SUMM: What's the purpose of the estimation?

[josef-pkt]

Why do we estimate? What are we using the results for?

• parameter estimates: how large is this effect? And is there even an effect?
• implied summary statistics: what's the average treatment effect on this subgroup? What's the average marginal effect? risk ratio, ...
• predicting mean: What's the expected value for this new observation?
• predicted distribution: What's this distributional property of a new observation? prediction confidence interval, tail probabilities, ...

Questions while pondering model diagnostics and goodness or lack of fit measures
e.g. Poisson and all LEF estimate the expected value consistently even if the rest is misspecified. But Poisson is not a good model for the distribution of overdispersed data.
When do we care about distributional fit diagnostic tests and measures?