Editing Consistent estimator (section)

== Establishing consistency ==

The notion of asymptotic consistency is very close, almost synonymous to the notion of convergence in probability. As such, any theorem, lemma, or property which establishes convergence in probability may be used to prove the consistency. Many such tools exist:

* In order to demonstrate consistency directly from the definition one can use the inequality {{sfn|Amemiya|1985|loc=equation (3.2.5)}}
:: <math>
    \Pr\!\big[h(T_n-\theta)\geq\varepsilon\big] \leq \frac{\operatorname{E}\big[h(T_n-\theta)\big]}{h(\varepsilon)},
  </math>
the most common choice for function ''h'' being either the absolute value (in which case it is known as [[Markov inequality]]), or the quadratic function (respectively [[Chebyshev's inequality]]).

* Another useful result is the [[continuous mapping theorem]]: if ''T<sub>n</sub>'' is consistent for ''θ'' and ''g''(·) is a real-valued function continuous at the point ''θ'', then ''g''(''T<sub>n</sub>'') will be consistent for ''g''(''θ''):{{sfn|Amemiya|1985|loc=Theorem 3.2.6}}
:: <math>
    T_n\ \xrightarrow{p}\ \theta\ \quad\Rightarrow\quad g(T_n)\ \xrightarrow{p}\ g(\theta)
  </math>

* [[Slutsky's theorem]] can be used to combine several different estimators, or an estimator with a non-random convergent sequence. If ''T<sub>n</sub>''&nbsp;→<sup style="position:relative;top:-.2em;left:-1em;">''d''</sup>''α'', and ''S<sub>n</sub>''&nbsp;→<sup style="position:relative;top:-.2em;left:-1em;">''p''</sup>''β'', then {{sfn|Amemiya|1985|loc=Theorem 3.2.7}}
:: <math>\begin{align}
  & T_n + S_n \ \xrightarrow{d}\ \alpha+\beta, \\
  & T_n   S_n \ \xrightarrow{d}\ \alpha \beta, \\
  & T_n / S_n \ \xrightarrow{d}\ \alpha/\beta, \text{ provided that }\beta\neq0
  \end{align}</math>

* If estimator ''T<sub>n</sub>'' is given by an explicit formula, then most likely the formula will employ sums of random variables, and then the [[law of large numbers]] can be used: for a sequence {''X<sub>n</sub>''} of random variables and under suitable conditions,
:: <math>\frac{1}{n}\sum_{i=1}^n g(X_i) \ \xrightarrow{p}\ \operatorname{E}[\,g(X)\,]</math>

* If estimator ''T<sub>n</sub>'' is defined implicitly, for example as a value that maximizes certain objective function (see [[extremum estimator]]), then a more complicated argument involving [[stochastic equicontinuity]] has to be used.{{sfn|Newey|McFadden|1994|loc=Chapter 2}}