Extreme values are often correlated over time, for example, in a financial time series,
and these values carry various risks. Max-stable processes such as maxima of moving
maxima (M3) processes have been recently considered in the literature to describe timedependent
dynamics, which have been difficult to estimate. This paper first proposes
a feasible and efficient Bayesian estimation method for nonlinear and non-Gaussian
state space models based on these processes and describes a Markov chain Monte Carlo
algorithm where the sampling efficiency is improved by the normal mixture sampler.
Furthermore, a unique particle filter that adapts to extreme observations is proposed
and shown to be highly accurate in comparison with other well-known filters. Our
proposed algorithms were applied to daily minima of high-frequency stock return data,
and a model comparison was conducted using marginal likelihoods to investigate the
time-dependent dynamics in extreme stock returns for financial risk management.
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