Discount Weighted Regression

A recursive procedure for discounted multiple linear regression is described, together with a discounted estimator for the residual variance. The method is simple to program and involves no matrix inversion. Two examples are given.     

Balanced credibility estimation

This paper shows how a multivariate credibility estimator should be adjusted to satisfy a linear balancing constraint.     

Discount Weighted Moving Averages

An estimator to replace exponentially weighted moving averages is described and examples of its use are given. It has similar limiting properties but includes the on-line estimation of the forecast error variance as an integral part. The procedures for starting off and intervention are given.     

Practical exponential smoothing credibility

Since some time so-called credibility estimators with geometric weights are of some practical importance. As alternatives one can use so-called exponential smoothing credibility estimators. In the present paper the second ones are prepared for practical application.     

Invariantly recursive credibility estimation

Necessary and sufficient conditions are given for recursivity of credibility estimators to be invariant against changes in term-length. A stationary invariant risk process is analysed, and a concept of asymptotic efficiency is developed.     

An unbayesed approach to credibility

It is shown how a classical credibility formula can be obtained by alternative methods, the Straub approach and the direct approach, none of which relies on the Bayes technique.     

Fuzzy integral on credibility measure

Fuzzy measure and credibility measure are two dissimilar concepts, and fuzzy integral based on fuzzy measure has been researched. This paper focuses on introduction of fuzzy integral on credibility measure and discusses its properties, since it has been not studied yet so far.     

Simple risk forecasts using credibility

Based on the credibility regression model of Hachemeister, we show how some risk characteristics of an insurance portfolio can be forecasted. Distribution free forecasts of individual pure and loaded risk premiums, mean risk, stability, fluctuation reserves and surplus are used to gain insight into the solvability problem.     

Non-linear regression in credibility theory

Let X be a random vector with distribution depending on a parameter treated as a random variable ?. The usual linear regression assumption is that E(X|?) can be displayed in the form yβ(?) where y is a fixed design matrix and β(?) an unknown vector. In the present paper we assume that E(X|?) is a rather arbitrary function ?(β(?)) of the unknown vector β(?) and we derive credibility approximations for β(?).     

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.     

Exponential smoothing and credibility theory

Some years ago Gerber & Jones (1975a, 1975b) investigated simple recursion formulas for the linear credibility estimator. In the present paper we reexamine such formulas and show their connection to models of modern times series analysis.     

供应链中供需协调及价格折扣订货模型研究

本文研究由单一供方和单一需方组成的供应链协调问题。通过制定一个合理的价格折扣范围,提出一种改进的订货策略—n阶段订货策略,使供应链整体利润最大化,同时使供需双方达到双赢。     

Time-risk Discount Valuation of Life Contracts

In this paper a new approach is developed to value life insurance contracts by means of the method of backward stochastic differential equation. Such a valuation may relax certain market limitations. Following this approach, the values of single decrement policies are studied and Thiele‘s-type PDEs for general life insurance contracts are derived.     

A gamma-minimax result in credibility theory

The statistical decision problem of experience ratemaking has been described by Bühlmann (1975) as a two-person game of the actuary (player 2) against (malevolent) nature (player 1). Within this framework premium calculation procedures are strategies of player 2 and some types of credibility formulae can be regarded as Bayesian strategies or in terms of statistical decision theory as (linear) Bayes estimators. Since the application of the Bayes principle of game theory to insurance ratemaking is not appropriate — usually the actuary has not enough information for identifying one single prior — it is quite natural to select a premium calculation procedure which is optimal according to the minimax principle of game theory or according to the so-called gamma-minimax principle. This principle is more adaptable than the minimax principle since it allows to take into account vague prior information. In this paper credibility formulae are derived which are gamma-minimax for special types of prior information similar to those in Bühlmann's paper.