This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a`best estimate'; reporting the variability of claims reserves; and use in dynamic financial analysis models.

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This paper extends the methods introduced in England & Verrall (2002), and shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or Bayesian techniques for clearly defined statistical models. A general procedure for bootstrapping is described, by extending the methods introduced in England & Verrall (1999), England (2002) and Pinheiro et al. (2003). The analogous Bayesian estimation procedure is implemented using Markov-chain Monte Carlo methods, where the models are constructed as Bayesian generalised linear models using the approach described by Dellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well known model introduced by Mack (1993). Mack's model is useful, since it can be used with data sets which exhibit negative incremental amounts. The techniques are illustrated with examples, and the resulting predictive distributions from both the bootstrap and Bayesian methods are compared.

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This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link:http://openaccess.city.ac.uk/3801/ Link to published version: http://dx.doi.org/10.1017/S1748499512000012 Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. is the row parameter in accident year i (related to the exposure of accident year i) and γ j is the column parameter for development period j (related to the runoff pattern), and to apply prior distributions to these parameters. We will assume that there is no prior knowledge about the runoff parameters, and we use non-informative prior distributions for γ j . By assuming informative prior distributions for the µ i 's we can incorporate external knowledge about the ultimate losses. We investigate a number of different formulations of these informative prior distributions, and examine the properties of the resulting posterior estimators. We also compare our results with the traditional BF method.An important observation will be that although we choose non-informative prior distributions for the parameters, their shapes may have a significant influence on the resulting claims reserves.Organization of the paper. In the remainder of this section we define the general Bayesian ODP Model and we discuss prediction in a Bayesian framework. In Sections 2 and 3 we then specify two different types of prior distributions (the uniform prior model with log link function and the gamma prior model). Parameter estimates, e.g. for γ j , are always denoted by γ j in the uniform prior model with log link function and with γ * j in the gamma prior model. In Section 4 we discuss parameter estimation via simulation methods and in Section 5 a numerical example 2 is provided. All the statements are proved in the appendix. Model assumptionsThe model assumptions are similar to those in the Bayesian claims reserving models presented in England-Verrall [4,5], Verrall [16] and Wüthrich-Merz [17], Section 4.4. We assume that the parameters are modeled through prior distributions and, conditional on these parameters, the incremental claims X i,j have independent ODP distributions for accident years i ∈ {0, . . . , I} and development years j ∈ {0, . . . , I}. The final development year is given by I and the observations at time I are given in the (upper) runoff triangleOur goal is to predict the future claims in the lower triangleModel 1.1 (Bayesian ODP model)• µ 0 , . . . , µ I , γ 0 , . . . , γ I , ϕ are independent positive random variables with joint density u(·).• Conditionally, given parameters Θ = (µ 0 , . . . , µ I , γ 0 , . . . , γ I , ϕ), are X i,j independent random variables withThe parameter µ i plays the role of the row parameter (related to the exposure of accident year i, see (1.2)), the γ j 's descri...

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It has been suggested that the rate of CD4 cell decline accelerates in parallel with decreasing numbers of cells; however, the statistical literature suggests the opposite. CD4 cells were counted about every 6 months in a cohort of 1264 human immunodeficiency virus-infected subjects (the Italian Seroconversion Study cohort). Kaplan-Meier analysis was used to estimate the time for CD4 cells to decline by 100 cells/mm3, conditional on reaching predefined levels. In addition, CD4 cell counts were modeled as a function of time since seroconversion in individuals with > or = 5 counts. Kaplan-Meier survival times for a 100 cell/mm3 decrease in CD4 cells increased as lower counts were reached (log rank test, P < .001). The shape of the overall fitted curve of the CD4 cell counts does not suggest an increasing rate of decline. Data from the Italian Seroconversion Study cohort do not show a general tendency for accelerating CD4 cell decline in association with lower counts.