Consistent estimator: Difference between revisions

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m Establishing consistency: sp -- match wikilink target
In Slutsky's Theorem, the first parameter Tn converges in distribution and all threes results converge in distribution.
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</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;">''pd''</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{pd}\ \alpha+\beta, \\
& T_n S_n \ \xrightarrow{pd}\ \alpha \beta, \\
& T_n / S_n \ \xrightarrow{pd}\ \alpha/\beta, \text{ provided that }\beta\neq0
\end{align}</math>