Modified Maximum Likelihood Estimation in First-Order Autoregressive Moving Average Models with some Non-Normal Residuals

When modeling time series data using autoregressive-moving average processes, it is a common practice to presume that the residuals are normally distributed. However, sometimes we encounter non-normal residuals and asymmetry of data marginal distribution. Despite widespread use of pure autoregressive processes for modeling non-normal time series, the autoregressive-moving average models have less been used. The main reason is the difficulty in estimating the autoregressive-moving average model parameters. The purpose of this study is to address this intricacy by approximating maximum likelihood estimators, which is particularly important from model selection perspective. Accordingly, the coefficients and residual distribution parameters of the first-order stationary autoregressive-moving average model with residuals that follow exponential and Weibull families, were estimated. Then based on the simulation study, the obtained theoretical results were investigated and it was shown that the modified maximum likelihood estimators were suitable estimators to estimate the first-order autoregressive-moving average model parameters in nonnormal mode. In a numerical example positive skewness of obtained residuals from fitting the first-order autoregressive-moving average model was shown. Following that, the parameters of candidate residual distributions estimated by modified maximum likelihood estimators and one of the estimated models for modeling the data was selected.