Showing papers in "Scandinavian Actuarial Journal in 1974"
TL;DR: In this paper, the authors make a contribution to the theory of demand for insurance, showing that, given a range of alternative possible insurance policies, the insured would prefer a policy offering complete coverage beyond a deductible.
Abstract: This report is intended as a contribution to the theory of demand for insurance. In many circumstances, it appears that, given a range of alternative possible insurance policies, the insured would prefer a policy offering complete coverage beyond a deductible. In an earlier paper (Arrow [1]; reprinted in Arrow [3], pp. 212-216), this argument was developed for the case where the risk being insured against was, effectively, loss of income. Recently, Ehrlich and Becker [4] have extended these results considerably, as well as analyzing other responses of the insured to the price of insurance, responses beyond the scope of this study. For some other related work, see Pashigian, Schkade, and Menefee [8], Smith [12], and Gould [6]. However, income is not the only uncertainty, especially in the context of health insurance, and only under special and unrealistic circumstances can it be held that the other uncertainties have income equivalents. Put loosely, the marginal utility of income will in general d...
299 citations
TL;DR: In this article, the probability of non-ruin in risk theory with an assumption that X(t), the accumulated claims during the interval (0, t), is a stochastic process with independent increments occurring at the event points of a stationary process.
Abstract: Summary A formula for U(w, t), the distribution function of the waiting time of a potential customer who joins a queue with a single server at epoch t after service commences without a queue was derived for dam theory by Gani & Prabhu (1959) and for queues by Benes (1960). Here we use it to calculate numerically the probability of non-ruin in risk theory with an assumption that X(t), the accumulated claims during the interval (0, t), is a stochastic process with independent increments occurring at the event points of a stationary process. The difficulties encountered are described in some detail and suggestions made for the attainment of three-decimal accuracy in U(w, t).
56 citations
Abstract: The question how a company should payout dividends has led to some controversy (see [2], p. 164). At first sight it seems that the payment of dividends could be reconciled with the safety of a company by the following rule: “At any payment date, determine the dividend such that the resulting probability of ruin is equal to some given level e.” But, as De Finetti points out (see [5]), repeated application of this rule leads to eventual ruin of the company (with probability one). Alternatively, he suggested that dividends could be determined in order to maximize the expected sum of the discounted dividends. This idea has inspired a series of writers, among others Borch, Morrill, Miyasawa (the reader will find references in [2], pp. 164–178, and [9], pp. 163–166).
45 citations
TL;DR: In this paper, Thorin has shown that the Wiener-hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
Abstract: During the latest few years much attention has been given to the study of the ruin problem of a risk business when the epochs of claims form a renewal process. The study of this problem was initiated by E. S. Andersen (1957). Thorin has then in a series of papers (Thorin, 1970, 1971a, 1971b) shown that the Wiener-Hopf technique, originally developed by Cramer (1955) in the case of a Poisson process, can be used in this more general case, and Takacs (1970) has derived results similar to those of Thorin by an entirely new technique.
43 citations
TL;DR: In this article, the authors give the rate of strong uniform convergence of the distribution estimators Fn (x) = ∫x−∞ f (u)du to the population cdf F(x), where cdf is the usual empirical distribution function.
Abstract: Bhattacharyya & Roussas (1970) have proposed that one estimates T(f) = ∫ ∞−∞f 2(u)du by estimates of the form T(fn ) where fn is a suitably chosen kernel estimator of the density f of the type studied by Rosenblatt (1956), Parzen (1962), and others. In this paper we first give the rate of strong uniform convergence of the distribution estimators Fn (x) = ∫x−∞ f (u)du to the population cdf F(x) = ∫x−∞ f(u)du. We then interpret T(f) as T(f, F) = ∫x−∞ f(u)dF(u) and estimate T(f, F) by T(fn , Fn ). Looking at the estimates in this fashion allows us to use both the properties of fn and Fn in establishing the rate of strong consistency of T(fn ) to T(f). We then consider the computationally simpler estimate where is the usual empirical distribution function. The rate of convergence remains the same.
32 citations
TL;DR: For non-arithmetic F(x) = ∫∞0 P(x + ct)dK(t), a slightly modified asymptotic formula is proved in this paper, where P(y), − ∞ < y < ∞ denotes the distribution function of the claim amounts and c denotes the gross risk premium per time unit.
Abstract: Summary In two previous papers [11] and [12] the author i.a. proved generalizations of Cramer's classical [4] asymptotic formula for the ruin probability for an infinite period proved by him when the epochs of claims form a Poisson process. However, these generalizations relied on a restriction as to the distribution function, K(t), t ⩾ 0, K(O) = 0, for the interoccurrence times between successive claims. In the present paper this restriction is relaxed for non-arithmetic F(x) = ∫∞0 P(x + ct)dK(t). For arithmetic F(x) a slightly modified asymptotic formula is proved. Here P(y), − ∞ < y < ∞ denotes the distribution function of the claim amounts and c denotes the gross risk premium per time unit. Of course, the restrictions on c and P(y) and −for c<0—on K(t) necessary for the existence of the positive constant R are still assumed. However, for functions not satisfying these conditions it is sometimes possible to give other asymptotic formulas. An example is given enclosing the case when P(y) is of Pareto type.
