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.

翻译：Rao间距检验是一种广泛使用的非参数方法，用于评估圆上的均匀性。然而，由于零分布不易计算，该方法在实际应用中的广泛适用性受到限制。因此，实践者传统上依赖于针对有限样本量预先计算的临界值表，这限制了该方法的灵活性和普适性。本文通过递归计算Rao间距检验统计量的高阶矩，并利用Gram-Charlier展开推导其零分布的高精度逼近，以解决这一局限性。该方法能够高效直接地计算任意样本量的p值，从而消除对现有临界值表的依赖。此外，我们证实即使对于当前表中未涵盖的大样本量，该方法仍保持准确有效，从而克服了一个重要的实际限制。与已发表的临界值及鞍点逼近法的比较评估表明，我们的方法在广泛的样本量范围内均实现了高度准确性。这些发现显著提升了Rao间距检验在理论研究和应用统计分析中的实用性与可用性。