Editing Consistent estimator (section)

== Definition ==
Formally speaking, an [[estimator]] ''T<sub>n</sub>'' of parameter ''θ'' is said to be '''weakly consistent''', if it [[convergence in probability|'''converges in probability''']] to the true value of the parameter:{{sfn|Amemiya|1985|loc=Definition 3.4.2}}
: <math>
    \underset{n\to\infty}{\operatorname{plim}}\;T_n = \theta.
  </math>
i.e. if, for all ''ε'' > 0
: <math>
    \lim_{n\to\infty}\Pr\big(|T_n-\theta| > \varepsilon\big) = 0.
  </math>

An [[estimator]] ''T<sub>n</sub>'' of parameter ''θ'' is said to be '''strongly consistent''', if it '''converges almost surely''' to the true value of the parameter:
: <math>
    \Pr\big(\lim_{n\to\infty}T_n = \theta\big) = 1.
  </math>

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 {{nowrap|{''p<sub>θ</sub>'': ''θ'' ∈ Θ}}} is a family of distributions (the [[parametric model]]), and {{nowrap|1=''X<sup>θ</sup>'' = {''X''<sub>1</sub>, ''X''<sub>2</sub>, … : ''X<sub>i</sub>'' ~ ''p<sub>θ</sub>''}}} is an infinite [[statistical sample|sample]] from the distribution ''p<sub>θ</sub>''. Let { ''T<sub>n</sub>''(''X<sup>θ</sup>'') } be a sequence of estimators for some parameter ''g''(''θ''). Usually, ''T<sub>n</sub>'' will be based on the first ''n'' observations of a sample. Then this sequence {''T<sub>n</sub>''} is said to be (weakly) '''consistent''' if {{sfn|Lehman|Casella|1998|page=332}}
: <math>
    \underset{n\to\infty}{\operatorname{plim}}\;T_n(X^{\theta}) = g(\theta),\ \ \text{for all}\ \theta\in\Theta.
  </math>

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: