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

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.

💡 Research Summary

The paper addresses a long‑standing practical limitation of Rao’s spacing test, a non‑parametric method for testing uniformity on the circle. Although the test is powerful, especially against multimodal alternatives, its adoption has been hampered because the null distribution of the test statistic (U_n) is analytically intractable. Consequently, practitioners have relied on pre‑computed critical‑value tables (e.g., Russell and Levitin 1995) that cover only a limited range of sample sizes (up to (n=1000)) and a few significance levels. When larger samples are encountered, software packages often substitute the (n=1000) critical values, which can lead to inaccurate inference.

The authors propose a two‑step solution. First, they derive a recursive formula for the (r)‑th raw moment of (U_n) (Theorem 1). By introducing the Pochhammer symbol and exploiting relationships with Stirling numbers of the second kind, they express (E(U_n^r)) as a linear combination of terms ((n)_j (n-j)^{n+r-1}) with coefficients (a(r)_j) that are generated recursively. This formulation enables the computation of moments of any order with computational complexity proportional to (r \times n), making it feasible even for very large (n).

Second, the authors employ the Gram‑Charlier expansion, an asymptotic series that corrects a normal approximation by incorporating higher‑order cumulants. Using the raw moments obtained from the recursion, they compute standardized cumulants (\kappa’_{{n},j}) and construct the expansion \