consistent estimator

Given a set of samples X1,,Xn from a given probability distribution f with an unknown parameter θΘ, where Θ is the parameter space that is a subset of m. Let U(=U(X1,,Xn)) be an estimatorMathworldPlanetmath of θ. Allowing the sample size n to vary, we get a sequence of estimators for θ:

U1 = U(X1),
Un = U(X1,,Xn),

We say that the sequence of estimators {Un} consistent (or that U is a consistent estimator of θ), if Ui converges in probability to θ for every θΘ. That is, for every ε>0,


for all θΘ.

Remark. Suppose U is an estimator of θ such that the sequence {Un} is consistent. If αnα and βnβm are two convergent sequences of constants with 0<|α|< and |β|<, then the sequence {Vn}, defined by Vn:=αnUn+βn, is consistent, V is an estimator of αθ+β.


First, observe that

|Vn-(αθ+β)| = |αnUn+βn-αθ-β|
= |αnUn-αnθ+αnθ-αθ|+|βn-β|
= |αn||Un-θ|+|αn-α||θ|+|βn-β|.

This implies

= P(|Un-θ|ε-|βn-β|-|αn-α||θ||αn|).

As n, |βn-β|0, |αn-α||θ|0, and |αn||α|0. So the last expression goes to 0 as n. Therefore,


and thus {Vn} is a consistent sequence of estimators of αθ+β. ∎

Title consistent estimator
Canonical name ConsistentEstimator
Date of creation 2013-03-22 15:26:34
Last modified on 2013-03-22 15:26:34
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 62F12
Defines consistent sequence of estimators