Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Accurate and Efficient Approximation of the Null Distribution of Rao's Spacing Test

Yoshiki Kinoshita, Aya Shinozaki, Toshinari Kamakura

TL;DR

This work solves a long-standing practical bottleneck in Rao's spacing test by deriving a recursive method to compute higher-order moments of the null distribution of $U_n$ and combining them with a Gram-Charlier expansion to approximate the CDF. The approach yields accurate p-values for arbitrary sample sizes, including very large $n$, and is shown to match exact values and outperform saddlepoint methods in benchmarks. Through theoretical derivations, explicit moment formulas, and cumulant-based density expansions, the method provides a flexible and scalable alternative to tabulated critical values. The practical impact is a more versatile nonparametric tool for circle uniformity that remains effective across diverse sample sizes and can be extended to more complex directional data settings in future work.

Abstract

Rao's spacing test is a widely used nonparametric method for assessing uniformity on the circle. However, its broader applicability in practical settings has been limited because the null distribution is not easily calculated. As a result, practitioners have traditionally depended on pre-tabulated critical values computed for a limited set of sample sizes, which restricts the flexibility and generality of the method. In this paper, we address this limitation by recursively computing higher-order moments of the Rao's spacing test statistic and employing the Gram-Charlier expansion to derive an accurate approximation to its null distribution. This approach allows for the efficient and direct computation of p-values for arbitrary sample sizes, thereby eliminating the dependency on existing critical value tables. Moreover, we confirm that our method remains accurate and effective even for large sample sizes that are not represented in current tables, thus overcoming a significant practical limitation. Comparative evaluations with published critical values and saddlepoint approximations demonstrate that our method achieves a high degree of accuracy across a wide range of sample sizes. These findings greatly improve the practicality and usability of Rao's spacing test in both theoretical investigations and applied statistical analyses.

Accurate and Efficient Approximation of the Null Distribution of Rao's Spacing Test

TL;DR

This work solves a long-standing practical bottleneck in Rao's spacing test by deriving a recursive method to compute higher-order moments of the null distribution of and combining them with a Gram-Charlier expansion to approximate the CDF. The approach yields accurate p-values for arbitrary sample sizes, including very large , and is shown to match exact values and outperform saddlepoint methods in benchmarks. Through theoretical derivations, explicit moment formulas, and cumulant-based density expansions, the method provides a flexible and scalable alternative to tabulated critical values. The practical impact is a more versatile nonparametric tool for circle uniformity that remains effective across diverse sample sizes and can be extended to more complex directional data settings in future work.

Abstract

Rao's spacing test is a widely used nonparametric method for assessing uniformity on the circle. However, its broader applicability in practical settings has been limited because the null distribution is not easily calculated. As a result, practitioners have traditionally depended on pre-tabulated critical values computed for a limited set of sample sizes, which restricts the flexibility and generality of the method. In this paper, we address this limitation by recursively computing higher-order moments of the Rao's spacing test statistic and employing the Gram-Charlier expansion to derive an accurate approximation to its null distribution. This approach allows for the efficient and direct computation of p-values for arbitrary sample sizes, thereby eliminating the dependency on existing critical value tables. Moreover, we confirm that our method remains accurate and effective even for large sample sizes that are not represented in current tables, thus overcoming a significant practical limitation. Comparative evaluations with published critical values and saddlepoint approximations demonstrate that our method achieves a high degree of accuracy across a wide range of sample sizes. These findings greatly improve the practicality and usability of Rao's spacing test in both theoretical investigations and applied statistical analyses.
Paper Structure (12 sections, 5 theorems, 42 equations, 5 tables)

This paper contains 12 sections, 5 theorems, 42 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Moments of Rao's Spacing Test
  3. Rao's Spacing Test Statistic
  4. Derivation of Higher-Order Moments
  5. Gram-Charlier Expansion of the CDF
  6. Comparison with Existing Methods
  7. Examples
  8. Simulation Study
  9. Discussion and Conclusion
  10. Proof of Theorem \ref{['col']}
  11. Gram-Charlier expansion
  12. Higher-Order Moments of Un

Key Result

Theorem 1

where $a_j^{(r)}$ satisfies: and for all $n\in\mathbb{N}_{\geq 2}$ and $r\in\mathbb{N}_{\geq 1}$.

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • proof
  • Definition 1: exponential partial Bell polynomials
  • Definition 2: exponential complete Bell polynomials