Consistent estimator
Definition taught in Estimation, part of Statistical inference
Consistent estimator
| Type | Definition |
| Taught in | Estimation |
| Part | Statistical inference |
| Worked examples | 1 |
A sequence of estimators T_n, one for each sample size n, is consistent for \theta when \mathbb{P}_\theta(|T_n-\theta|>\varepsilon)\to0 as n\to\infty, for every \varepsilon>0 and every \theta.
Examples
A Poisson mean
Question page: A Poisson mean
Let X_1,\ldots,X_n be a random sample from \operatorname{Poisson}(\theta). Find the maximum likelihood estimator of \theta, show that it is unbiased, and find its mean squared error. Is it consistent?
The log-likelihood is \sum x_i\log\theta-n\theta-\sum\log x_i!, with derivative \sum x_i/\theta-n. Setting it to zero gives \widehat\theta=\overline X; the second derivative -\sum x_i/\theta^2 is negative, so it is a maximum. (If every x_i=0 the likelihood e^{-n\theta} is largest at \theta=0, which is again \overline x.)
\mathbb{E}\overline X=\theta, so the estimator is unbiased, and its mean squared error is its variance, \operatorname{Var}\overline X=\theta/n, because a Poisson variance equals its mean. That tends to zero, so \overline X is consistent, by Chebyshev’s inequality or by the law of large numbers.
Used in
- Taught in Estimation, where the derivation and examples are.
- All 5 questions on Estimation
See also
Estimator · Method of moments · Sample correlation coefficient · Likelihood function · Maximum likelihood · Invariance of maximum likelihood · Unbiased estimator · Mean squared error