Inference for autoregressive and moving average models with extreme value distribution via simulation study

Time series analysis has emerged as one of the most important statistical discipline and it has been applied in different fields over the years. Literature reviews show that independent identical distributed Gaussian random variables is not suitable for modelling extreme events. We evaluate the impact of dependence on the parameter estimates of Autoregressive (AR) and Moving Average (MA) processes with Gumbel distributed innovation. The performance of the parameter estimates of the Gumbelgeneralised Pareto distribution fitted to the autoregressive and moving average processes and their respective cluster maxima is also assess. The extension of time series to extreme value theory can be achieved by inducing time dependence in the underlying state of an extreme value process. Extreme values occur in clusters in the presence of dependence. Gumbel distribution, a member of the family of the generalised extreme value distribution is the possible limit for the entire range of tail behaviour between polynomial decrease and essentially a finite endpoint and it is known to fit well in many situations. It is important to make general statements that characterises time series extreme models over a range of sample sizes with varying degree of dependence. Such general characterisation for a given model is useful for the extremal behaviour of physical processes. To achieve our objectives, a stationary autoregressive and moving average models with Gumbel distributed innovation is proposed and we characterise the short-term dependence among maxima, arising from light-tailed Gumbel distribution over a range of sample sizes with varying degrees of dependence. Dependence is induced through a linear filter operation. The linear filter operation takes a weighted sum of past innovations. The estimate of the maximum likelihood of the parameters of the Gumbel autoregressive and Gumbel moving average processes and their respective residuals are evaluated. Gumbel-AR(1) and Gumbel-MA(1) was fitted to the Gumbel-generalised Pareto distribution and we evaluate the performance of the parameter estimates fitted to the cluster maxima and the original series. Ignoring the effect of dependence leads to overestimation of the location parameter of the Gumbel-AR(1) and Gumbel-MA(1) processes respectively. The estimate of the location parameter of the autoregressive process using the residuals gives a better estimate. The estimate of the scale parameter perform marginally better for the original series than the residual estimate. The degree of clustering increases as dependence is enhance for both the AR and MA processes. The Gumbel-AR(1) and Gumbel-MA(1) are fitted to the Gumbel-generalised Pareto distribution show that the estimates of the scale and shape parameters fitted to the cluster maxima perform better as sample size increases, however, ignoring the effect of dependence leads to an underestimation of the parameter estimates of the scale parameter. The shape parameter of the original series gives a superior estimate compare to the threshold excesses fitted to the Gumbel-generalised Pareto distribution.

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