World Library  
Flag as Inappropriate
Email this Article

Consistency (statistics)

Article Id: WHEBN0000610028
Reproduction Date:

Title: Consistency (statistics)  
Author: World Heritage Encyclopedia
Language: English
Subject: Generalized estimating equation, High-dimensional statistics, Statistical theory, Resampling (statistics), Fixed effects model
Collection: Statistical Terminology, Statistical Theory
Publisher: World Heritage Encyclopedia

Consistency (statistics)

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 with unlimited data 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 1
  • Tests 2
  • Classification 3
  • Sparsistency 4
  • See also 5
  • References 6


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.

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.


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]


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.


Let \mathbf{b} be a vector and define the support supp(\mathbf{b}) = \{i : \mathbf{b}_i \neq 0\} where \mathbf{b}_i is the ith element of \mathbf{b} . Let \hat{\mathbf{b}} be an estimator for \mathbf{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(\hat{\mathbf{b}}) = supp(\mathbf{b})) \rightarrow 1 as n\rightarrow \infty .[3]

See also


  1. ^ a b 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. ^
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.