[Paper Review] Tables of Quantiles of the Distribution of the Empirical Chiral Index in the Case of the Uniform Law and in the Case of the Normal Law

Michel Petitjean|arXiv (Cornell University)|May 20, 2020

Radioactive Decay and Measurement Techniques4 references1 citations

TL;DR

This paper provides Monte Carlo simulations of the empirical chiral index distribution under the uniform and normal laws, tabulating critical quantiles (K0.90, K0.95, K0.98, K0.99) for sample sizes from 3 to 10,000. The chiral index, computed via correlation between ordered and reverse-ordered samples, enables location- and scale-free symmetry testing, with results showing decreasing quantile values as sample size increases, converging toward zero for symmetric distributions.

ABSTRACT

The empirical distribution of the chiral index is simulated for various sample sizes for the uniform law and and for the normal law. The estimated quantiles $K_{0.90}$, $K_{0.95}$, $K_{0.98}$, and $K_{0.99}$, are tabulated for use in symmetry testing in the uniform case and in the normal case.

Motivation & Objective

• To provide empirical critical values for the chiral index under the uniform and normal distributions to support symmetry testing.
• To enable location- and scale-free inference by demonstrating that the chiral index distribution is invariant under location and scale shifts.
• To offer a computationally simple, pocket-calculator-friendly method for computing the chiral index as a measure of asymmetry.
• To support hypothesis testing for symmetry by tabulating estimated quantiles (K0.90, K0.95, K0.98, K0.99) across diverse sample sizes.
• To facilitate practical application of the chiral index in statistical testing by providing reproducible, simulation-based quantile tables.

Proposed method

• The chiral index is computed as χ = (1 + r_m)/2, where r_m is the minimal correlation between the ordered and reverse-ordered sample sequences.
• Monte Carlo simulations generate 10,000 chiral index values per sample size n for both U(0,1) and N(0,1) distributions.
• Quantiles K0.90, K0.95, K0.98, and K0.99 are estimated as midpoints of the 9000th–9001st, 9500th–9501st, 9800th–9801st, and 9900th–9901st order statistics, respectively.
• Simulations are repeated 100 times to compute mean quantile estimates and their standard errors, reported in Tables 1 and 2.
• A long-period pseudo-random number generator (NAG g05saf) is used without reinitialization to ensure consistency across simulations.
• The chiral index is computed using a simple two-step algorithm: sorting data in increasing order and correlating it with the reverse-ordered sequence.

Experimental results

Research questions

• RQ1 What are the critical quantiles of the empirical chiral index distribution under the uniform law for various sample sizes?
• RQ2 What are the critical quantiles of the empirical chiral index distribution under the normal law for various sample sizes?
• RQ3 How does the chiral index behave under symmetric distributions, and can it serve as a robust, scale-free symmetry test?
• RQ4 How stable are the estimated quantiles across repeated Monte Carlo simulations, and what are their standard errors?
• RQ5 Can the chiral index be computed efficiently even with minimal computational tools, such as a pocket calculator?

Key findings

• For the uniform distribution, the 95th percentile of the chiral index (K0.95) decreases from 0.231 at n=3 to 0.00205 at n=1000, indicating convergence to zero under symmetry.
• For the normal distribution, the 95th percentile (K0.95) decreases from 0.229 at n=4 to 0.00205 at n=1000, showing similar convergence behavior.
• At n=1000, the 99th percentile (K0.99) is 0.002977 for the normal law and 0.002977 for the uniform law, indicating near-zero values under symmetry.
• The estimated standard deviations of the quantiles (SK0.90 to SK0.99) decrease with increasing sample size, reflecting improved precision.
• For n=10,000, the 99th percentile (K0.99) is 0.000317 for both distributions, confirming asymptotic convergence to zero.
• The chiral index can be computed in O(n log n) time using a simple algorithm: sort the data, reverse it, compute the correlation, and apply χ = (1 + r)/2.