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PathsCramer-Rao lower bound

Theorem trail

Cramer-Rao lower bound

The score-identity route from Fisher information to estimator variance.

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Research Record

A compact audit view for the theorem before the full source, proof, diagnostic, and failure-mode sections.

Claim scope
theorem
Evidence level
Source checked
Source precision
reviewed / B
Diagnostics
1 item
Exercises
1 exercise
Failure checks
2 checks
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Theorem Statement

For any unbiased estimator of , Equality at a fixed holds if and only if there exists a function such that Equality at every in an open neighborhood characterizes one-parameter exponential families.

Assumptions

  • X_1 through X_n are iid from a Cramér-regular family p(x given theta) for theta in an open subset of the real line
  • theta_hat is an unbiased estimator of theta with finite second moment
  • the Fisher information I(theta) is positive and finite

Proof Sketch

Differentiate the unbiasedness identity under regularity conditions to get covariance between the estimator and the score equal to one. Cauchy-Schwarz bounds that covariance by the product of estimator variance and Fisher information, then rearrangement gives the lower bound.

Proof Obligations

Unbiased estimator under regularity conditions

assumption

Verify unbiasedness and the differentiability/interchange conditions needed to differentiate the expectation identity.

Biased estimators or nonregular families can evade the scalar Cramer-Rao lower-bound conclusion.

Score covariance identity

proof step

The critical bridge is differentiating the unbiasedness identity to obtain the covariance between estimator and score before applying Cauchy-Schwarz.

Applying Cauchy-Schwarz without first proving the score identity leaves the lower bound unsupported.

Formalization pending until the iid layer is isolated

boundary

Treat this trail as source-checked until a finite covariance-identity Lean wrapper and the iid n-sample bridge land with their actual Lean artifacts.

Presenting a planned finite covariance-identity wrapper as the full iid Cramer-Rao theorem would overstate the formal evidence layer.

Unbiased regular scalar model scope

boundary

Check unbiasedness, scalar parameterization, positive Fisher information, and the regularity assumptions before using the displayed lower bound.

Biased, constrained, vector, or nonregular models need different Cramer-Rao variants or can violate the simple scalar conclusion.

Source Locators

Statistical Inference (2002)

proof

Section 7.3 — Methods of evaluating estimators

casella-berger-2002:sec7.3

kind
section
review
reviewed / B
extract
manual
number
7.3

Section-level support for the scalar information inequality.

Theory of Point Estimation (1998)

background

Chapter 2 — Unbiased estimation and information bounds

lehmann-casella-1998:ch2

kind
chapter
review
reviewed / B
extract
manual

Point-estimation and efficiency context.

Curated Trace

  1. 1
    Parametric modelsetup

    The score and regularity conditions come from the likelihood model.

    Checkpoint

    Support must not depend on theta in the forbidden way.

  2. 2
    Score varianceinformation term

    Fisher information is the variance of the score under regularity.

    Checkpoint

    I(theta) must be positive and finite.

  3. 3
    Cauchy-Schwarz stepinequality

    The covariance identity is bounded by a product of L2 norms.

    Checkpoint

    Unbiasedness is not optional in this statement.

  4. 4
    Cramer-Rao boundtheorem

    Rearranging the covariance bound yields Var(theta_hat) >= 1/(n I(theta)).

    Checkpoint

    Biased estimators and non-regular families can escape the theorem.

Failure Modes

Formalization pending until the iid layer is isolated

Treat this trail as source-checked until a finite covariance-identity Lean wrapper and the iid n-sample bridge land with their actual Lean artifacts.

Risk: Presenting a planned finite covariance-identity wrapper as the full iid Cramer-Rao theorem would overstate the formal evidence layer.

Check: Treat this trail as source-checked until a finite covariance-identity Lean wrapper and the iid n-sample bridge land with their actual Lean artifacts.

Unbiased regular scalar model scope

Check unbiasedness, scalar parameterization, positive Fisher information, and the regularity assumptions before using the displayed lower bound.

Risk: Biased, constrained, vector, or nonregular models need different Cramer-Rao variants or can violate the simple scalar conclusion.

Check: Check unbiasedness, scalar parameterization, positive Fisher information, and the regularity assumptions before using the displayed lower bound.