The paper solves the loss reserving problem using Kalman recursions in linear statespace models. In particular, if one orders claims data from run-off triangles to time series with missing observations, then state space formulation can be applied for projections or interpolations of IBNR (Incurred But Not Reported) reserves. Namely, outputs of the corresponding Kalman recursion algorithms for missing or future observations can be taken as the IBNR projections. In particular, by means of such recursive procedures one can perform effectively simulations in order to estimate numerically the distribution of IBNR claims which may be very useful in terms of setting and/or monitoring of prudency level of loss reserves. Moreover, one can generalize this approach to the multivariate case of several dependent run-off triangles for correlated business lines and the outliers in claims data can be also treated effectively in this way. Results of a numerical study for several sets of claims data (univariate and multivariate ones) are presented.

This work deals with prediction of IBNR reserve under a different data ordering of the non-cumulative runoff triangle. The rows of the triangle are stacked, resulting in a univariate time series with several missing values. Under this ordering, two approaches entirely based on state space models and the Kalman filter are developed, implemented with two real data sets, and compared with two well-established IBNR estimation methods — the chain ladder and an overdispersed Poisson regression model. The remarks from the empirical results are: (i) computational feasibility and efficiency; (ii) accuracy improvement for IBNR prediction; and (iii) flexibility regarding IBNR modeling possibilities.

This paper presents a Bayesian approach using Markov chain Monte Carlo methods and the generalized-t (GT) distribution to predict loss reserves for the insurance companies. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserves. To develop a more robust model for irregular claims, this paper extends the conventional normal error distribution to the GT distribution which nests several heavy-tailed distributions including the Student-t and exponential power distributions. It is shown that the GT distribution can be expressed as a scale mixture of uniforms (SMU) distribution which facilitates model implementation and detection of outliers by using mixing parameters. Different models for the mean function, including the log-ANOVA, log-ANCOVA, state space and threshold models, are adopted to analyze real loss reserves data. Finally, the best model is selected according to the deviance information criterion (DIC).

In this paper our object is to show that a certain class of nonstationary random processes can locally be approximated by stationary processes. The class of processes we are considering includes the time-varying autoregressive conditional heteroscedastic and generalised autoregressive conditional heteroscedastic processes, amongst others. The measure of deviation from stationarity can be expressed as a function of a derivative random process. This derivative process inherits many properties common to stationary processes. We also show that the derivative processes obtained here have alpha-mixing properties.