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).

Definition

An estimator ${\displaystyle \scriptstyle {\hat {\theta ))}$ is called an extremum estimator, if there is an objective function ${\displaystyle \scriptstyle {\hat {Q))_{n))$ such that

${\displaystyle {\hat {\theta ))={\underset {\theta \in \Theta }{\operatorname {arg\;max} ))\ {\widehat {Q))_{n}(\theta ),}$

where Θ is the parameter space. Sometimes a slightly weaker definition is given:

${\displaystyle {\widehat {Q))_{n}({\hat {\theta )))\geq \max _{\theta \in \Theta }\,{\widehat {Q))_{n}(\theta )-o_{p}(1),}$

where o_p(1) is the variable converging in probability to zero. With this modification ${\displaystyle \scriptstyle {\hat {\theta ))}$ 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. The theory only specifies the properties that the objective function has to possess, and so selecting a particular objective function only requires verifying that those properties are satisfied.

Consistency

If the parameter space Θ is compact and there is a limiting function Q₀(θ) such that: ${\displaystyle \scriptstyle {\hat {Q))_{n}(\theta )}$ converges to Q₀(θ) in probability uniformly over Θ, and the function Q₀(θ) is continuous and has a unique maximum at θ = θ₀ then ${\displaystyle \scriptstyle {\hat {\theta ))}$ is consistent for θ₀.^[1]

The uniform convergence in probability of ${\displaystyle \scriptstyle {\hat {Q))_{n}(\theta )}$ means that

${\displaystyle \sup _{\theta \in \Theta }{\big |}{\hat {Q))_{n}(\theta )-Q_{0}(\theta ){\big |}\ {\xrightarrow {p))\ 0.}$

The requirement for Θ to be compact can be replaced with a weaker assumption that the maximum of Q₀ was well-separated, that is there should not exist any points θ that are distant from θ₀ but such that Q₀(θ) were close to Q₀(θ₀). Formally, it means that for any sequence {θ_i} such that Q₀(θ_i) → Q₀(θ₀), it should be true that θ_i → θ₀.

Asymptotic normality

Assuming that consistency has been established and the derivatives of the sample ${\displaystyle Q_{n))$ satisfy some other conditions,^[2] the extremum estimator converges to an asymptotically Normal distribution.

Examples