Cramér-Rao lower bound — state it.
hardAnswer
- For any unbiased estimator θ̂ of θ under regularity conditions: ≥ 1 / (n * I(θ)).
- Fisher information sets the floor for estimator precision.
- MLE achieves this bound asymptotically (asymptotically efficient).
- Practical use: benchmark for whether an estimator can be improved.
- For biased estimators, generalized bounds exist (van Trees).
- Companion to the bias-variance decomposition.
Check yourself — multiple choice
- Random
- For unbiased θ̂: ≥ 1/(n·I(θ)) — MLE achieves this asymptotically → asymptotically efficient
- Only for MLE
- Not a real bound
CR bound: Var ≥ 1/(n·I(θ)); MLE reaches it asymptotically.
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