CRLB Analytical Derivation Engine
Computes Fisher information I_n(θ) and evaluates the theoretical Cramér-Rao Lower Bound 1/I_n(θ).
Statistics & Probability / Cramér-Rao Bound & Efficiency
FREE TO EXPLORELABORATORY 25 · STATISTICS & PROBABILITY
Cramér-Rao Lower Bound, Fisher Information Matrix, and Minimum Variance Unbiased Estimators (MVUE)
Investigate the theoretical limits of statistical precision, derive the Cramér-Rao Lower Bound (CRLB) via Cauchy-Schwarz inequality, evaluate estimator efficiency, and verify that sample mean achieves the CRLB bound. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EMPIRICAL ESTIMATOR VARIANCE VAR(Θ̂)—
SAMPLING TELEMETRY
Ready to sample with CRLB Bound = 1.0.
Adjust CRLB Bound, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
The CRLB is not attainable for all parametric distributions; some models have MVUEs whose variance strictly exceeds 1/I_n. The empirical regression of estimator variance against theoretical CRLB confirms 100% efficiency attainment with linear slope m = 1.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Unbiased Estimator Generator
Synthesizes competing unbiased estimators for Gaussian mean μ (sample mean x̄, mid-range, trimmed mean).
Monte Carlo Variance Profiler
Evaluates empirical sampling variances across 50,000 replicated trials to test the lower bound barrier.
Cauchy-Schwarz Covariance Auditor
Verifies the correlation equality condition |Cov(T, S)| = √[Var(T) Var(S)] for efficiency attainment.
Rao-Blackwell Conditioning Engine
Conditions naive unbiased estimators on complete sufficient statistics to attain the unique MVUE.
CRLB Efficiency Regression Unit
Performs linear regression of empirical estimator variance against theoretical CRLB to confirm unit slope m = 1.000.
FOLLOW THE METHOD
A reason behind every step.
Investigate the theoretical limits of statistical precision, derive the Cramér-Rao Lower Bound (CRLB) via Cauchy-Schwarz inequality, evaluate estimator efficiency, and verify that sample mean achieves the CRLB bound. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units².
01
Configure Gaussian Estimation Model
Specify true Gaussian mean μ_0 = 0.0 and known variance σ^2 = 1.0 on the CRLB simulation dashboard.
02
Calculate Theoretical Cramér-Rao Bound
Compute single-observation Fisher information I_1(μ) = 1/σ^2 = 1.0. For sample size n, verify that CRLB = 1 / (n I_1) = 1 / n.
03
Simulate Sample Mean Estimator Performance
Draw 50,000 Monte Carlo datasets of size n = 1.0. Evaluate sample mean μ̂ = x̄ and compute its empirical variance across all replications.
04
Test Competing Estimators against the Bound
Compute the sample median and mid-range estimators. Confirm that their sampling variances strictly exceed the CRLB barrier (Var(median) ≈ 1.57 / n).
05
Step Theoretical CRLB Setting Across Scaled Values
Increment the theoretical lower bound across standardized values x ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the corresponding estimator variance.
06
Perform Linear Regression on Estimator Efficiency
Fit a linear regression model y = mx to the coordinate pairs (x = CRLB, y = Var(θ̂)). Confirm that empirical slope m = 1.0000 proves 100% efficiency attainment.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Cramér-Rao Bound & Efficiency mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x
| # | Scaled Theoretical CRLB Bound 1/I_n (units²) | Empirical Estimator Variance Var(θ̂) (Var(θ̂)) | Theoretical Lower Bound 1/I_n | Observed Estimator Variance Var(θ̂) | Remove |
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