Additive models with p-order autoregressive skew-normal errors for modeling trend and seasonality in time series

In this article, we propose an additive model in which the random error follows a skew-normal p-order autoregressive (AR) process where the systematic component is approximated by cubic and cyclic cubic regression splines. The maximum likelihood estimators are calculated through the expectation-maximization (EM) algorithm with analytic expressions for the E and M-steps. The effective degrees of freedom concerning the non parametric component are estimated based on a linear smoother. The smoothing parameters are estimated by minimizing the Bayesian information criterion. The conditional quantile residuals are used to construct simulated confidence bands for assessing departures from the error assumptions. Also, we use the same residuals to construct graphs of the autocorrelation and partial autocorrelation functions to verify the AR structure's adequacy for the errors. We then perform local influence analysis based on the conditional expectation of the complete-data log-likelihood function. A simulation study is carried out to evaluate the efficiency of the EM algorithm. Finally, the method is illustrated by using a real dataset of the average weekly cardiovascular mortality in Los Angeles.