Most time series observed in practice exhibit time-varying trend (first-order) and autocovariance (second-order) behaviour. Differencing is a commonly-used technique to remove the trend in such series, in order to estimate the time-varying second-order structure (of the differenced series). However, often we require inference on the second-order behaviour of the original series, for example, when performing trend estimation. In this article, we propose a method, using differencing, to jointly estimate the time-varying trend and second-order structure of a nonstationary time series, within the locally stationary wavelet modelling framework. We develop a wavelet-based estimator of the second-order structure of the original time series based on the differenced estimate, and show how this can be incorporated into the estimation of the trend of the time series. We perform a simulation study to investigate the performance of the methodology, and demonstrate the utility of the method by analysing data examples from environmental and biomedical science.

翻译：在实际中观测到的多数时间序列显示时间变化趋势(第一顺序)和自动变化(第二顺序)行为。差异是用来消除这种序列趋势的一种常用技术,以便估计(不同序列)时间变化第二顺序结构(不同序列),然而,我们常常要求对原始序列第二顺序行为进行推断,例如在进行趋势估计时。在本条中,我们提出一种方法,利用差异来在地方固定波浪建模框架内,共同估计非静止时间序列的时间变化趋势和第二顺序结构。我们根据差异估计,为最初时间序列的第二顺序结构开发一个基于波盘的估测器,并表明如何将这一方法纳入时间序列趋势的估计中。我们进行模拟研究方法的绩效,并通过分析环境和生物医学的数据实例,展示该方法的效用。