Editing Consistent estimator (section)

{{Short description|Statistical estimator converging in probability to a true parameter as sample size increases}}{{broader|Consistency (statistics)}}

[[Image:Consistency of estimator.svg|thumb|250px|{''T''<sub>1</sub>, ''T''<sub>2</sub>, ''T''<sub>3</sub>, ...} is a sequence of estimators for parameter ''θ''<sub>0</sub>, the true value of which is 4. This sequence is consistent: the estimators are getting more and more concentrated near the true value ''θ''<sub>0</sub>; at the same time, these estimators are biased. The limiting distribution of the sequence is a degenerate random variable which equals ''θ''<sub>0</sub> with probability 1.]]

In [[statistics]], a '''consistent estimator''' or '''asymptotically  consistent estimator''' is an [[estimator]]—a rule for computing estimates of a parameter ''θ''<sub>0</sub>—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates [[convergence in probability|converges in probability]] to ''θ''<sub>0</sub>. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to ''θ''<sub>0</sub> converges to one.

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''<sub>0</sub>, 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]].