Practically, the statistical properties of time-variant time series are always changing, and the parameters of the modeled probability distribution are time-dependent.
One simple way I used is to update the estimation using only recent data with maximum likelihood or momentum method, but the problem is how to define the size of the recent data? Furthermore, if I impose weights for recent datas (i.e., big weights for the most recent data and small weights for old datas), then how to impose?
Maybe, there are some good methods to deal with parameter estimation of time-variant probability distribution. I find a simlar but specific question Maximum Likelihood Estimator for Poisson with time-dependent Parameter, but no one answered there.