Showing papers in "Scandinavian Actuarial Journal in 1972"
TL;DR: Recently, Malik [6, 7] has done classical maximum likelihood estimation as well as Bayesian estimation of the index parameter v of a Pareto distribution with density function as mentioned in this paper, and
Abstract: Recently, Malik [6, 7] has done classical maximum likelihood estimation as well as Bayesian estimation of the index parameter v of a Pareto distribution with density function
62 citations
TL;DR: In this paper, it is shown that if the characteristic function of a distribution is known in the whole space of the distribution, then the distribution can be uniquely determined in every continuity point.
Abstract: If the characteristic function ƒ(u) of a distribution F(x) is known in the whole space of u, F(x) is uniquely determined in every continuity point. This is not the case, if ƒ(u) is given only in a finite part of the space. The question then arises, if it is possible to determine F(x) within certain limits.
50 citations
TL;DR: In this article, the exact distribution of Y (X 1 X 2 X 2, X m /X m+1... X k ) where X 1, X, X k are independently distributed as generalized Gamma variates is given.
Abstract: This article gives the exact distribution of Y=(X 1 X 2 ... X m /X m+1 ... X k ) where X 1 ..., X k are independently distributed as generalized Gamma variates. The case when (X 1, ..., X k ) is a simple random sample of size k from a generalized Gamma distribution, is discussed in detail and the other cases are mentioned briefly. The techniques of inverse Mellin transform and Calculus of residues are used in deriving the results. Different representations of the density of Y are also given and a particular case is verified.
31 citations
TL;DR: In this article, the author has described a method for determining optimum sample sizes in multivariate stratified surveys, and studied some of the mathematical properties of the optimum solution, which is similar to ours.
Abstract: In [1] the author has described a method for determining optimum sample sizes in multivariate stratified surveys. In this paper we study some of the mathematical properties of the optimum solution.
28 citations
TL;DR: Life table probabilities are developed for the multiple decrement table, a model in which each cause of death acts in the presence of other causes or is ‘dependent’, and for the associated single decrements table, which shows the operation of each cause as though it acted alone or ‘independently’.
Abstract: Life table probabilities are developed for the multiple decrement table, a model in which each cause of death acts in the presence of other causes or is ‘dependent’, and for the associated single decrement table, which shows the operation of each cause as though it acted alone or ‘independently’. The decrement for the ith cause, , in the multiple decrement table is the number of deaths due to the ith cause out of l 0 births; our method is provided by (12). The ‘independent’ decrement for the ith cause is calculated as though no other cause were operating, from the beginning of the table, with arbitrary radix ; Equation (21) shows its close relation to deaths in the single decrement table for all causes together (4). The modification needed is to increase the r in the program for an ordinary life table by an amount that allows for the loss of exposure to the ith cause because of operation of other causes.
24 citations
TL;DR: In this article, it is shown that when making a decision concerning the probability distribution of a random variable by means of observing this random variable, one is recommended by the statisticians to consider certain functions of the operating characteristic (O.C.) of the decision function as measures of the reliability of the actual decision made.
Abstract: It is a fact that when one is making a decision concerning the probability distribution of a random variable by means of observing this random variable, one is recommended by the statisticians to consider certain functions of the operating characteristic (O.C.) of the decision function as measures of the reliability of the actual decision made. For instance, the confidence coefficient of an interval estimator will as a rule be regarded as a measure of our confidence in the interval.
8 citations
TL;DR: This paper derived moments of order statistics from the Pareto distribution without the transformation of the ordered variates to the unordered ones, which is a known transformation in the theory of multiple integrals.
Abstract: The evaluation of multiple integrals which occur in order statistics distribution theory is involved due to the fact that the integration is to be carried on over an ordered range of variables of integration. This difficulty is sometimes completely obviated by transforming the ordered variates to the unordered ones. Several such transformations are available in the Theory of Multiple Integrals. In previous papers [2, 3] the author used one such transformation, and gave alternative simplified proofs of several known results in the distribution theory of order statistics from the exponential and the power function distributions. In this paper we use such a known transformation to derive moments (and distributions if necessary) of order statistics from the Pareto distribution. Malik [4] has derived moments of order statistics from this distribution without the transformation of the ordered variates to the unordered ones. The process of direct integration used by Malik becomes complicated for dealing...
