Uniformity Tests via Overlapping Spacings

• The paper establishes a rigorous asymptotic framework for uniformity tests based on sum-functions of overlapping spacings, extending classical methods to multidimensional settings.
• It demonstrates that the test performance is governed by the correlation between overlapping and disjoint spacings, with explicit Pitman efficacy formulas quantifying efficiency.
• The study shows that increasing the spacing order improves discrimination power and reduces variance, moving detection rates closer to the parametric n⁻¹/² limit under local alternatives.

Tests for uniformity based on sum-functions of overlapping spacings form a central class of nonparametric test statistics, extending the classic spacings philosophy from univariate to multidimensional, discrete, and circular settings. These tests have recently been the focus of advanced asymptotic studies, particularly when the order of overlapping spacings grows with the sample size, and their efficacy strongly depends on connections with disjoint spacings statistics. Applications range from classical goodness-of-fit in one dimension to uniformity testing on spheres, high-dimensional data, and spatial point processes.

1. Statistical Framework for Higher Order Overlapping Spacings

The principle underlying tests based on sum-functions of overlapping spacings is to use the ordered sample $(X_1, ..., X_n)$ (typically after applying a probability integral transform to reduce the problem to $[0,1]$) and consider $m$-order overlapping spacings: $D_{k,m} = X_{k+m} - X_k, \quad 0 \leq k \leq n-m$ where $m$ may increase with $n$. The prototypical test statistic takes the form

$V_{n,m} = \sum_{k=0}^{n-1} h(n D_{k,m})$

with $h(\cdot)$ a chosen function (e.g., $h(x)=x^2$ for the Greenwood statistic) (Mirakhmedov, 26 Aug 2025). Overlapping spacings differ from disjoint spacings by their dependence structure: overlapping spacings share sample points, leading to more data reuse and stronger correlation between terms.

A corresponding "disjoint spacings" version partitions the data into non-overlapping blocks: $V_{n,m}^* = \sum_{k=0}^{N} h(n D_{k\cdot m,m})$ where $N = n/m - 1$, and $D_{k\cdot m,m}$ are $m$-step disjoint spacings.

Overlapping spacings allow the order $m$ to diverge with $n$, typically $m = o(n)$, to achieve improved discrimination between uniformity and alternatives.

2. Asymptotic Distribution and Local Power

Under the null hypothesis of uniformity, statistics $V_{n,m}$ can be shown to be asymptotically normal when properly centered and normalized (Mirakhmedov, 26 Aug 2025, Mirakhmedov, 15 Apr 2024): $$\frac{V_{n,m} - n\mathcal{A}_{0,n}}{\sigma_m\sqrt{n}} \xrightarrow{d} N(0,1)$$ where $\mathcal{A}_{0,n} = E[h(Z_{0,m})]$ (with $Z_{0,m}$ being the sum of $m$ standard exponentials) and $\sigma_m^2$ depends on the variance and covariances of $h(Z_{0,m})$ with its nearby shifts, as well as a correction for the mean-variance relationship created by overlap: $$\sigma_m^2 = \mathrm{Var}[h(Z_{0,m})] + 2\sum_{j=1}^{m-1}\mathrm{Cov}(h(Z_{0,m}), h(Z_{j,m})) - m^2\tau_m^2$$ with $\tau_m = m^{-1}\mathrm{Cov}(h(Z_{0,m}), Z_{0,m})$ (Mirakhmedov, 26 Aug 2025, Mirakhmedov, 15 Apr 2024).

