| There continues to be unfading interest in using parametric max-stable processes to study time dependence and clustered extremes in time series data. However, this comes with some difficulties largely due to inability to find enough models that fit to data directly without transforming the data and the barriers in estimating the often large number of model parameters.;In this work, we study the use of sparse maxima of moving maxima (M3) process with random effects and hidden Frechet type shocks from which we get a max-linear model. In this setting, the model is applicable to cases of tail dependence or independence depending on the parameter values. Some of the fine properties of the model include mirroring the dependence structure in real data, dealing with the undesirable signature patterns found in most parametric max-stable processes and being directly applicable to real data.;In the multivariate setting, we employ a sparse multivariate maxima of moving maxima (M4) process as the univariate setting does with the M3 process. A copula structure is added to our model, specifically to model the joint distribution of the random Frechet type shocks.;For parameter estimation, we use Monte Carlo methods and extend application to high-frequency financial time series data. Finally, since our model has a latent Markov process, we investigate the estimation of the latent process using discrete distribution similar to the well studied Sequential Monte Carlo (SMC) methods.;Keywords: Extreme value theory; max-stable processes; time series; Bayesian inference; max-linear models; latent process estimation; high-frequency financial data. |