16 citations
TL;DR: In this paper, it was shown that the independent random vectors x and y have multinomial (negative mUltinomial) distributions with the same parameter vector o, and the other parameters being respectively m and n if and only if the conditional distribution of x given x + y is multivariate hypergeometric (multivariate inverse hypergeometry) distribution with parameters m + n = N and x+ y = N
Abstract: The intent of this paper is to show that the independent random vectors x and y have multinomial (negative mUltinomial) distributions with the same parameter vector o, and the other parameters being respectively m and n if and only if the conditional distribution of x given x + y is multivariate hypergeometric (multivariate inverse hypergeometric) distribution with parameters m + n = N and x + y = N
7 citations
TL;DR: The first passage time processes of Brownian Motion with positive drift are of considerable importance, particularly in life testing or life-time situation as a natural consequence as discussed by the authors, and have been use...
Abstract: The first passage time processes of Brownian Motion with positive drift are of considerable importance, particularly in life-testing or life-time situation as a natural consequence. It has been use...
7 citations
TL;DR: In this article, the authors apply the Basu form (1958) of the Rao-Blackwell theorem to the RHC method and the estimator proposed by Rao et al. (1962).
Abstract: In this note we apply the Basu form (1958) of the Rao-Blackwell theorem to the RHC-method and the estimator proposed by Rao et al. (1962). The admissible estimator derived is suitable, however, only for small samples. In consequence, a simple approximation is derived following lines somewhat similar to those used by Hartley & Rao (1962). In terms of the approximation certain results on efficiency are derived which supplement those given previously in the literature.
7 citations
TL;DR: In this paper, a reinsurer has free reserves of amount U at his disposal and a portfolio characterised by the distribution function Fx (z; µ σ2), where X is a stochastic variable describing the accumulated loss during a certain time interval; µ, and σ 2 = V are the expected value and the variance of X respectively.
Abstract: Suppose a (re)insurer has free reserves of amount U at his disposal and a portfolio characterised by the distribution function Fx (z; µ σ2). X is a stochastic variable describing the accumulated loss during a certain time interval; µ, and σ2) = V are the expected value and the variance of X respectively.
7 citations
TL;DR: In this paper, a sequence of maximum likelihood estimators based on independent but not necessarily identically distributed random variables is shown to be consistent under certain assumptions, and some examples are given to show that these assumptions are easy to verify and not very restrictive.
Abstract: A sequence of maximum likelihood estimators based on a sequence of independent but not necessarily identically distributed random variables is shown to be consistent under certain assumptions. Some examples are given to show that these assumptions are easy to verify and not very restrictive.
TL;DR: In this paper, the authors report on some experiments which they have made with the aid of simulation technique and compare different strategies, which an insurance company could use for its decisions regarding the premium level of the company.
Abstract: In this paper I am going to report on some experiments which I have made with the aid of simulation technique. The purpose of the experiments was to compare different strategies, which an insurance company could use for its decisions regarding the premium level of the company.
TL;DR: In this paper, a particle starts a random walk from the point U on the χaxis and each step it takes during its random walk along the axis is a random variable with mean value λ, standard deviation a and coefficient of skewness γ.
Abstract: A particle starts a random walk from the point U on the χaxis. Each step it takes during its random walk along the χaxis is a random variable with mean value λ, standard deviation a and coefficient of skewness γ.
TL;DR: In this article, the authors assume that the mean value of the distribution is zero, i.e. that φ'(0) = 0, and they use the following formula where The dash on the summation sign indicates that the term corresponding to k = 0 is missing.
Abstract: Given a characteristic functionφ(t) we want to calculate the corresponding distribution function. For the sake of simplicity we will assume that the mean value of the distribution is zero, i.e. that φ'(0) =0. For these calculations we will use the following formula where The dash on the summation sign indicates that the term corresponding to k = 0 is missing.
TL;DR: In this article, the authors studied the large-sample properties of the joint distribution of {X 2 N1,..., X 2 Nr} when these cannot be assumed to be independent.
Abstract: Let {X 2 N1, ..., X 2 Nr} be statistics of the type X 2 = Σ({O} − E)2/ E for testing mUltiple hypotheses in a categorical array of N objects. This study is concerned with the large-sample properties of the joint distribution of {X 2 N1, ..., X 2 Nr} when these cannot be assumed to be independent. The limiting form of the joint distribution is given explicitly under quite general conditions for a large class of joint distributions involving Pearson's statistics. In the case of tests for goodness of fit of simple hypotheses, the rate of convergence to this limiting form is given precisely in terms of multidimensional Esseen type bounds.
