Showing papers in "Scandinavian Actuarial Journal in 1970"

An extension of the renewal equation and its application in the collective theory of risk

Abstract: Let us consider the renewal equation where z(x) and the proper probability distribution F(x) on (0,∞) are given. Let µ = ∫0 ∞ x dF(x), the case µ = ∞ is not excluded. Then the following theorem is equivalent to the renewal theorem (see Feller [2]). Theorem 1.1. If z is directly Riemann integrable and F is not arithmetic, then . The defective renewal equation is of great importance for applications. There L(x) is a defective probability distribution, L ∞ < 1. We have (see [2]) Theorem 1.2. If z(∞) = lim z(x), x → ∞, exists, then .

On some distributions in time connected with the collective theory of risk

Abstract: The title of this paper might as well have been “On the distribution in time of certain first passages in a Poisson process (in the proper sense) with nonzero mean and two barriers, one reflecting or absorbing and the other one absorbing”.

Bayesian estimation of the Paretian index

Abstract: It is well known that a Pareto distribution provides reasonably good fit to distributions of income and of property values. Seal [13] conceived the idea of using the discrete Pareto distribution, to represent the distribution of multiplicate life insurance policies among the policy holders of one or more offices. For detailed arguments on the existence of such distributions in economic life the reader is referred to the discussions by Davis [1], Hagstroem [2, 3], and Mandlebrot [8].

Stochastic processes and practical working models or why is the Polya process approach defective in modem practice and how cope with its deficiencies

Abstract: 1. One of the principal aims of mathematical statistics is the creation of theoretical models of probabilistically governed events or series of events. These models are mostly constructed by bringing together in suitable arrangements different elements from the tool-kit of general methods and premanufactured simple pieces of work at hand and known by the research worker.

On the transient behaviour of the GI/G/1 waiting-times

Abstract: Consider a single-server queuing system, where the arrival intervals Ti and the service-times Ui of consecutive customers form two independent sequences of independent and equally distributed random variables. Assume that customers arriving when the server is busy line up and that they are then served in order of arrival. Let Wn be the waiting-time of the nth customer and suppose that the server is idle at the start, i.e. W1 = 0. Put W = lim n ∞ Wn when the limit exists. Furthermore, let Fn (⋅) be the c.d.f. of Wn and put EWn =ω n .

Probability distributions of aggregate sickness durations

Abstract: The statistic that is obviously appropriate for calculating the premium to be charged for sickness indemnity insurance is the so-called “sickness rate”, namely the age-specific average number of days of covered sickness per person per annum. In fact most sickness experiences since early in the nineteenth century have been limited to the calculation of sickness rates for various covered durations of sickness. However if one is interested in the probability distribution of the length of sickness, as one would be if one wished to calculate the risk reserve required on a given portfolio of sickness indemnity contracts, it is necessary to measure both the probability of becoming sick in a specified interval and that of remaining sick for various periods thereafter. An early example of the calculation of these two different types of probability is the Leipzig sickness society's publication of its data on the first 34 weeks of sickness collected in the period 1887-1905. 1

On risk reserve processes

Abstract: It is supposed that a company deals with insurance and annuities and the risk reserve varies according to a compound recurrent process. In this paper mathematical methods are given for finding the probability that the risk reserve does not become depleted in a time interval of length t. For the case where the risk reserve varies according to a compound Poisson process, this probability has been found by H. Cramer. The present paper contains a generalization of Cramer's result.

A characterization of the Pareto distribution

Abstract: It is well known that certain distributions may be characterized by properties of suitable statistics. Kac [5] proved that if random variables X and Y are independent, then X + Y and X − Y are independent if, and only if, both X and Y have normal distributions with a common variance. Characterization of the normal distribution by the independence of the sample mean and the sample variance is given by Kawata & Sakamoto [6]. A characterization of the gamma distribution has been proved by Hogg [3] and Lukacs [7]. Characterization of the exponential distribution is considered by Ferguson [1] and Govindarajulu [3]. Several theorems based on properties of some suitable statistics characterizing normal, Poisson and gamma distributions have been given by Lukacs [8]. Fisz [2] has also characterized some probability distributions, in particular the power density function. The purpose of this note is to characterize the Pareto distribution whose probability density function is given by

On a generalization of Gautschi's inequality

Abstract: In this note a generalization of Gautschi's inequality satisfied by the gamma function is obtained.

Some density functions of a counting variable in the simple random walk

Abstract: Extract Let 〈Xi 〉 be a sequence of completely independent and identically distributed random variables such that Xi = + 1, −1 with probabilities p (0 (µ + λ)n holds for only finitely many n-values, where λ is any positive number less than 2 to avoid triviality. Define the indicator variables Yn ≡ Yn (p, λ) by Yn = 1 if Sn > (µ + λ)n and Yn = 0 otherwise. The counting variable of interest here is N∞ ≡ ∞ (p, λ) and is defined by . Hence, P{N∞ < ∞} = 1, that is, N∞ is an honest random variable or N∞ has a proper distribution. The following theorem provides some exact density functions of N∞ for selected combinations of p and λ and thus elucidates the nature of chance fluctuations of sums of Bernoulli random variables with respect to the “finitely man...

On some analogues of Gautschi's inequality satisfied by the beta and the hypergeometric functions

Abstract: In this note, some analogues of Gautschi's inequality satisfied by the beta and the hypergeometric functions are obtained along the lines of the present author's previous paper [1].

On a computer-oriented approach to actuarial calculations

Abstract: The literature about the use of electronic computers in life insurance companies is very extensive. But still there is not much written about the possibility to by-pass commutation functions in making actuarial calculations. Most actuaries seem to use basically the same methods nowadays as they used twenty years ago when they had to do all calculations by hand or with the aid of very simple mechanical devices. But as a consequence of the increasing speed of generally used data processing equipment there will probably be a change in this attitude because of the great flexibility that can be reached by more advanced methods.

A note on the behaviour of the “tail” of the mixture-poisson distribution

Abstract: A mixture-Poisson distribution is defined by where U(x) is a distribution function concentrated on (0, ∞). This distribution has been applied as a model of the number of claims occurring in an insurance business during a certain period of time.

Moments of the bivariate logarithmic series distribution

Abstract: In this paper we discuss the structure of the moments of the bivariate logarithmic series distribution (BLSD) and then point out the similarity between the structure of the moments of the BLSD and that of the cumulants of the bivariate negative binomial distribution (BNBD). Similar results for the univariate logarithmic series distribution are given by Wani (1967) and Patil & Wani (1965).

Mortality in the general population and among the insured in Sweden in 1967

Abstract: The Research Council for Actuarial Science and Insurance Statistics, in everyday Swedish shortened to “Forsakringstekniska forskningsnamnden”, was founded in 1950 by the Swedish life offices to promote statistical and actuarial research in those fields, where the individual companies had inadequate resources. One of the main tasks of the Council was to study the mortality among the insured.