning the case with a negative gross risk premium are also included. I.a. the Cramer asymptotic formula for the ruin probability for an infinite period is generalized also in that case. Further information about the contents of the paper may be inferred from the following section headings. The paragraph 5.4 and the sections 6–7 are found in Part II to appear in SAT 1971:3–4.
32 citations
TL;DR: In this paper, some properties of a point process, which has been proposed by Cramer [4] as a model of the claims arising in an insurance company, have been studied by Cox in a different context.
Abstract: In this paper we are going to study some properties of a point process, which has been proposed by Cramer [4] as a model of the claims arising in an insurance company. This process has been studied by Cox in a different context. A few elementary results, concerning moments, are given in Cox and Lewis [2].
20 citations
TL;DR: In this paper, a model for the risk business of an insurance company, where the claims are assumed to be located in time according to a Poisson process and where the claim distributions are Γ-distributed, is considered.
Abstract: A model for the risk business of an insurance company, where the claims are assumed to be located in time according to a Poisson process and where the claims are Γ-distributed, is considered. Various approximations of the ruin probability are compared with its exact value. Finally some limit results, when the variance of the Γ-distribution tends to infinity, are discussed.
20 citations
TL;DR: In this paper, it was shown that a simultaneous maximum likelihood estimation of θ and τ1 τ2, τ3, τ4, τ5, τ6, τ7, τ8, τ9, τ10, τ11, τ12, τ13, τ14, τ15, τ16, τ17, τ18, τ19, τ20, τ21, τ22, τ23, τ24, τ25, τ26, τ27, τ28, τ29, τ30, τ31, τ32, τ33,
Abstract: Let X 1, X 2, ... be vector-valued random variables and let the distribution of X i depend on two parameters θ and τ i where θ has the same value for all i while the value of τ i changes with i. Following Neyman & Scott [8] we shall denote θ a structural parameter and τ i an incidental parameter. It was shown by Neyman & Scott that a simultaneous maximum likelihood estimation of θ and τ1 τ2, ... , on may lead to an inconsistent estimation of θ. Methods for obtaining consistent estimates for a structural parameter in the presence of infinitely many incidental parameters have been suggested by Neyman & Scott [8], Kiefer & Wolfowitz [4] and Andersen [2]. In Andersen [2] the consistent solution was obtained as a conditional maximum likelihood estimator given minimal sufficient statistics T i = T(X i); i = 1, 2, ... for the incidental parameters. As shown in Andersen [1] a minimal sufficient statistic for τ1 in the presence of θ will in general depend on θ. The conditional maximum likelihood method re...
8 citations
TL;DR: In this paper, the authors distinguish between the above "gross" rates and the "net" rates, where p(y) refers to the probability that a new-born girl will be alive at age y.
Abstract: The age-specific (female) fertility rates ƒ(y) are usually defined by applying the conventional definition of the birth rate to the sub-population consisting of y-aged females. Without further notification only live-born girls are taken into account; in the cases where both live-born girls and boys are considered the rates may be indicated as “total”. We distinguish between the above “gross” rates and the “net” rates p(y)ƒ(y), where p(y) refers to the probability that a new-born girl will be alive at age y. Apart from proportionality factors these interdependent rates often show only small differences in a given situation.
6 citations
3 citations
TL;DR: In Norwegian group life insurance, the sums assured of this group term life assurance are, in principle, stipulated for one year at a time, and natural premiums are paid for the single-year risk at the beginning of each contract year as mentioned in this paper.
Abstract: (1 A) Norwegian group life insurance does not have the multiplicity of benefits provided by American [4] or even Swedish [1, 2, 3, 7, 12] insurers. The only coverage given 1 is a straight death benefit for groups of lives, such as the employees of a firm or the members of a professional or vocational association. The sums assured of this group term life assurance are, in principle, stipulated for one year at a time, and natural premiums are paid for the single-year risk at the beginning of each contract year. Renewal of a group policy at the end of a contract year is usually quite automatic, however, and the insurer will commonly assume that the same sums will be in force until otherwise notified. Still, the whole contract is really a single-year transaction which can be terminated (or where the tariff can be changed) at the end of any contract year without necessitating any settlement of future obligations.
