Invariance of maximum likelihood
Proposition taught in Estimation, part of Statistical inference
Invariance of maximum likelihood
| Type | Proposition |
| Taught in | Estimation |
| Part | Statistical inference |
| Worked examples | 1 |
If \widehat\theta is the maximum likelihood estimator of \theta and g is a function, then the maximum likelihood estimator of g(\theta) is g(\widehat\theta).
Examples
An exponential rate
Question page: An exponential rate
Let X_1,\ldots,X_n be a random sample from the exponential distribution with rate \lambda, density \lambda e^{-\lambda x} for x>0.
Find the maximum likelihood estimator of \lambda, and check that it is a maximum.
Find the method-of-moments estimator of \lambda.
Use invariance to find the maximum likelihood estimator of the median lifetime (\log 2)/\lambda.
- The log-likelihood is
\log L(\lambda)=n\log\lambda-\lambda\sum_{i=1}^nx_i.
Its derivative n/\lambda-\sum x_i is zero at \lambda=n/\sum x_i=1/\overline x. The second derivative, -n/\lambda^2, is negative, so this is a maximum. The estimator is \widehat\lambda=1/\overline X.
\mathbb{E} X=1/\lambda. Setting 1/\lambda=\overline x gives the same estimator, 1/\overline X.
The median lifetime m solves 1-e^{-\lambda m}=1/2, so m=(\log2)/\lambda. By invariance its maximum likelihood estimator is (\log2)/\widehat\lambda=\overline X\log2.
Used in
- Taught in Estimation, where the derivation and examples are.
- All 5 questions on Estimation
See also
Maximum likelihood · Estimator · Method of moments · Sample correlation coefficient · Likelihood function · Unbiased estimator · Mean squared error · Consistent estimator