In this paper additive models with p-order autoregressive conditional symmetric errors based on penalized regression splines are proposed for modeling trend and seasonality in time series. The aim with this kind of approach is try to model the autocorrelation and seasonality properly to assess the existence of a significant trend. A backfitting iterative process jointly with a quasi-Newton algorithm are developed for estimating the additive components, the dispersion parameter and the autocorrelation coefficients. The effective degrees of freedom concerning the fitting are derived from an appropriate smoother. Inferential results and selection model procedures are proposed as well as some diagnostic methods, such as residual analysis based on the conditional quantile residual and sensitivity studies based on the local influence approach. Simulations studies are performed to assess the large sample behavior of the maximum penalized likelihood estimators. Finally, the methodology is applied for modeling the daily average temperature of San Francisco city from January 1995 to April 2020.

在本文中，具有p的加性模型提出了基于惩罚回归样条的二阶自回归条件对称误差，用于对时间序列的趋势和季节进行建模。这种方法的目的是尝试对自相关性和季节性进行正确建模，以评估是否存在显着趋势。开发了一种与拟牛顿算法相结合的反向拟合迭代过程，用于估计加性成分，色散参数和自相关系数。有关拟合的有效自由度是从适当的平滑器得出的。提出了推论结果和选择模型程序，以及一些诊断方法，例如基于条件分位数残差的残差分析和基于局部影响方法的敏感性研究。进行模拟研究以评估最大惩罚似然估计器的大样本行为。最后，该方法适用于模拟1995年1月至2020年4月旧金山市的每日平均温度。