Parameter estimation
A mathematical model contains parameters controlling rates, strengths or scales. Parameter estimation uses data to infer plausible values of these unknown quantities.
Write a model prediction as
\[y(t)=f(t;\theta),\]where \(\theta\) is the parameter vector. Observations \(y_i\) are compared with predictions \(f(t_i;\theta)\).
Objective
An estimator chooses parameter values according to a specified statistical criterion, such as minimising squared residuals or maximising a likelihood.
Point estimates are not enough
Different parameter values may fit almost equally well. Estimates should therefore be accompanied by information about uncertainty and identifiability.
Key idea. Parameter estimation is an inverse problem: observations are known and model parameters are inferred from them.