TL;DR: In this article, the gamma and beta functions and the hyper geometric function with the last argument unity with respect to the last-argument unity were investigated and some interrelationships between them were established.
Abstract: In this note some interrelationships between, and inequalities involving, the gamma and beta functions and the hyper geometric function with the last argument unity are established.
TL;DR: In this article, the joint density function hn =hn (x m (n); θ) of the X jn's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2,..., θ k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k⩽m( n).
Abstract: Let X m(n) =(X j , n, ..., X j m,n ) be a subset of observations of a sample Xn = (X1n X 2n ... , X nn ). Here the Xjn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function hn =hn (x m (n); θ) of the X jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k⩽m(n).
TL;DR: In this article, the Levy parameters of the inverse gaussian distribution are obtained and the asymptotic behavior at infinity of Bessel processes and the inverse Gaussian distribution corresponds to no random passage time.
Abstract: The Levy parameters of the inverse gaussian distribution are obtained. Indices for inverse gaussian processes and Bessel processes are computed and used to compare small time sample path properties of the two classes of stochastic processes. The asymptotic behavoir at infinity of inverse gaussian and Bessel processes is discussed. It is shown that the inverse gaussian distribution corresponds to no random passage time.
TL;DR: In this paper, it was shown that S t is a non-decreasing continuous function with an existing second differential for all t ⩾ 0. And P t is also a continuous function having the same second differential.
Abstract: We indicate by S t a non-decreasing function having the differential dSt υ defined for all t⩾0 (S0=0); P t a non-decreasing continuous function with an existing second differential for all t ⩾0.
TL;DR: The two-parameter Pareto distribution of a random variable (r.v.) X with probability density function (p.d.f.) was proposed in this article, where
Abstract: The two-parameter Pareto distribution of a random variable (r.v.) X with probability density function (p.d.f.)
TL;DR: The generalized gamma distribution having the density function was introduced by Stacy (1962) who studied some of its properties as mentioned in this paper, and it has been shown that certain functions of a normal variable have Stacy's gamma form.
Abstract: The generalized gamma distribution having the density function was introduced by Stacy (1962), who studied some of its properties. As observed by Stacy, many standard distributions are special cases of (1). For example d=p=1 gives the exponential, p=1 gives the gamma p=1 and d=n/2 (n is a positive integer) gives the chi-squared, and d=p gives the Weibull distribution. Furthermore, certain functions of a normal variable— viz., its positive even powers, its modulus, and all positive powers of its modulus, have Stacy's gamma form.
TL;DR: The problem of χ2 tests of a linear hypothesis H0 for matched samples in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). as mentioned in this paper presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c − 1) functional restrictions.
Abstract: The problem of χ2 tests of a linear hypothesis H0 for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c −1) functional restrictions: F 1(p) = 0, ... , F C−1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ2 and the modified minimum χ2 (=χ*2) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).
TL;DR: At the beginning of 1973 new technical bases for calculating premiums and reserves for actual cases (the ‘sickness reserve’) regarding Long term individual sickness insurance have been adopted in Sweden.
Abstract: At the beginning of 1973 new technical bases for calculating premiums and reserves for actual cases (the ‘sickness reserve’) regarding Long term individual sickness insurance have been adopted in Sweden. The former technical bases being valid from 1965 have been described by the author in the 1969 volume of this journal (Dillner, 1969).
TL;DR: In this article, the distribution of any linear combination of a finite number of truncated exponential variates from possibly n distinct populations is obtained by using the Laplace transform, and the distribution is demonstrated in a compact form which is quite suitable for computational purposes.
Abstract: The distribution of any linear combination of a finite number of truncated exponential variates from possibly n distinct populations is obtained by using the Laplace transform. The distribution is demonstrated in a compact form which is quite suitable for computational purposes. The results are exemplified. Finally, a brief remark on the distribution of the product of truncated exponential variates is also added.
TL;DR: In this paper, it is shown that the constant n in (l) is the best possible, in the sense that where L(F, G) = sup {t: ϕ(s) = ψ (s),|s| ⩽ t}, the product in (2) being interpreted as 0 whenever any factor is O.
Abstract: It is a well-known fact (see e.g. Feller, 1971, p. 506) that two distinct characteristic functions ϕ and ψ can coincide in some finite interval [-L, L]. However, as was shown by Esseen (1945, p. 30), the corresponding distribution functions F and G must then satisfy The purpose of the present note is to show that the constant n in (l) is the best possible, in the sense that where L(F, G) = sup {t: ϕ(s) = ψ (s),|s| ⩽ t}, the product in (2) being interpreted as 0 whenever any factor is O. On account of (1), it clearly suffices to prove that sup F, G L(F, G)ϱ(F, G) ⩾ π.