Original Sample Cache Buffer
Holds the primary empirical observation dataset X = {x_1, x_2, ..., x_n} of sample size n = 50.
Statistics & Probability / Bootstrap Resampling
FREE TO EXPLORELABORATORY 10 · STATISTICS & PROBABILITY
Non-Parametric Bootstrap Resampling and Percentile Confidence Intervals
Implement Bradley Efron's non-parametric bootstrap resampling algorithm, construct percentile and BCa confidence intervals for non-linear estimators, and verify Monte Carlo standard error convergence MCSE ∝ B^(-1/2). Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
MONTE CARLO STANDARD ERROR MCSE—
SAMPLING TELEMETRY
Ready to sample with Replications B = 100.0.
Adjust Replications B, 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 non-parametric bootstrap assumes observations are independent; it fails for time series or spatial data without block adjustments. The empirical regression of ln(MCSE) versus ln(B) confirms the theoretical Monte Carlo simulation rate with an empirical gradient of m = -0.5000 and R² = 1.000000.
Statistical tools & sampling modules.
Uniform Resampling Engine with Replacement
Draws bootstrap pseudo-samples X* = {x*_1, ..., x*_n} of size n uniformly with replacement using discrete PRNG.
Non-Linear Estimator Kernel
Evaluates complex target estimators θ̂* (e.g., sample correlation, trimmed mean, ratio of variances) across each bootstrap replicate.
Bootstrap Replicate Distribution Cache
Accumulates and sorts B bootstrap estimates θ̂*_1 ≤ θ̂*_2 ≤ ... ≤ θ̂*_B to construct empirical bootstrap distribution F̂_B.
Percentile & BCa Confidence Interval Resolver
Calculates (1 - α) percentile intervals [θ̂*_{(α/2)}, θ̂*_{(1 - α/2)}] and bias-corrected accelerated (BCa) adjusted intervals.
Monte Carlo Convergence Regression Engine
Performs linear regression on ln(MCSE) versus ln(B) to confirm the theoretical -0.500 (-1/2) simulation convergence rate.
FOLLOW THE METHOD
A reason behind every step.
Implement Bradley Efron's non-parametric bootstrap resampling algorithm, construct percentile and BCa confidence intervals for non-linear estimators, and verify Monte Carlo standard error convergence MCSE ∝ B^(-1/2). Standardized parameter settings: 100.0, 200.0, 400.0, 800.0, 1600.0, 3200.0 replicates.
01
Acquire and Cache Original Sample Dataset
Generate an empirical dataset of size n = 50 from a skewed log-normal distribution and compute the target statistic θ̂ (e.g., 20% trimmed mean).
02
Execute Primary Bootstrap Resampling for B = 100
Draw B = 100 independent bootstrap samples of size n = 50 with replacement. For each pseudo-sample, evaluate the bootstrap estimate θ̂* and store it in memory.
03
Compute Bootstrap Standard Error and Confidence Limits
Calculate the sample standard deviation of the B = 100 estimates to obtain the bootstrap standard error SE_boot. Extract empirical 2.5th and 97.5th percentiles to form a 95% confidence interval.
04
Evaluate Monte Carlo Standard Error (MCSE)
Estimate the simulation error of the bootstrap quantile estimate, noting that the simulation uncertainty of the empirical percentile scales as MCSE ∝ B^(-1/2).
05
Step Bootstrap Replicates Across B = 100 to 3200
Increment the number of bootstrap replicates across B ∈ {100, 200, 400, 800, 1600, 3200}, observing the progressive smoothing of the bootstrap histogram and stabilization of interval endpoints.
06
Execute Log-Log Regression on Simulation Error
Fit a linear regression line to coordinates (x = ln(B), y = ln(MCSE)). Confirm that the empirical gradient m = -0.5000 rigorously verifies the B^(-1/2) Monte Carlo convergence rate.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Bootstrap Resampling mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(MCSE) = -0.5000 · ln(B) + 1.5000
| # | Bootstrap Replications B (replicates) | Monte Carlo Standard Error MCSE (MCSE) | Natural Log of Bootstrap Replicates ln(B) | Natural Log of Monte Carlo SE ln(MCSE) | Remove |
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