Answer First
The maximum likelihood estimator is the most efficient estimator because, under standard regularity conditions, it achieves the Cramér–Rao lower bound asymptotically. This means no unbiased estimator can have a smaller variance than the MLE in large samples.
Problem Setup
Suppose we observe independent data \(X_1, X_2, \ldots, X_n\) from a distribution with density \(f(x \mid \theta)\). The likelihood is:
\[ L(\theta) = \prod_{i=1}^{n} f(X_i \mid \theta) \]
The log‑likelihood is:
\[ \ell(\theta) = \sum_{i=1}^{n} \log f(X_i \mid \theta) \]
The MLE is defined as:
\[ \hat{\theta}_{MLE} = \arg\max_{\theta} \ell(\theta) \]
The Fisher information is:
\[ I(\theta) = -E\left[ \frac{\partial^2}{\partial \theta^2} \log f(X \mid \theta) \right] \]
Step-by-Step Solution
1. Expand the log‑likelihood around the true parameter
A Taylor expansion shows that the MLE behaves like the solution to a linear approximation of the score equation.
2. Use the score function
The score is the derivative of the log‑likelihood:
\[ U(\theta) = \frac{\partial}{\partial \theta} \ell(\theta) \]
The MLE satisfies \(U(\hat{\theta}) = 0\).
3. Apply the central limit theorem to the score
The score has mean zero and variance \(n I(\theta)\). By the central limit theorem, the score is asymptotically normal.
4. Solve the linearized score equation
Solving the Taylor expansion yields:
\[ \sqrt{n}(\hat{\theta}_{MLE} – \theta) \xrightarrow{d} N\left(0, I(\theta)^{-1}\right) \]
5. Compare to the Cramér–Rao lower bound
The Cramér–Rao lower bound states that any unbiased estimator satisfies:
\[ Var(\hat{\theta}) \geq \frac{1}{n I(\theta)} \]
The MLE achieves this bound asymptotically, making it the most efficient estimator.
Intuition
The MLE chooses the parameter value that makes the observed data most probable. Because it aligns with the shape of the likelihood function, it naturally uses all available information in the sample. Fisher information measures how much information the data contain about the parameter, and the MLE extracts this information optimally.
Common Exam Mistakes
- Thinking the MLE is always unbiased (it is often biased in small samples).
- Confusing consistency with efficiency.
- Believing the MLE is always efficient (only under regularity conditions).
- Ignoring the role of Fisher information.
Why This Matters
MLEs form the foundation of modern statistics, econometrics, and machine learning. Their efficiency makes them the default choice for parametric estimation, and their asymptotic properties justify likelihood‑based inference, confidence intervals, and hypothesis tests.
Final Summary
The maximum likelihood estimator is the most efficient estimator because it achieves the Cramér–Rao lower bound asymptotically. Under regularity conditions, the MLE uses all available information in the data, making it the optimal estimator for large samples.
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