I'm using maximum likelihood estimation to fit a model to time series data. I later want to use the model to do forecasting. Let $L$ be the likelihood function, $$L_i = L(f_k(\theta), data_i).$$ It depends on some distribution $f_k$ of the parameter(s) $\theta$ and it depends on the observed data for timeseries i. I use MLE to find the optimal $\hat{\theta}$ for several different distributions $f_k$ and want to to pick the best model, i.e $f_k$ in combination with $\hat{\theta}$. For example I want to compare $f_k$ being a lognormal distribution or $f_k$ being a gamma distribution. How would I use the Akaike information criterion(AIC) to do this? I assume that I can calculate and use (AIC) to pick the best type of distribution $f_k$ and $\hat{\theta}$ for the same $data_i$. Is this correct?
If I wanted to find the best $f_k$ for several time series $data_i$ for i 1 to N. Would it bet better to take the $f_k$ which has had most often the lowest AIC? Or would I instead take $f_k$ with the lowest average AIC?
My confusion comes from the fact, that I can construct a likelihood function for the whole dataset which would be $L = \prod_i L_i$. Since $ \ln L = \sum_i \ln(L_i)$ the AIC for the whole dataset essentially is the sum of the individual AICs. However, this can easily be dominated by a single time series for which a certain $f_k$ does not fit well even if it performs much better on all other data time series.
