Periodic autoregressive (PAR) time series with finite variance is considered as one of the most common models of second-order cyclostationary processes. However, in the real applications, the signals with periodic characteristics may be disturbed by additional noise related to measurement device disturbances or to other external sources. Thus, the known estimation techniques dedicated for PAR models may be inefficient for such cases. When the variance of the additive noise is relatively small, it can be ignored and the classical estimation techniques can be applied. However, for extreme cases, the additive noise can have a significant influence on the estimation results. In this paper, we propose four estimation techniques for the noise-corrupted PAR models with finite variance distributions. The methodology is based on Yule-Walker equations utilizing the autocovariance function. It can be used for any type of the finite variance additive noise. The presented simulation study clearly indicates the efficiency of the proposed techniques, also for extreme case, when the additive noise is a sum of the Gaussian additive noise and additive outliers. The proposed estimation techniques are also applied for testing if the data corresponds to noise-corrupted PAR model. This issue is strongly related to the identification of informative component in the data in case when the model is disturbed by additive non-informative noise. The power of the test is studied for simulated data. Finally, the testing procedure is applied for two real time series describing particulate matter concentration in the air.

翻译：有限方差的周期性自回归（PAR）时间序列被视为二阶循环平稳过程中最常见的模型之一。然而，在实际应用中，具有周期特征的信号可能受到与测量设备扰动或其他外部源相关的附加噪声的干扰。因此，针对PAR模型的已知估计技术在此类情况下可能效率低下。当加性噪声的方差相对较小时，可将其忽略并应用经典估计方法。然而在极端情形下，加性噪声会对估计结果产生显著影响。本文针对具有有限方差分布的噪声污染PAR模型提出了四种估计技术。该方法的理论基础是利用自协方差函数的Yule-Walker方程，适用于任意类型的有限方差加性噪声。仿真研究表明，所提技术在极端情形下（包括加性噪声为高斯加性噪声与加性异常值之和时）仍具有有效性。所提估计方法还可用于检验数据是否对应于噪声污染PAR模型，该问题与数据中有效成分的识别密切相关（当模型受到非信息性加性噪声干扰时）。通过模拟数据检验了检验统计量的检验功效，最后将检验程序应用于描述空气中颗粒物浓度的两组真实时间序列。