Can you reach the precision floor?

Change the experiment, repeat it, and compare empirical spread with the exact bound.

Sampling distribution · one estimate per trial

Simulated estimatesEfficient mean referenceChosen estimator, exact

Empirical variance—across repeated trials

Exact estimator variance—theoretical, not fitted

Unbiased CRB—σ² / n, using all data

A simulated variance slightly below the CRB is Monte Carlo fluctuation, not a violation.

Independent N(μ, σ²) data

Samples per trial, n 16Measurement noise, σ 2.0EstimatorRandom seed

Truth μ = 1 stays fixed.
Parameter changes replay 2,000 trials from the chosen seed.

Can you “beat” the bound by shrinking toward zero?

Watch the variance fall, then move the true mean away from the shrinkage target.

Exact sampling distributions · no Monte Carlo

Unbiased sample meanShrinkage estimatorTrue μ

Variance—α²σ² / n

Squared bias—(1 − α)²μ²

Total MSE—variance + squared bias

Tα = α X̄ · target = 0

Keep-data factor, α 0.55True mean, μ 0.35

n = 16 and σ = 2, so the unbiased variance bound is 0.25.

Fixed α; no prior is assumed. A lower MSE here is a pointwise bias–variance tradeoff, not a violation of the unbiased CRB.

Drag any anchor or the target. The Fisher matrix and its inverse update immediately.

Known anchors · local information at the true target

Anchors A1–A4True target PCRB ellipse

Unit-Mahalanobis CRB ellipse; not a confidence region.

Range noise, σ (m)0.60Also estimate a common range offset βSelect / move without draggingx (m)0.00y (m)0.00