Compound Binomial Model Research Articles

On the first time of ruin in the bivariate compound Poisson model

This paper considers a bivariate compound Poisson model for a book of two dependent classes of insurance business. We focus on the ruin probability that at least one class of business will get ruined. As expected, general explicit expressions for this bivariate ruin probability is very difficult to obtain. In view of this, we introduce the so-called bivariate compound binomial model which can be used to approximate the finite-time survival probability of the assumed model. We then study some simple bounds for the infinite-time ruin probability via the association properties of the bivariate compound Poisson model. We also investigate the impact of dependence on the infinite-time ruin probability by means of multivariate stochastic orders.

On a class of discrete time renewal risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of the distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. A recursive formula is derived for the expected discounted penalty (Gerber-Shiu) function, which can be used to analyze many quantities associated with the time of ruin, e.g., the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.

Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.

Ruin probability in the continuous-time compound binomial model

The continuous-time compound binomial model, introduced in this paper, is a continuous-time version of the compound binomial model in discrete time. Its skeleton chain is coincided with the compound binomial model and its limiting case is the compound Poisson model. We set the risk model in the framework of PDMP and get a exponential martingale by virtue of the extended generator (coupled with a discrete part) of the PDMP. The theory of change of measures is developed for this model. Some results are derived for the ruin probabilities, such as the general expressions for ruin probabilities, Lundberg bounds and Cramér–Lundberg approximations and etc. All the results are parallel to the corresponding ones in the compound Poisson risk model. And a closed-form expression for the ruin probability is obtained under some special cases.

Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.

Compound binomial risk model in a markovian environment

In this paper, we propose a compound binomial model defined in a markovian environment which is an extension to the compound binomial model presented by Gerber (1988) [Mathematical fun with the compound bionomial process. ASTIN Bull. 18, 109–123; Mathematical fun with ruin theory. Ins. Math. Econ. 7, 15–23]. An algorithm is presented for the computation of the aggregate claim amount distribution for a fixed time period. We focus on infinite-time ruin probabilities and propose a numerical algorithm to compute their numerical values. Along the same lines as Gerber’s compound binomial model which can be used as an approximation to the classical risk model, we will see that the compound binomial model defined in a markovian environment can approximate the risk model based on a particular Cox model, the marked Markov modulated Poisson process. Finally, we compare via stochastic ordering theory our proposed model to two other risk models: Gerber’s compound binomial model and a mixed compound binomial model. Numerical examples are provided.

The discrete stationary renewal risk model and the Gerber–Shiu discounted penalty function

This paper considers the stationary and the ordinary discrete renewal risk models. The main result is an expression of the Gerber–Shiu discounted penalty function in the stationary model in terms of the corresponding Gerber–Shiu function in the ordinary model. Subsequently, this relationship is considered in more detail in both the discount free case and under the compound binomial model. The latter case may be viewed as a discrete analog of the classical Poisson model. Simplifications of the general relationship are obtained, and a connection between the defective joint cumulative distribution functions of the surplus prior to ruin and the deficit at ruin in the stationary and the ordinary renewal risk models is established. Moreover, the defective probability function of the claim causing ruin is derived in the compound binomial case.

Use of a correlated compound-binomial model to assess absence of non-counting noise in Pu-isotope ratios measured by AMS at LLNL

Methods for error analysis pertaining to isotopic ratio measures made by accelerator mass spectrometry (AMS) typically focus on accuracy/precision of detected counts of a single rare isotope (e.g. 14C), relative to a current strength indicating the abundance of an isotope present in far greater amounts (e.g. 13C). Because AMS analysis of the relative abundance of actinide isotopes involves comparing detected counts that each correspond to a relatively rarely occurring isotope, error analysis is complicated by correlations between isotope-specific counts and by isotope-specific dead times, that affect each pair of isotope-specific counts. A new method of error analysis described here was developed specifically to estimate and characterize error in AMS measures of actinide isotope ratios. The method models a series of detected pairs of isotope counts as correlated compound-binomial (censored trinomial contingency) data, provides an approximately unbiased moment-method estimator of a common isotopic proportion (or ratio) corresponding to any data-pair series, and provides a corresponding homogeneity test for isotopic proportions observed within any data-pair series. Applications of this method to AMS measures of the relative abundance of plutonium isotopes, in samples analyzed at the Lawrence Livermore National Laboratory Center for AMS, revealed that observed measurement errors were nearly all attributable to modeled counting errors.

Exact expressions and upper bound for ruin probabilities in the compound Markov binomial model

The compound Markov binomial model was first proposed by Cossette et al. [Scandinavian Actuarial Journal (2003) 301] to introduce time-dependence in the aggregate claim amount increments. As pointed out in [Scandinavian Actuarial Journal (2003) 301], this model can be seen as an extension to Gerber’s compound binomial model. In this paper, we pursue the analysis of the compound Markov binomial model by first showing that the conditional infinite-time ruin probability is a compound geometric tail. Based on this property, an upper bound and asymptotic expression for ruin probabilities are then provided. Finally, special cases of claim amount distributions are considered which allow the exact calculation of ruin probabilities.

