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In statistics and econometrics, extremum estimators is 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).
Contents
Definition
An estimator
where Θ is the possible range of parameter values. Sometimes a slightly weaker definition is given:
where op(1) is the variable converging in probability to zero. With this modification
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 when one selects a particular objective function, he or she only has to verify that those properties are satisfied.
Consistency
If the set Θ is compact and there is a limiting function Q0(θ) such that:
The uniform convergence in probability of
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 {θi} such that Q0(θi) → Q0(θ0), it should be true that θi → θ0.
Asymptotic normality
Assuming that consistency has been established and the derivatives of the sample