Consistent estimator: Difference between revisions

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In practice one constructs an estimator as a function of an available sample of [[sample size|size]] ''n'', and then imagines being able to keep collecting data and expanding the sample ''ad infinitum''. In this way one would obtain a sequence of estimates indexed by ''n'', and consistency is a property of what occurs as the sample size “grows to infinity”. If the sequence of estimates can be mathematically shown to converge in probability to the true value ''θ''₀, it is called a consistent estimator; otherwise the estimator is said to be '''inconsistent'''.

Consistency as defined here is sometimes referred to as '''weak consistency'''. When we replace convergence in probability with [[almost sure convergence]], then the estimator is said to be '''strongly consistent'''. Consistency is related to [[bias of an estimator|bias]]; see [[#Bias versus consistency|bias versus consistency]].

== Definition ==

Formally speaking, an [[estimator]] ''T_n'' 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.

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</math>

An [[estimator]] ''T_n'' of parameter ''θ'' is said to be '''strongly consistent''', if it '''converges almost surely''' 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 {{nowrap|{''p_θ'': ''θ'' ∈ Θ}}} is a family of distributions (the [[parametric model]]), and {{nowrap|1=''X^θ'' = {''X''₁, ''X''₂, … : ''X_i'' ~ ''p_θ''}}} is an infinite [[statistical sample|sample]] from the distribution ''p_θ''. Let { ''T_n''(''X^θ'') } be a sequence of estimators for some parameter ''g''(''θ''). Usually ''T_n'' will be based on the first ''n'' observations of a sample. Then this sequence {''T_n''} is said to be (weakly) '''consistent''' if {{sfn|Lehman|Casella|1998|page=332}}

: <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_θ'': ''θ'' ∈ Θ}}} is a family of distributions (the [[parametric model]]), and {{nowrap|1=''X^θ'' = {''X''₁, ''X''₂, … : ''X_i'' ~ ''p_θ''}}} is an infinite [[statistical sample|sample]] from the distribution ''p_θ''. Let { ''T_n''(''X^θ'') } be a sequence of estimators for some parameter ''g''(''θ''). Usually, ''T_n'' will be based on the first ''n'' observations of a sample. Then this sequence {''T_n''} 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:

== Examples ==

=== Sample mean of a normal random variable ===

Suppose one has a sequence of [[Independence (probability theory)|statistically independent]] observations {''X''₁, ''X''₂, ...} from a [[Normal distribution|normal ''N''(''μ'', ''σ''²)]] distribution. To estimate ''μ'' based on the first ''n'' observations, one can use the [[sample mean]]: ''T_n''&nbsp;=&nbsp;(''X''₁ + ... + ''X_n'')/''n''. This defines a sequence of estimators, indexed by the sample size ''n''.

From the properties of the normal distribution, we know the [[sampling distribution]] of this statistic: ''T''_''n'' is itself normally distributed, with mean ''μ'' and variance ''σ''²/''n''. Equivalently, <math style="vertical-align:-.3em">\scriptstyle (T_n-\mu)/(\sigma/\sqrt{n})</math> has a standard normal distribution:

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2\left(1-\Phi\left(\frac{\sqrt{n}\,\varepsilon}{\sigma}\right)\right) \to 0

</math>

as ''n'' tends to infinity, for any fixed {{nowrap|''ε'' > 0}}. Therefore, the sequence ''T_n'' of sample means is consistent for the population mean&nbsp;''μ'' (recalling that <math>\Phi</math> is the [[Cumulative distribution function|cumulative distribution]] of the standard normal distribution).

== Establishing consistency ==

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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_n'' is consistent for ''θ'' and ''g''(·) is a real-valued function continuous at the point ''θ'', then ''g''(''T_n'') 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’sSlutsky's theorem]] can be used to combine several different estimators, or an estimator with a non-random convergent sequence. If ''T_n''&nbsp;→<sup style="position:relative;top:-.2em;left:-1em;">''d''</sup>''α'', and ''S_n''&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, \\

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=== Unbiased but not consistent ===

An estimator can be [[biased estimator|unbiased]] but not consistent. For example, for an [[iid]] sample {''x''{{su|b=1}},..., ''x{{su|b=n}}''} one can use ''T{{su|b=n}}''(''X'') = ''x''{{su|b=n}} as the estimator of the mean E[''xX'']. Note that here the sampling distribution of ''T{{su|b=n}}'' is the same as the underlying distribution (for any ''n,'' as it ignores all points but the last),. soSo E[''T{{su|b=n}}''(''X'')] = E[''xX''] andfor any ''n,'' hence it is unbiased, but it does not converge to any value.

However, if a sequence of estimators is unbiased ''and'' converges to a value, then it is consistent, as it must converge to the correct value.

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* [[Regression dilution]]

* [[Statistical hypothesis testing]]

* [[Instrumental variables estimation]]

== Notes ==

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| publisher = [[Harvard University Press]]

| isbn = 0-674-00560-0

| ref = CITEREFAmemiya1985

| url-access = registration

| url = https://archive.org/details/advancedeconomet00amem

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| publisher = Elsevier Science

| isbn = 0-444-88766-0

| ref = CITEREFNeweyMcFadden1994

}}

* {{SpringerEOM| title=Consistent estimator |id=C/c025240 |first=M. S. |last=Nikulin}}