In finance models that use historical returns for inputs, including option pricing models, forecasting and portfolio optimization, only the statistical moments of the returns distribution, $\mu$ and $\sigma$ (expected value, or mean, and standard deviation), are used as inputs because the moments summarize a return series' probability distribution (pdf). How strong is the argument that the user would be better off, and would get more accurate results, in using the data's entire pdf, instead of only $\mu$ and $\sigma$?
And would using the entire pdf also be better than models that try to extend to the third and fourth moments (skewness, kurtosis)? given that you could even create a distribution of the rolling skewness and rolling kurtosis of a return series, i.e. the distribution of each moment