Definition

where Θ is the possible range of parameter values. Sometimes a slightly weaker definition is given:

where o_p(1) is the variable converging in probability to zero. With this modification $\backslash scriptstyle\backslash hat\backslash theta$ 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 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: $\backslash scriptstyle\backslash hat\{Q\}\_n(\backslash theta)$ converges to Q₀(θ) in probability uniformly over Θ, and the function Q₀(θ) is continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 and has a unique maximum at θ = θ₀. If these conditions are satisfied then $\backslash scriptstyle\backslash hat\backslash theta$ is consistent

Consistent estimator

In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...

for θ₀.

The uniform convergence in probability of $\backslash scriptstyle\backslash hat\{Q\}\_n(\backslash theta)$ means that

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 , it should be true that .

Examples

• Maximum likelihood
  Maximum likelihood
  In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
  
   estimator uses the objective function

  
  where f(·|θ) is the density function

  Probability density function

  In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

  
   of the distribution from where the observations are drawn.
  
  

  Generalized method of moments

  Generalized method of moments

  In econometrics, generalized method of moments is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data...

  
   estimator is defined through the objective function

  
  where g(·|θ) is the moment condition of the model.
  
  

  Minimum distance

  Minimum distance estimation

  Minimum distance estimation is a statistical method for fitting a mathematical model to data, usually the empirical distribution.-Definition:...

  
   estimator
  

  m-estimator

  M-estimator

  In statistics, M-estimators are a broad class of estimators, which are obtained as the minima of sums of functions of the data. Least-squares estimators and many maximum-likelihood estimators are M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new...

  
  s
  

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