Extremum estimator

In statistics and econometrics, extremum estimators are a wide class of estimators for parametric models that are calculated through maximization (or minimization) of a certain objective function, which depends on the data.

The general theory of extremum estimators was developed by Amemiya (1985).

Sometimes a slightly weaker definition is given: where op(1) is the variable converging in probability to zero.

doesn't have to be the exact maximizer of the objective function, just be sufficiently close to it.

The theory of extremum estimators does not specify what the objective function should be.

There are various types of objective functions suitable for different models, and this framework allows us to analyse the theoretical properties of such estimators from a unified perspective.

satisfy some other conditions,[2] the extremum estimator converges to an asymptotically Normal distribution.

{\displaystyle \scriptscriptstyle {\hat {\theta }}} — When the parameter space Θ is not compact ( Θ = R in this example), then even if the objective function is uniquely maximized at θ ₀ , this maximum may be not well-separated, in which case the estimator $\scriptscriptstyle {\hat {\theta }}$ will fail to be consistent.