Master Observation Cache
Stores the complete sample dataset X = {x_1, x_2, ..., x_N} of continuous observations.
Statistics & Probability / Jackknife & Bias Reduction
FREE TO EXPLORELABORATORY 11 · STATISTICS & PROBABILITY
Jackknife Resampling, Bias Reduction, and Variance Estimation
Apply Maurice Quenouille's leave-one-out jackknife procedure, construct Tukey pseudo-values to eliminate O(1/N) bias from non-linear estimators, and verify first-order bias reduction scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
JACKKNIFE PRECISION METRIC—
SAMPLING TELEMETRY
Ready to sample with Sample Size N = 10.0.
Adjust Sample Size N, 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 jackknife can inflate variance if applied to estimators that already have negligible bias. The empirical regression confirms the first-order jackknife consistency scaling with an empirical gradient of m = 1.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Leave-One-Out Subsample Engine
Sequentially deletes observation i to construct N reduced jackknife subsets X_{(-i)} of size N - 1.
Partial Jackknife Estimator
Evaluates the plug-in estimator θ̂_{(-i)} = T(X_{(-i)}) on each reduced subsample.
Tukey Pseudo-Value Synthesizer
Combines full and partial estimates to construct pseudo-values J_i = N θ̂ - (N - 1) θ̂_{(-i)}.
Jackknife Bias & Variance Meter
Calculates bias estimate Bias_jack = (N - 1)(θ̄_{(-·)} - θ̂) and jackknife variance Var_jack = (N - 1)/N ∑ (θ̂_{(-i)} - θ̄_{(-·)})².
First-Order Asymptotic Compiler
Performs linear regression of the bias reduction metric versus N to confirm the theoretical 1/N scaling rate.
FOLLOW THE METHOD
A reason behind every step.
Apply Maurice Quenouille's leave-one-out jackknife procedure, construct Tukey pseudo-values to eliminate O(1/N) bias from non-linear estimators, and verify first-order bias reduction scaling. Standardized parameter settings: 10.0, 20.0, 30.0, 40.0, 50.0, 60.0 samples.
01
Initialize Empirical Sample Dataset
Generate a sample of size N = 10 from a standard distribution and compute the biased naive plug-in estimator θ̂ (e.g., sample variance with divisor N).
02
Execute Leave-One-Out Deletion Iterations
For each observation i from 1 to N, construct the reduced dataset X_{(-i)} omitting x_i and compute the partial estimator θ̂_{(-i)}.
03
Construct Tukey Pseudo-Values
For each i, evaluate the Tukey pseudo-value J_i = N θ̂ - (N - 1) θ̂_{(-i)}. Verify that the arithmetic average J̄ = (1/N) ∑ J_i equals the bias-corrected jackknife estimator θ̂_jack.
04
Quantify First-Order Bias Elimination
Confirm mathematically that if the original estimator has bias E[θ̂] = θ + a_1/N + O(1/N²), the jackknife estimator eliminates the a_1/N term, leaving residual bias of order O(1/N²).
05
Step Sample Size Across N = 10 to 60
Repeat the leave-one-out procedure across standardized sample sizes N ∈ {10, 20, 30, 40, 50, 60}, monitoring the decay of estimation bias and variance.
06
Perform Asymptotic Linear Regression
Fit a linear regression model y = mx + b to the data coordinates (x = N, y = Metric). Confirm that the empirical gradient m = 1.0000 verifies exact first-order consistency.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Jackknife & Bias Reduction mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x
| # | Sample Size N (samples) | Jackknife Precision Metric (precision) | Sample Size N | Bias Reduction Metric | Remove |
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