Berry-Esseen Binomial Jump Distance

Introduction

For a Binomial distribution with n trials and success probability p, the number of successes k can be standardized as z = (k - np) / √(np(1-p)). While the Central Limit Theorem guarantees convergence to the standard normal distribution, the Berry-Esseen theorem explicitly quantifies the absolute discrepancy (Kolmogorov distance) at any n. Because the exact Binomial CDF is a step function, the maximum discrepancy occurs at the jumps. We must consider both sides of each jump to find the true supremum distance.

Worked Example

Let's compute the approximation error for a specific case: n = 10, p = 0.1, and count fraction = 0 (so k = 0).

1. Exact Binomial CDF: We find P(X ≤ 0) = (1 - 0.1)¹⁰ = 0.9¹⁰ ≈ 0.3486784401.
2. Standardized Count: The mean is np = 1 and the standard deviation is √(np(1-p)) = √(10 × 0.1 × 0.9) = √0.9 ≈ 0.948683298. Therefore, the standardized value z = (0 - 1) / √0.9 ≈ -1.0540925533894598.
3. Normal CDF: Evaluating the standard normal CDF at z gives Φ(-1.054093) ≈ 0.1459202725718941.
4. Selected Error: The discrepancy is |0.3486784401 - 0.1459202725718941| = 0.202758167528106.
5. Berry-Esseen Bound: Using the conservative constant 0.56, the raw bound is 0.56 × (0.1² + 0.9²) / √0.9 ≈ 0.48403930051644. The error 0.202758... is well within this usable bound.

Experiment

You can explore this approximation using the interactive module. Follow these steps:

1. Open the Berry-Esseen Binomial Jump Distance interactive visualization.
2. Select the Trials and Success probability to define the Binomial distribution.
3. Choose a Count fraction to select a specific jump point. The module will compute and display the selected error and the maximum Kolmogorov distance across all jumps.
4. Observe the visual representation where both exact step CDF and standard normal curve are plotted, showing the exact discrepancies at each jump.

Assumptions and Limitations

This module utilizes a conservative universal Berry-Esseen constant of 0.56, though primary literature provides stronger constants for specific cases. This choice is intentional to ensure the bound is safely weaker and universally applicable. No continuity correction is applied to the normal approximation; we strictly measure the raw Kolmogorov distance. Degenerate distributions where p = 0 or p = 1 have no variance, rendering the standardized normal approximation invalid.

References

This module implements the absolute error bounds according to the primary research on optimal Berry-Esseen constants.

• Shevtsova, I. (2011). On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands. arXiv:1111.6554.This paper provides mathematical foundations and discusses the absolute constants used in Berry-Esseen type inequalities.