Given a time series data $\{X_t\}_{t = 0}^\infty$, what does its moving average and moving standard deviation estimate when there is no assumption that $\mathbb{E}[X_t] = \text{const}, \forall t$? Suppose that $X_t\sim\mathcal{N}(\mu_t, \sigma^2)$, then its $N$-step MA is $N^{-1}\sum_{s = 0}^{N - 1}X_{t - s} \sim \mathcal{N}(N^{-1}\sum_{s = 0}^{N - 1}\mu_{t - s}, Var[N^{-1}\sum_{s = 0}^{N - 1}X_{t - s}])$. It seems that the MA can not be directly used in estimating the mean of $X_t$ with no additional assumption on $\mu_t$. I was wondering are there any other statistics like MA can be used to estimate $\mu_t$ given $\mu_s, s\in\{0,\cdots,t-1\}$ when we assume that $\mu_t = f(t;\mu_{t-1}, \mu_{t-2},\cdots,\mu_0)$ ($f$ is not trival)?
Thank you!
