A lower bound on the variance (or covariance matrix) of any unbiased estimator of a parameter, expressed in terms of the Fisher information: for a scalar parameter, Var(theta_hat) >= 1 / I(theta); in multivariate form the covariance matrix is bounded below by the inverse Fisher information matrix under regularity conditions.

Principle

Derived from the Cauchy–Schwarz inequality applied to the score function, the information inequality quantifies how the curvature of the log-likelihood (Fisher information) limits achievable precision of unbiased estimators.

Demonstration

Estimating the mean of a normal distribution with known variance: the sample mean is unbiased and attains Var = sigma^2 / n, which equals the Cramér–Rao bound computed from Fisher information, illustrating attainability when an efficient estimator exists.

Misapplication

Using the bound for biased estimators, for parameters at the boundary of the parameter space, or when regularity conditions (differentiability under the integral sign, finite Fisher information) fail; computing Fisher information incorrectly yields misleading bounds.

Consequence

Gives a fundamental efficiency benchmark: if an unbiased estimator attains the bound it is called efficient; the bound guides estimator comparison and asymptotic theory (e.g., MLE attains the bound asymptotically under regularity).

Reversal

For models where the bound is not attainable, or when one allows bias to reduce mean squared error, the Cramér–Rao bound is neither achievable nor the sole criterion; in Bayesian settings posterior or Bayes risk criteria replace the frequentist bound.

Boundary

Valid under regularity and differentiability conditions for parametrized families and for unbiased estimators; it does not apply to all estimators, can be loose for finite samples, and fails in singular models or models with infinite Fisher information.

Semantic Tension

Competes with other lower bounds (e.g., Bhattacharyya, Hammersley–Chapman–Robbins) and with bounds that permit biased estimators or minimax criteria; tension arises between parametric optimality and practical attainability.

Synthesis

The Cramér–Rao Lower Bound uses Fisher information to set a theoretical lower limit on the variance of unbiased estimators under regularity; it provides a benchmark for efficiency and asymptotic optimality while admitting cases where it is unattainable or inappropriate.