# Selecting a forecasting model with AIC: some questions

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.