5 citations
TL;DR: In this article, it was shown that the probability of a service completion is directly proportional to the number of customers momentarily present in the system, and that this criteria could be proved in the more general case in which the probability differential of a Service Completion is directly proportionally proportional to customers momentarily being present.
Abstract: It is well known that if in a queueing situation the arrivals occur in a Poisson stream with intensity λ, and probability differential of a service completion is μd t+o(dt), then in the steady state the successive epochs of service completion also occur in a Poisson stream with intensity λ, that of arrivals. Here, we shall show that this criteria could be proved in the more general case in which the probability differential of a service completion is directly proportional to the number of customers momentarily present in the system.
4 citations
TL;DR: In this article, a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0.
Abstract: Let X 1, X 2,... be a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0.
4 citations
TL;DR: In this article, the authors compare the two methods for two distributions of claims where the number of claims is a Poisson variate, and present a comparison of their approximate figures with the exact figures for the claim distribution.
Abstract: Let X 1, X 2,... be a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0. An alternate method of approximating Ψ(u, T) was presented in [10] by Olof Thorin and exemplified in [11] by Nils Wikstad. One of the purposes of this paper is to compare the two methods for two distributions of claims where the number of claims is a Poisson variate. The paper will also discuss the advantages and disadvantages of the two methods. We will also present a comparison of our approximate figures with the exact figures for the claim distribution
3 citations
TL;DR: In this paper, a multistage multiple decision procedure is developed to test if a number of sets of independent variables are binomially distributed and if so is the case, test if they are homogeneous with respect to the binomial parameter p. The binomial distributions are either untruncated or truncated in different ways.
Abstract: A multistage multiple decision procedure is developed to test if a number of sets of independent variables are binomially distributed and if so is the case, to test if they are homogeneous with respect to the binomial parameter p. The homogeneity hypothesis is decomposed in a hierarchy of homogeneity hypotheses of the nested type. The binomial distributions are either untruncated or truncated in different ways. In the multiple procedure the goodness of fit test is included as an integrated part. For those test situations which cannot be considered as two-sample tests, chi-square tests are applied throughout. Small-sample and large-sample versions of all tests are given, also of the goodness of fit tests. Special merits of the procedure are its power to exclude deviating and heterogeneous sets of variables and to separate different sets of variables where the variables within each set are homogeneous with respect to p, but the different sets have different values of p. The significance level of th...
TL;DR: In this paper, the authors characterised the Pareto distribution using the properties of the order statistics of a random sample of size N. A similar characterisation theorem has been recently proved by Malik [2].
Abstract: In this paper we characterise the Pareto distribution using the properties of the order statistics of a random sample of size N. A similar characterisation theorem has been recently proved by Malik [2].
TL;DR: In this article, the Variationsrechnung hat zur Aufgabe, unter den Funktionen F(x), die einer gewissen Klasse C angehoren, diejenige zu finden, welche ein Funktional J[F] zum Maximum oder Minimum macht.
Abstract: Die Variationsrechnung hat zur Aufgabe, unter den Funktionen F(x), die einer gewissen Klasse C angehoren, diejenige zu finden, welche ein Funktional J[F] zum Maximum oder Minimum macht. In der allgemeinen Variationsrechnung enthalt die Definition der Klasse C gewohnlich nur derartige Bedingungen, die erforderlich sind, urn die Existenz von J[F] zu sichern, z. B. Kontinuitat oder Differentierbarkeit bis zu einer gewissen Ordnung, und ausserdem eventuell gewisse “Nebenbedingungen” von der Form Jv [F] = cv (v = 0, 1, ..., n).
TL;DR: In this article, the authors studied the local behavior of Poisson distributed events and found that the Poisson distribution is only suitable to represent actually realized random processes within a limited range of time and space.
