The Cramér-Rao Lower Bound (CRLB) sets the theoretical minimum variance for unbiased estimators. It's a crucial tool in statistical inference, helping us evaluate and compare estimator performance against a benchmark.

Applying the CRLB involves calculating Fisher Information and using it to assess estimator efficiency. This process allows us to identify the best possible unbiased estimator and make informed choices when selecting estimation methods for different scenarios.

Understanding the Cramér-Rao Lower Bound

Cramér-Rao Lower Bound definition

• CRLB establishes theoretical minimum variance for unbiased estimators
• Mathematically expressed as $Var(\hat{\theta}) \geq \frac{1}{I(\theta)}$
• Serves as performance benchmark for evaluating estimator efficiency
• Helps identify best possible unbiased estimator (minimum variance unbiased estimator)
• Enables comparison of estimator variances to theoretical minimum (sample mean vs sample median)

Derivation of CRLB

1. Identify probability density function of distribution (normal, exponential)

2. Calculate log-likelihood function

3. Determine score function (first derivative of log-likelihood)

4. Compute Fisher Information

   • Negative expected value of second derivative of log-likelihood
   • Or expected value of squared score function

5. Invert Fisher Information to obtain CRLB

• Fisher Information measures parameter information in sample
• Represented as $I(\theta) = E[(\frac{\partial}{\partial\theta} \log f(X;\theta))^2]$
• Regularity conditions ensure CRLB validity
  • Differentiability and integrability of likelihood function

Applying the Cramér-Rao Lower Bound

Efficiency assessment with CRLB

• Efficiency ratio compares CRLB to actual estimator variance
• Expressed as $e(\hat{\theta}) = \frac{CRLB(\theta)}{Var(\hat{\theta})}$
• Efficiency ranges from 0 to 1, with 1 indicating full efficiency
• Lower efficiency suggests potential for estimator improvement
• Minimum Variance Unbiased Estimator achieves CRLB, considered most efficient

Estimator comparison using CRLB

• Relative efficiency compares variances of two estimators
• Asymptotic efficiency examines efficiency as sample size approaches infinity
• Trade-offs in estimator selection consider bias vs variance, efficiency vs robustness
• Estimator comparisons include MLE vs Method of Moments, sample mean vs sample median
• Some estimators may not be efficient for small samples but become efficient asymptotically