Showing papers in "Scandinavian Actuarial Journal in 1976"
TL;DR: It is shown that to any set of bonus rules, there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963, and the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules.
Abstract: Summary This paper deals with the problem of designing experience rating systems of the bonus type, commonly used in automobile insurance. On the basis of a simple model the mean squared deviation between a policy's expected claim amount and its premium in the nth insurance period as n→∞, is taken as a measure of the efficiency of a bonus system. It is shown that to any set of bonus rules (which determines the bonus class transitions of the policies), there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963. Thus the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules. Examples are given of comparison between different bonus rules. In one example the present Norwegian bonus system is compared to alternative systems. Comments are made on earlier papers on bonus systems. The credibility theoretic foundation is laid in a separate section.
77 citations
TL;DR: In this article, it has been shown that the single cause failure time distributions conditional on the cause of death, each in presence of the remaining causes, are the same as the distribution of observable failure time, regardless of the cause.
Abstract: Suppose that death (or any non-repetitive event) can occur due to various causes, each having its own failure time. Assuming independence of failure times and proportional hazard rates over the whole range of time, some authors have shown that the single cause failure time distributions conditional on the cause of death, each in presence of the remaining causes, are the same as the distribution of observable failure time, regardless of the cause. It has been shown in the present article, that this result is also valid without the assumption of independence (section 3). It has also been suggested (Section 5), that in the case of dependent failure times, a conditional limiting distribution as T α→∞ could represent the failure time distribution when cause C α is eliminated. Three examples (trivariate exponential, bivariate Gompertz, and US Life Table 1959–61 data) are given as illustrations.
50 citations
TL;DR: In this paper, the authors apply the theory of integro-differential inequalities to obtain bounds on probability of ruin over finite and infinite times respectively, and apply these results to queuing theory.
Abstract: Summary Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. These equations contain terms which cause difficulty in obtaining analytical solutions. One partial way out of this difficulty is to omit the awkward terms, or modify them, to produce a more tractable equation. The price to be paid for this is that the equation becomes a differential or integro-differential inequality. The paper concentrates on applying the theory of such inequalities to obtain bounds on probability of ruin over finite and infinite times respectively. By these methods it is possible to 1. (1) sharpen Lundberg's inequality; 2. (2) order (in some cases) probabilities of ruin over infinite time in accordance with the properties of different claim size distributions; 3. (3) give an upper bound for finite-time probability of non-ruin in terms only of probabilities involving (a) zero initial reserves, and (b) infinite time. Finally, the application of these results to queuing theory is i...
31 citations
TL;DR: In this article, strong convergence of to Δ was established under less assumptions than those of Schuster (1974) and established the asymptotic normality of established, which is the same rate of strong convergence to Δ as in this paper.
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.
28 citations
TL;DR: In this paper, two special classes of covariance matrices are considered which give simplified computations for linear forecasts without continued reinversion of the matrix, and the optimal coefficients in the forecast can be computed in advance for every time period by simple closed formulas.
Abstract: Two special classes of covariance matrices are considered which give simplified computations for linear forecasts without continued reinversion of the matrix. In the first class, the optimal coefficients in the forecast can be computed in advance for every time period by simple closed formulas. In the second class, which is a generalization of the first, the optimal coefficients are obtained through a simple first-order linear recursive relation between forecasts of successive time periods. Collective risk forecasting models which give rise to these classes of covariances are presented.
18 citations
TL;DR: In this paper, a dynamic programming algorithm is used to determine the optimal behavior of Scandinavian, Swiss and German drivers, as a function of two parameters: the claim frequency and the discount factor.
Abstract: Summary The introduction of a bonus-malus premium system in automobile insurance induces the insured drivers to defray the cost of the cheap claims themselves, in order to avoid any future increase in premiums. We analyse this “hunger for bonus” and use a dynamic programming algorithm to determine the optimal behaviour of Scandinavian, Swiss and German drivers, as a function of two parameters: the claim frequency and the discount factor. We then define an efficiency concept for a bonus-malus system and study the influence of the hunger for bonus on this notion.
17 citations
TL;DR: In this article, a systematic presentation of time-continuous stable population theory in modern probabilistic dress is presented, where life-time births of an individual are represented by an inhomogeneous Poisson process stopped at death, and an aggregate of such processes on the individual level constitutes the population process.
Abstract: This paper contains a systematic presentation of time-continuous stable population theory in modern probabilistic dress. The life-time births of an individual are represented by an inhomogeneous Poisson process stopped at death, and an aggregate of such processes on the individual level constitutes the population process. Forward and backward renewal relations are established for the first moments of the main functionals of the process and for their densities. Their asymptotic convergence to a stable form is studied, and the stable age distribution is given some attention. It is a distinguishing feature of the present paper that rigorous proofs are given for results usually set up by intuitive reasoning only.
16 citations
TL;DR: In this paper, it was shown that the conditional distribution of R = max(X 1, X 2)-min(X 2, X 1), given R > 0, is the same as the distribution of X 1.
Abstract: Let X 1, X 2 be independent identically distributed positive integer valued random variables. H the X i 's have a geometric distribution, then the conditional distribution of R = max(X 1, X 2)-min(X 1, X 2), given R > 0, is the same as the distribution of X 1. This property is shown to characterize the geometric distribution.
16 citations
TL;DR: In this paper, the authors consider the case where the claims occur at timepoints which are generated by a point process N(t), t⩾0 and assume that the sequences (T 1, T 2,...) and ((X 1, Y 1), (X 2, Y 2),...) are independent and identically distributed with distribution function G.