Abstract: The probability law of Poisson, which is specified by the formula assigns to every non-negative integer n the probability Φ(n, u). This distribution has many applications; it is successfully used to describe and explain various random phenomena observed in practical experiments. On the other hand, it plays a very important role in the modern theory of probability. In view of these facts it seems worthwhile to investigate closely the random processes related with the Poisson distribution. In this article we study the local behavior of Poisson distributed events. To our knowledge such problems have not yet found widespread attention. It may therefore be advantageous to indicate shortly the usefulness of such investigations: It is well-known that the Poisson distribution is only suitable to represent actually realized random processes within a limited range of time and space [6, 23, 29, p. 79]. The restriction of the scope of the investigation to intervals makes allowance for this fact. Furthermore ...
TL;DR: In this article, the authors developed a similar procedure for the Supervision of Insurance, discounting dividends, with the purpose of the present paper is to develop a procedure for discounting the dividends.
Abstract: Optimal policies for Insurance over an infinite future were considered in [4] with the use of a potential function. The purpose of the present paper is to develop a similar procedure for the Supervision of Insurance, discounting dividends.
TL;DR: In this article, the authors consider an insurance scheme where claims do not necessarily arise as a stationary process and demonstrate the power of very straightforward branching process methods outside their traditional realm of application, and show that there is, for an expanding portfolio, only one premium size which is fair in the sense that if the premium is larger than that, then the profit of the insurer grows infinite with time, whereas a smaller premium leads to his inevitable ruin.
Abstract: The purpose of this short note is to demonstrate the power of very straightforward branching process methods outside their traditional realm of application. We shall consider an insurance scheme where claims do not necessarily arise as a stationary process. Indeed, the number of policy-holders is changing so that each of them generates a random number of new insurants. Each one of these make claims of random size at random instants, independently but with the same distribution for different individuals. Premiums are supposed equal for all policy-holders. It is proved that there is, for an expanding portfolio, only one premium size which is fair in the sense that if the premium is larger than that, then the profit of the insurer grows infinite with time, whereas a smaller premium leads to his inevitable ruin. (Branching process models for the development of the portfolio may seem unrealistic. However, they do include the classical theory, where independent and identically distributed claims arise ...
TL;DR: In this article, the authors proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions of the form F 0(x) = Φ[(x - μ0)/σ0], where μ0 is location parameter, σ0 scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2.
Abstract: In [5] S. Holm proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions (d.f.) of the form F 0(x) = Φ[(x - μ0)/σ0], where μ0 is location parameter, σ0 scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2. If μ0 and (σ0 are known, the hypothesis H 0 is: H 0: H(x) = F 0(x) = Φ[(x−μ0)/σ0], while the three possible alternatives are H 1: H(x) > F 0(x) H 2: H(x) < F 0(x) H 3: H(x) ≠ F 0(x).
TL;DR: In this article, a comparison of two policies of paying premiums by applying the principle of minimum variance has been made, where the technique used is different from the one used in this paper.
Abstract: Recently Dharmadhikari [1] has obtained some interesting results about comparison of two policies of paying premiums by applying the principle of minimum variance. Here we prove some more results by applying the same principle. The technique used is different.
TL;DR: In this paper, the authors investigated the situation when the company has several departments dealing with different types of insurance, and the determination of optimal reinsurance procedures over an infinite future was investigated.
Abstract: A previous paper [3] dealt with determination of optimal reinsurance procedures over an infinite future, when the company conducts only one type of insurance. Here, we shall investigate the situation when the company has several departments dealing with different types of insurance.
TL;DR: In this paper, the distribution function of the total amount of claims in an interval of length t is defined, where t is the probability of not being ruined in the period (0, t) when the initial risk reserve is x.
Abstract: We use the same notations as Seal [1] and put F(x, t) = the distribution function of the total amount of claims in an interval of length t. u(x, t) = probability of not being ruined in the period (0, t) when the initial risk reserve is x. π1 = premium income per unit of time.
TL;DR: In this article, the authors present a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory, where they use the concepts of utility and subjective probability to quantify an insurance company's preferences for consequences and its opinions about the occurrence of events.
Abstract: In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. F...