CLT for ECDF Across Multiple Points

At each of 10 different x-values, visualize how √n·(F̂ₙ(x) - F(x)) / √[F(x)·(1-F(x))] converges to N(0,1). Each x-position shows 5 overlapping histograms (colored by sample size n) plus the theoretical normal curve in cyan.

🎯 What to Look For:

  • Convergence Pattern: As n increases (red → blue), histograms get closer to the cyan normal curve
  • Variance Effect: Middle quantiles (F(x) ≈ 0.5) have maximum variance, extreme quantiles have less
  • Universal Shape: Despite different distributions, all standardized ECDFs converge to the same N(0,1)
  • Sample Size Impact: Larger n = tighter distribution = better approximation to true CDF
Initializing 3D scene...
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n = 10
n = 25
n = 50
n = 100
n = 500
Standard Normal (Theory)

Theory: CLT for the Empirical CDF at Multiple Points

Pointwise CLT for the ECDF: For each fixed point x, the standardized empirical CDF converges:

Zₙ(x) = √n · (F̂ₙ(x) - F(x)) / √[F(x)·(1-F(x))] →ᵈ N(0, 1)

This visualization samples 10 different x-values across the distribution's support. At each x-value (positioned along the horizontal axis):

  • We run many simulations for each sample size n
  • For each simulation, we compute Zₙ(x) - the standardized deviation from the true CDF
  • The histograms show the empirical distribution of Zₙ(x), all centered at 0
  • As n increases, the distributions tighten and converge to N(0,1) (cyan curve)

Key observations: The variance F(x)·(1-F(x)) is maximized when F(x) = 0.5 (median), so convergence may appear different at different quantiles even though all are standardized to have variance 1.