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In [[statistics]], '''consistency''' of procedures, such as computing confidence intervals or conducting hypothesis tests, is a desired property of their behaviour as the number of items in the data set to which they are applied increases indefinitely. In particular, consistency requires that the outcome of the procedure should identify the underlying truth.<ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9 (entries for consistency, consistent estimator, consistent test)</ref>
Use of the term in statistics derives from Sir [[Ronald Fisher]] in 1922.<ref>Upton, G.; Cook, I. (2006) ''Oxford Dictionary of Statistics'', 2nd Edition, OUP. ISBN 978-0-19-954145-4</ref>


Use of the terms ''consistency'' and ''consistent'' in statistics is restricted to cases where essentially the same procedure can be applied to any number of data items. In complicated applications of statistics, there may be several ways in which the number of data items may grow. For example, records for rainfall within an area might increase in three ways: records for additional time periods; records for additional sites with a fixed area; records for extra sites obtained by extending the size of the area. In such cases, the property of consistency may be limited to one or more of the possible ways a sample size can grow.


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==Estimators==
{{Main|Consistent estimator}}
A [[consistent estimator]] is one for which, when the estimate is considered as a [[random variable]] indexed by the number ''n'' of items in the data set, as ''n'' increases the estimates [[Convergence of random variables|converge]] to the value that the estimator is designed to estimate.  
 
{{Main|Fisher consistency}}
An estimator that has [[Fisher consistency]] is one for which, if the estimator were applied to the entire population rather than a sample, the true value of the estimated parameter would be obtained.
 
==Tests==
{{Main|Statistical hypothesis testing}}
A [[Statistical hypothesis testing#Definition of terms|consistent test]] is one for which the [[statistical power|power]] of the test for a fixed untrue hypothesis increases to one as the number of data items increases.<ref name=Dodge/>
 
==Classification==
In [[statistical classification]], a consistent classifier is one for which the probability of correct classification, given a training set, approaches, as the size of the training set increases, the best probability theoretically possible if the population distributions were fully known.
 
==Sparsistency==
Let <math> \mathbf{b} </math> be a vector and define the support <math> supp(\mathbf{b}) = \{i : \mathbf{b}_i \neq 0\} </math> where <math>\mathbf{b}_i </math> is the <math>i</math>th element of <math>\mathbf{b} </math>. Let <math> \hat{\mathbf{b}} </math> be an estimator for <math> \mathbf{b} </math>. Then sparsistency is the property that the support of the estimator converges to the true support as the number of samples grows to infinity. More formally, <math> P(supp(\hat{\mathbf{b}}) = supp(\mathbf{b})) \rightarrow 1 </math> as <math> n\rightarrow \infty </math>.<ref>http://normaldeviate.wordpress.com/2013/09/11/consistency-sparsistency-and-presistency/</ref>
 
==References==
{{reflist}}
 
[[Category:Statistical theory]]
[[Category:Statistical terminology]]

Revision as of 00:09, 7 November 2013

In statistics, consistency of procedures, such as computing confidence intervals or conducting hypothesis tests, is a desired property of their behaviour as the number of items in the data set to which they are applied increases indefinitely. In particular, consistency requires that the outcome of the procedure should identify the underlying truth.[1] Use of the term in statistics derives from Sir Ronald Fisher in 1922.[2]

Use of the terms consistency and consistent in statistics is restricted to cases where essentially the same procedure can be applied to any number of data items. In complicated applications of statistics, there may be several ways in which the number of data items may grow. For example, records for rainfall within an area might increase in three ways: records for additional time periods; records for additional sites with a fixed area; records for extra sites obtained by extending the size of the area. In such cases, the property of consistency may be limited to one or more of the possible ways a sample size can grow.

Estimators

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A consistent estimator is one for which, when the estimate is considered as a random variable indexed by the number n of items in the data set, as n increases the estimates converge to the value that the estimator is designed to estimate.

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. An estimator that has Fisher consistency is one for which, if the estimator were applied to the entire population rather than a sample, the true value of the estimated parameter would be obtained.

Tests

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A consistent test is one for which the power of the test for a fixed untrue hypothesis increases to one as the number of data items increases.[1]

Classification

In statistical classification, a consistent classifier is one for which the probability of correct classification, given a training set, approaches, as the size of the training set increases, the best probability theoretically possible if the population distributions were fully known.

Sparsistency

Let b be a vector and define the support supp(b)={i:bi≠0} where bi is the ith element of b. Let b^ be an estimator for b. Then sparsistency is the property that the support of the estimator converges to the true support as the number of samples grows to infinity. More formally, P(supp(b^)=supp(b))→1 as n→∞.[3]

References

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  1. ↑ 1.0 1.1 Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entries for consistency, consistent estimator, consistent test)
  2. ↑ Upton, G.; Cook, I. (2006) Oxford Dictionary of Statistics, 2nd Edition, OUP. ISBN 978-0-19-954145-4
  3. ↑ http://normaldeviate.wordpress.com/2013/09/11/consistency-sparsistency-and-presistency/
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