Empirical Step Function Generator
Sorts sample points X_{(1)} ≤ ... ≤ X_{(N)} and constructs right-continuous step function F_N(x) = (1/N) ∑ 1_{X_i ≤ x}.
Statistics & Probability / Empirical CDF & DKW Bound
FREE TO EXPLORELABORATORY 03 · STATISTICS & PROBABILITY
Empirical Cumulative Distribution Function and Dvoretzky–Kiefer–Wolfowitz Inequality
Construct the empirical cumulative distribution function (ECDF), verify the Glivenko-Cantelli theorem, and establish non-parametric confidence bands using the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
DKW BAND HALF-WIDTH Ε_N—
SAMPLING TELEMETRY
Ready to sample with Sample Size N = 25.0.
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
DKW confidence bands are conservative in the distribution tails where F(x)(1 - F(x)) approaches zero, where local variance is much smaller than 1/4. The empirical regression of ln(ε_N) versus ln(N) yields a gradient of m = -0.5000, confirming the theoretical N^(-1/2) square-root rate with R² = 1.000000.
Statistical tools & sampling modules.
Glivenko-Cantelli Supremum Meter
Evaluates the uniform Kolmogorov distance D_N = sup_x |F_N(x) - F(x)| between sample ECDF and true CDF.
DKW Confidence Band Calculator
Applies the Massart-tightened Dvoretzky-Kiefer-Wolfowitz bound to construct simultaneous confidence bands F_N(x) ± ε_N.
Kolmogorov-Smirnov Statistic Engine
Computes exact one-sample and two-sample KS test statistics and corresponding p-values under the Brownian bridge limit.
Quantile Inversion Resolver
Inverts the empirical distribution function to estimate sample quantiles Q(p) = inf{x: F_N(x) ≥ p}.
Power-Law Regression Compiler
Performs linear regression on ln(ε_N) versus ln(N) to verify theoretical N^(-1/2) scaling.
FOLLOW THE METHOD
A reason behind every step.
Construct the empirical cumulative distribution function (ECDF), verify the Glivenko-Cantelli theorem, and establish non-parametric confidence bands using the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality. Standardized parameter settings: 25.0, 50.0, 100.0, 200.0, 400.0, 800.0 samples.
01
Generate Random Sample Array
Draw an initial sample of size N = 25 from a standard normal distribution N(0, 1) and sort the sample array in ascending order to establish order statistics.
02
Construct the Empirical Cumulative Distribution Function
Evaluate the empirical step function F_N(x) across a fine grid of 1,000 points spanning x ∈ [-3.5, +3.5], noting steps of magnitude 1/N at each data point.
03
Compute the DKW Confidence Band Half-Width
Set confidence level 1 - α = 0.95 and calculate the DKW critical half-width ε_N = √[ln(2/α) / (2N)] = √[ln(40) / (2 · 25)] = 0.2716.
04
Construct Simultaneous Non-Parametric Envelope
Overlay upper bound U(x) = min(1, F_N(x) + ε_N) and lower bound L(x) = max(0, F_N(x) - ε_N), verifying that the true distribution function F(x) lies entirely within the envelope.
05
Step Through Geometric Sample Sizes N = 25 to 800
Iterate sample sizes across N ∈ {25, 50, 100, 200, 400, 800}, measuring the uniform constriction of the confidence band envelope.
06
Perform Log-Log Regression on Band Half-Width
Fit a linear regression model to the coordinates (x = ln(N), y = ln(ε_N)). Confirm that the empirical gradient m = -0.5000 validates the N^(-1/2) square-root rate.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Empirical CDF & DKW Bound mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(ε_N) = -0.5000 · ln(N) + 1.2000
| # | Sample Size N (samples) | DKW Band Half-Width ε_N (margin) | Natural Log of Sample Size ln(N) | Natural Log of Band Half-Width ln(ε_N) | Remove |
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