Recently Markov-modulated compound Poisson models have gained its popularity in modelling insurance claims in the actuarial science literature. A Markov-modulated compound Poisson model can provide a realistic and flexibile way to model aggregate insurance claims by incorporating the impact of hidden states of an economy on claim frequencies and claim sizes. However, in practice, the Markov chain in the model is not observable. It is of practical interest to develop some methods to estimate the hidden state of the Markov chain and other unknown model parameters of the Markov- modulated compound Poisson model. This paper considers this important issue. We shall develop filters and smoothers for the hidden state of the economy underlying the Markov-modulated compound Poisson model. In general, we consider the case when both the stochastic intensity and the distribution of the claim sizes of the compound Poisson process depend on the hidden Markov chain. The filter and smoother provide an optimal way to estimate the insurance claims model in the "mean- squared-error" sense. We shall also develop estimators for the unknown model parameters of the Markov-modulated marked point process using the robust filter-based and smoother-based EM algorithms.
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