Hans U. Gerber

Some Results for Discrete Unimodality

Abstract In a classical theorem, Ibragimov demonstrated the strong unimodality of log-concave probability density functions. Comparable results for lattice distributions are exhibited and their potential significance is suggested.

The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain

A martingale argument is used to derive the generating function of the number of i.i.d. experiments it takes to observe a given string of outcomes for the first time. Then, a more general problem can be studied: How many trials does it take to observe a member of a finite set of strings for the first time? It is shown how the answer can be obtained within the framework of hitting times in a Markov chain. For these, a result of independent interest is derived.

On the probability of ruin in the presence of a linear dividend barrier

1. Introduction and Summary We shall consider the model that was introduced in Gerber (1974). Let {St } denote the compound Poisson process of the aggregate claims (given by the Poisson parameter λ...

Survival patterns in cystic fibrosis

Abstract This paper describes the Data Registry System of the Cystic Fibrosis Foundation. Through this system an annual up-dated and corrected data base has been prepared for cystic fibrosis patients seen at U.S.A. Centers from 1966 through 1972. From the data base, a large number of life table studies are calculated. A study is specified by its data base, the observation period, analysis year (age year or year since diagnosis), the Centers included, and the condition of patients at birth, and possible additional factors such as sex, age at diagnosis, or race. The annual rates of mortality are measured by a force of mortality technique. Two techniques have been used to appraise the mortality of particular groups, graphing the cumulative survival function for the group together with the cumulative survival function for a comparison group, and calculation of mortality ratios of actual to expected deaths (by an applicable standard mortality table). This latter approach lends itself readily to tests of significance. For the purpose of calculating mortality ratios, two standard tables have been prepared on the basis of the 1972 Data Base and the 1966–1972 observations. The first of these standard tables is for use with age year studies, and the second is for years since diagnosis studies. The paper comments on a number of mortality studies by calendar year, by sex, by condition at birth, by age at diagnosis, and of survival from diagnosis of patients diagnosed in 1966–1972. The calendar year studies indicate an improvement trend in the survival experience from 1966 to 1972 but a change may be occurring. The study by sex exhibits a higher female mortality. The 7% of patients who have meconium ileus at birth show a significantly higher first-year mortality, and a somewhat lower survival rate thereafter even if the first year of life experience is excluded. Mortality data for groups specified by age at diagnosis are given. For patients diagnosed during 1966–1972, there is some indication of better survival than for patients diagnosed prior to 1966. The continuation of these mortality studies on an annual basis should further our understanding of survival patterns for cystic fibrosis patients.

On the representation of additive principles of premium calculation

Abstract Two characterizations of mixtures of Esscher principles in terms of additivity combined with other conditions are given.

The dilemma between dividends and safety and a generalization of the Lundberg-Cramér formulas

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).

SOME PRACTICAL CONSIDERATIONS IN CONNECTION WITH THE CALCULATION OF STOP-LOSS PREMIUMS

Publisher Summary This chapter discusses some practical considerations in connection with the calculation of stop-loss premiums. A premium calculation principle is a rule that assigns a premium to any risk. In mathematical terms, a risk is a random variable given by its supposedly sufficiently regular distribution. The exponential principle involves the evaluation of the moment generating function of the risk at an argument. The net premium principle and the exponential principle are additive and iterative. Under a mild continuity condition, these two principles can be characterized by certain properties. Also, the exponential principle fits into the framework of the collective theory of risk. The exact calculation of a stop-loss premium is feasible if the claim amount distribution is arithmetic with a sufficiently large span. If F is an arbitrary compound Poisson distribution, the exact calculation of the stop-loss premium can be an extensive procedure and can lead to considerable round-off errors.

A Characteristic Property of the Poisson Distribution

Abstract Suppose that N is a nonnegative, integer-valued random variable and that the conditional distribution of N 1, ···, N m (given N) is multinomial. Then, according to a result by Moran and Renyi, N 1 and N 2 are independent if and only if N has a Poisson distribution. A new proof is given that uses only the generating function of a single variable.

On Optimal Cancellation of Policies

One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

The Wiener process with drift between a linear retaining and an absorbing barrier

The Wiener process with constant drift is modified by a time-dependent retaining barrier that increases at a constant rate and by an absorbing barrier at zero. Explicit expressions in terms of series expansions are derived for the Laplace transform and the probability density function of the time of absorption.