I have some time series data that shows autocorrelation and seasonality. I want to fit a distribution to it. To use maximum likelihood it needs to be uncorrelated. I first fit a GARCH and an ARIMA model then use \begin{align} \hat{y}_t=\frac{y_t-\mu_t}{\sigma_t} \end{align} as a filter to get rid of the correlation ($\sigma_t$ is from GARCH and $\mu_t$ is from ARIMA). Then I successfully fit a distribution to the transformed data $\hat{y}_t$ and obtain the CDF.
My question is how do I convert the result back to the original scale? i.e. how do I get the fitted CDF (of $y_t$) on the original scale?
Many thanks!