Use of a correlated compound-binomial model to assess absence of non-counting noise in Pu-isotope ratios measured by AMS at LLNL

Methods for error analysis pertaining to isotopic ratio measures made by accelerator mass spectrometry (AMS) typically focus on accuracy/precision of detected counts of a single rare isotope (e.g. 14 C), relative to a current strength indicating the abundance of an isotope present in far greater amounts (e.g. 13 C). Because AMS analysis of the relative abundance of actinide isotopes involves comparing detected counts that each correspond to a relatively rarely occurring isotope, error analysis is complicated by correlations between isotope-specific counts and by isotope-specific dead times, that affect each pair of isotope-specific counts. A new method of error analysis described here was developed specifically to estimate and characterize error in AMS measures of actinide isotope ratios. The method models a series of detected pairs of isotope counts as correlated compound-binomial (censored trinomial contingency) data, provides an approximately unbiased moment-method estimator of a common isotopic proportion (or ratio) corresponding to any data-pair series, and provides a corresponding homogeneity test for isotopic proportions observed within any data-pair series. Applications of this method to AMS measures of the relative abundance of plutonium isotopes, in samples analyzed at the Lawrence Livermore National Laboratory Center for AMS, revealed that observed measurement errors were nearly all attributable to modeled counting errors.

Ruin Probabilities in the Compound Markov Binomial Model

In this paper, we present a compound Markov binomial model which is an extension of the compound binomial model proposed by Gerber (1988a, b) and further examined by Shiu (1989) and Willmot (1993). The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences. Recursive formulas are provided for the computation of the ruin probabilities over finite- and infinite-time horizons. A Lundberg exponential bound is derived for the ruin probability and numerical examples are also provided.

Problèmes de ruine en théorie du risque à temps discret avec horizon fini

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

Problèmes de ruine en théorie du risque à temps discret avec horizon fini

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

Ruin probabilities for time-correlated claims in the compound binomial model

In this paper we consider the ruin probability for a risk process with time-correlated claims in the compound binomial model. It is assumed that every main claim will produce a by-claim but the occurrence of the by-claim may be delayed. Recursive formulas for the finite time ruin probabilities are obtained and explicit expressions for ultimate ruin probabilities are given in two special cases.

Discounted probabilities and ruin theory in the compound binomial model

The aggregate claims are modeled as a compound binomial process, and the individual claim amounts are integer-valued. We study f(x, y; u), the “discounted” probability of ruin for an initial surplus u, such that the surplus just before ruin is x and the deficit at ruin is y. This function can be used to calculate the expected present value of a penalty that is due at ruin, and, if it is interpreted as a probability generating function, to obtain certain information about the time of ruin. An explicit formula for f(x, y; 0) is derived. Then it is shown how f(x, y; u) can be expressed in terms of f(x, y; 0) and an auxiliary function h(u) that is the solution of a certain recursive equation and is independent of x and y. As an application, we use the asymptotic expansion of h(u) to obtain an asymptotic formula for f(x, y; u). In this model, certain results can be obtained more easily than in the compound Poisson model and provide additional insight. For the case u=0, expressions for the expected present value of a payment of 1 at ruin and the expected time of ruin (given that ruin occurs) are obtained. A discrete version of Dickson’s formula is provided.

Estimators of Conditional Scale‐Score Standard Errors of Measurement: A Simulation Study

This paper describes four procedures previously developed for estimating conditional standard errors of measurement for scale scores: the IRT procedure (Kolen, Zeng, & Hanson. 1996), the binomial procedure (Brennan & Lee, 1999), the compound binomial procedure (Brennan & Lee, 1999), and the Feldt‐Qualls procedure (1998). These four procedures are based on different underlying assumptions. The IRT procedure is based on the unidimensional IRT model assumptions. The binomial and compound binomial procedures employ, as the distribution of errors, the binomial model and compound binomial model, respectively. By contrast, the Feldt‐Qualls procedure does not depend on a particular psychometric model, and it simply translates any estimated conditional raw‐score SEM to a conditional scale‐score SEM. These procedures are compared in a simulation study, which involves two‐dimensional data sets. The presence of two category dimensions reflects a violation of the IRT unidimensionality assumption. The relative accuracy of these procedures for estimating conditional scale‐score standard errors of measurement is evaluated under various circumstances. The effects of three different types of transformations of raw scores are investigated including developmental standard scores, grade equivalents, and percentile ranks. All the procedures discussed appear viable. A general recommendation is made that test users select a procedure based on various factors such as the type of scale score of concern, characteristics of the test, assumptions involved in the estimation procedure, and feasibility and practicability of the estimation procedure.

On s-convex stochastic extrema for arithmetic risks

Recently, Denuit and Lefèvre (Insurance: Mathematics and Economics 20 (1997) 197–213) have introduced a class of discrete s-convex stochastic orderings for comparing arithmetic risks in actuarial sciences inter alia. The present paper is concerned with the construction of the extremal distributions with respect to these orderings. Firstly, the general problem of bounding such risks is studied in some details. Then, improved extrema are obtained for the case where the risks are known to have a decreasing density function. For illustration, the results are applied to derive bounds for the probability of ruin in the compound binomial risk model.

Conditional Scale-Score Standard Errors of Measurement under Binomial and Compound Binomial Assumptions

This article develops two procedures for estimating individual-level conditional standard errors of measurement (SEMs) for scale scores, assuming tests consist of dichotomously scored items. Using binomial model assumptions, one procedure provides a scale-score analogue of Lord’s raw-score SEM. Assuming a compound binomial model, another procedure provides a scale-score analogue of Feldt’s raw-score SEM. These two procedures are compared to a polynomial procedure and a procedure developed by Feldt and Qualls using data from the Iowa Tests of Basic Skills.

Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefèvre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.

072038 (M20, M11, M13) Some comments on the compound binomial model : Dickson D.C.M., The University of Melbourne, Astin Bulletin, Vol. 24, No. 1, 1994, pp. 33–45

072038 (M20, M11, M13) Some comments on the compound binomial model : Dickson D.C.M., The University of Melbourne, Astin Bulletin, Vol. 24, No. 1, 1994, pp. 33–45