2 citations
TL;DR: The main object of as discussed by the authors is to derive Bayes and minimax tests for some common testing problems and study the size of these tests as function of the sample size and the loss ratio.
Abstract: The main object of the paper is to derive Bayes and minimax tests for some common testing problems and to study the size of these tests as function of the sample size and the loss ratio.
2 citations
TL;DR: For any continuous univariate population with finite variance there is a mathematical relation which expresses the variate-value z as a convergent series of Legendre polynomials in (2F-1), where F≡F(z) is the distribution function of the population, and the coefficients in this series are the expectations of homogeneous linear forms in the order statistics of random samples from the population as mentioned in this paper.
Abstract: For any continuous univariate population with finite variance there is a mathematical relation which expresses the variate-value z as a convergent series of Legendre polynomials in (2F—1), where F≡F(z) is the distribution function of the population, and the coefficients in this series are the expectations of homogeneous linear forms in the order statistics of random samples from the population. The relation is well adapted for estimating the median and other percentile points when nothing more is known about the population, but a random sample from it is available. The variances of these estimates can be estimated from the data. A somewhat similar relation which expresses z as a series of Chebyshev polynomials is also discussed briefly. Finally a modification of the Legendre polynomial relation enables prior knowledge of a finite extremity of the population range to be used, and a numerical illustration is given.
TL;DR: In this article, Buhlmann has deduced the characteristic function of S t = ∑ i=1 N S t (i), from assumptions with respect to the characteristic functions of s t, i = 1, 2... N, which have been differently formulated in different connections.
Abstract: 1. In Mathematical methods in Risk Theory [3]. 1 Buhlmann has in several connections deduced the characteristic function of S t = ∑ i=1 N S t (i), from assumptions with respect to the characteristic functions of S t (i), i = 1, 2 ... N, which have been differently formulated in different connections. S t is a random function, which constitutes the risk process of a main group of insurances, and S t (i), a random function constituting the risk process of the ith sub-group of the main group, for i = 1, 2 ... N, where some postulates with regard to the mutual independence of the S t (i)'s have been made.
TL;DR: In this article, the renewal process is based on the assumption that τ n are mutually independent variables, equally distributed with the distribution function Prob [τ n ⩽ τ] = K(τ), n = 1, 2..., where K( τ) shall be defined below.
Abstract: 1. Renewal models applied to the risk theory In this note the interval between the (n—1)th, and the nth event in a random process—the interoccurrence time—will be denoted by τ n for n = 2, 3 ... , and the interval from the starting point of the process to the time point of the first event by τ1. τ has here been chosen for the notations, in order to differentiate from t, usually used for the expected number of claims; the mean of n in a renewal process is, for finite values of τ, different from τ, and for τ→∞ linear in τ; τ is not necessarily measured on the absolute parameter scale. The renewal process is based on the assumptions that τ n are mutually independent variables, equally distributed with the distribution function Prob [τ n ⩽ τ] = K(τ) , n = 1, 2 ... , where K(τ) shall be defined below. The complex Fourier transform of dK(τ) shall be designated by k(z).
TL;DR: In this paper, the evaluation of joint life endowments and joint-life annuities is performed by substituting a single life (u) for (x) and (y) and altering the force of interest, provided that and with the same value of the parameter c( > 1).
Abstract: 1. If (x) and (y) are lives whose remaining lifetimes are stochasticallyindependent, and if the mortality of each of the lives is given by a Makeham expression, then as a well known fact (see e.g. P. F. Hooker & L. H. Longley-Cook, Life and Other Contingencies, Cambridge 1957, vol. II, pp. 137&138) the evaluation of joint-life endowments and joint-life annuities on the lives (x) and (y) may be performed by substituting a single life (u) for (x) and (y) and altering the force of interest, provided that and with the same value of the parameter c( > 1).