Consistency (statistics)
Encyclopedia
In statistics
, consistency of procedures such as confidence intervals or hypothesis tests involves 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.
Use of the term in statistics derives from Sir Ronald Fisher
in 1922.
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.
is one which, considered as a random variable
indexed by the number of items in the data-set, converges
to the value that the estimation procedure is designed to estimate.
An estimator that has Fisher consistency
is one for which, if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained.
of the test for a fixed untrue hypothesis increases to one as the number of data items increases.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, consistency of procedures such as confidence intervals or hypothesis tests involves 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.
Use of the term in statistics derives from Sir Ronald Fisher
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
in 1922.
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
A consistent estimatorConsistent estimator
In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...
is one which, considered as a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
indexed by the number of items in the data-set, converges
Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...
to the value that the estimation procedure is designed to estimate.
An estimator that has Fisher consistency
Fisher consistency
In statistics, Fisher consistency, named after Ronald Fisher, is a desirable property of an estimator asserting that if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained...
is one for which, if the estimator were calculated using the entire population rather than a sample, the true value of the estimated parameter would be obtained.
Tests
A consistent test is one for which the powerStatistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
of the test for a fixed untrue hypothesis increases to one as the number of data items increases.