Abstract: In an insurance business claims occur at timepoints which are generated by a point process N(t), t⩾0. Multiple points are not allowed. Number the claims in the order they occur. Let the nth claim occur at T n and let the distribution function of T n be F n . Let Y n be the amount of the n-th claim and assume that this claim and its amount is reported at T n + X n . Suppose that the sequences (T 1, T 2, ...) and ((X 1, Y 1), (X 2, Y 2), ...) are independent and that the variables (X 1, Y 1), (X 2, Y 2), ... are independent and identically distributed with distribution function G.
11 citations
TL;DR: The need for random-numbers in simulation in nuclear physics, communications, risk theory, queue systems, industrial organization and many other related and similar topics is beyond question.
Abstract: The need for random-numbers in simulation in nuclear physics, communications, risk theory, queue systems, industrial organization and many other related and similar topics is beyond question. Sooner or later the calculation part of these problems grows beyond human powers and a computer has to be involved. When using this instrument, spending only microseconds on the arithmetic operations, we cannot spend seconds on generating random numbers with the classical tools: coins, dice, tombolas or something like that. However, many other methods are known which can be adapted for computer applications.
8 citations
TL;DR: In this article, the authors considered a collective risk model (R(t), 0≤t<∞) formed by a linear combination of four stochastic processes, and the first process was a compound Poisson one which portrayed the claims.
Abstract: Part I of this paper considered a collective risk model {R(t), 0≤t<∞} formed by a linear combination of four stochastic processes. The first process was a compound Poisson one which portrayed the claims. The other three processes were Ornstein-Uhlenbeck processes which served as models for deviations in assumptions about investment performance, operating expenses, and lapse expenses. We now remove the restriction that the claims process be a compound Poisson one. Furthermore, more general and meaningful values for the parameters in the Ornstein-Uhlenbeck processes are allowed. Basic properties of the R(t) process are presented. Probabilities of extreme deviations for the R(t) process are discussed, with several detailed examples.
TL;DR: In this paper, the authors exploit the results of Lloyd (1952) to obtain optimal linear estimators based on order statistics of location or/and scale parameter (s) of a continuous univariate data distribution.
Abstract: Summary An estimator which is a linear function of the observations and which minimises the expected square error within the class of linear estimators is called an “optimal linear” estimator. Such an estimator may also be regarded as a “linear Bayes” estimator in the spirit of Hartigan (1969). Optimal linear estimators of the unknown mean of a given data distribution have been described by various authors; corresponding “linear empirical Bayes” estimators have also been developed. The present paper exploits the results of Lloyd (1952) to obtain optimal linear estimators based on order statistics of location or/and scale parameter (s) of a continuous univariate data distribution. Related “linear empirical Bayes” estimators which can be applied in the absence of the exact knowledge of the optimal estimators are also developed. This approach allows one to extend the results to the case of censored samples.
TL;DR: In this paper, it was shown that the Gegenbeispiele conjecture has no solution, if m 2 > 2S, and a unique solution if k ⩾ 2 and m 2 < 2S.
Abstract: Summary In a paper in Biometrika, Anscombe (1950) considered the question of solving the equation with respect to x. Here “Log” denotes the natural logarithm, while N s , where N k >0 and N s =0 for s>k, denotes the number of items ⩾s in a sample of independent observations from a population with the negative binomial distribution and m denotes the sampling mean: it can in the case k ⩾ 2 be shown that the equation (*) has at least one root. In vain search for “Gegenbeispiele”, Anscombe was led to the conjecture (l.c., 367) that (*) has no solution, if m 2 > 2S, and a unique solution, if k ⩾ 2 and m 2 < 2S. In the latter case, x equals the maximum-likelihood estimate of the parameter ϰ. In the present paper it will, after some preliminaries, be shown that the equation (*) has no solution, if k=l, or if k⩾2 and m 2 ⩾ 2S, whereas (*) has a unique solution, if k ⩾ 2 and m 2 < 2S.
TL;DR: In this article, the authors consider an insurance portfolio and let X be a random variable equal to the yearly net result of the portfolio, i.e. premiums earned minus cost of claims and minus operational costs.
Abstract: Consider an insurance portfolio and let X be a random variable equal to the yearly net result of the portfolio, i.e. premiums earned minus cost of claims and minus operational costs.
TL;DR: In this paper, the authors present the definitions and modus operandi of renewal (point, or recurrent event, processes) processes in the stationary community of lives, where deaths at any epoch are immediately replaced by an equivalent number of births.
Abstract: It was the Swiss actuary Chr. Moser who, in lectures at Bern University at the turn of the century, gave the name “self-renewing aggregate” to what Vajda (1947) has called the “unstationary community” of lives, namely where deaths at any epoch are immediately replaced by an equivalent number of births. It was Moser too (1926) who coined the expression “steady state” for the stationary community in which the age distribution at any time follows the life table (King, 1887). With such a distinguished actuarial history, excellently summarized by Saxer (1958, Ch. IV), it behoves every actuary to know at least the definitions and modus operandi of today's so-called renewal (point), or recurrent event, processes.
TL;DR: In this article, an empirical Bayes approach to interval estimation of an unknown parameter λ of a univariate data distribution is formulated, with special reference to some specific parametric forms of the data distributions and prior distributions.
Abstract: Summary An empirical Bayes approach to interval estimation of an unknown parameter λ of a univariate data distribution is formulated, with special reference to some specific parametric forms of the data distributions and prior distributions. The connection with the notion of tolerance limits is traced.
TL;DR: In this article, the problems caused by inflation in life insurance business are quite different from the problems within non-life business and will not be dealt with here, as discussed in this paper.
Abstract: This paper deals exclusively with non-life insurance business. The problems caused by inflation in life insurance business are quite different from the problems caused within non-life business and will not be dealt with here.