Definition
Loosely speaking, an estimator Tn of parameter θ is said to be consistent, if it converges in probability to the true value of the parameter:
A more rigorous definition takes into account the fact that θ is actually unknown, and thus the convergence in probability must take place for every possible value of this parameter. Suppose {pθ: θ ∈ Θ} is a family of distributions (the parametric model), and Xθ = {X1, X2, … : Xi ~ pθ} is an infinite sample from the distribution pθ. Let { Tn(Xθ) } be a sequence of estimators for some parameter g(θ). Usually Tn will be based on the first n observations of a sample. Then this sequence {Tn} is said to be (weakly) consistent if
This definition uses g(θ) instead of simply θ, because often one is interested in estimating a certain function or a sub-vector of the underlying parameter. In the next example we estimate the location parameter of the model, but not the scale:
Read more about this topic: Consistent Estimator
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