Under local alternatives of the form $f(x) = 1 + (nm)^{-1/4} l_n(x)$, the expectation of $V_{n,m}$ is shifted by

$$\mathcal{A}_{1,n} = E[h(Z_{0,m})] + \frac{\sigma_m^*\sqrt{m+1}}{\sqrt{2n}} \mu_m(h) \lVert l \rVert_2^2$$

where $\sigma_m^{*2} = \mathrm{Var}[\varphi(Z_{0,m})]$ for $\varphi(u) = h(u) - E[h(Z_{0,m})] - (u-m)\tau_m$, and the efficacy is quantified by

$$e_m^2(h) = \frac{(m+1) \sigma_m^{*2}}{2\sigma_m^2} \mu_m^2(h)$$

with

$$\mu_m(h) = \mathrm{corr}(\varphi(Z_{0,m}), (Z_{0,m} - m)^2)$$

The power is then given by

$$\text{Power} \approx \Phi\left(e_m(h)\lVert l \rVert_2^2 - u_\alpha\right)$$

with $u_\alpha = \Phi^{-1}(1-\alpha)$.

The key parameter $\mu_m(h)$—correlation between the centered function of the spacing and its quadratic deviation—drives the local asymptotic power (Mirakhmedov, 26 Aug 2025).

3. Comparison and Role of Disjoint Spacings Statistics

A critical insight is that the asymptotic power of overlapping spacings tests is governed by the efficacy of the statistics based on disjoint spacings. Under similar Lyapunov-type conditions, disjoint spacings statistics $V_{n,m}^*$ satisfy

$$E[V_{n,m}^*] = N\mathcal{A}_{0,n}(1+o(1)), \qquad \mathrm{Var}(V_{n,m}^*) = N\sigma_m^{*2}(1+o(1))$$

with efficacy under local alternatives

$e_m^{*2}(h) = \frac{m+1}{2m}\mu_m^2(h)$

The Pitman relative efficiency between overlapping and disjoint spacings test is

$$PE(V_{n,m}(h), V_{n,m}^*(h)) = \lim_{n\to\infty} \frac{m \sigma_m^{*2}(h)}{\sigma_m^2(h)} \geq 1$$

Thus, overlapping spacings tests are at least as efficient (typically more so): for the Greenwood statistic, the disjoint test requires approximately $1.5$ times the sample size to reach equivalent power (Mirakhmedov, 26 Aug 2025). This dependence reveals overlapping spacings tests can be calibrated and understood via their disjoint analogues.

4. Effect of Increasing Spacing Order

Permitting the order $m$ of spacings to increase with $n$ (subject to $m = o(n)$) significantly impacts the rate at which local alternatives can be discriminated. With fixed $m$, alternatives detectable at $n^{-1/4}$ are at the statistical limit, but if $m$ diverges, the rate improves to $(nm)^{-1/4}$, approaching the parametric $n^{-1/2}$ rate in the limit. This improvement is apparent in the efficacy expressions, for instance: $e_m^2(h) \to \frac{3}{4} \text{ for the Greenwood statistic (as %%%%36%%%%)}$ Hence, higher-order overlapping spacings “average over” larger sample intervals, reducing variance and enhancing the test's ability to detect local deviations from uniformity.

A plausible implication is that practitioners can tune $m$ to balance power and stability, achieving higher sensitivity for alternatives with finer-grained deviations as $m$ increases—subject to computational constraints and the requirement $m=o(n)$.

5. Practical Relevance and Pitman Efficiency

These findings provide detailed guidance for designing and analyzing uniformity tests based on sum-functions of overlapping spacings. The core facts are:

• Overlapping spacings tests admit a normal limit for a wide class of functions $h$ and for orders $m=o(n)$ (Mirakhmedov, 15 Apr 2024, Mirakhmedov, 26 Aug 2025).
• Their asymptotic local power depends strongly on correlation structures rooted in disjoint spacings statistics, directly quantifiable and interpretable via $\mu_m(h)$ and Pitman ARE formulas.
• The choice of $h$ (Greenwood for quadratic, Moran for log spacings, etc.) determines the direction of maximal efficacy, and standard choices yield locally most powerful tests within the flexible class of overlapping spacings statistics (Singh et al., 2021).
• For fixed $m$, efficiency is limited, but exercises in the recent literature show that increasing $m$ sharpens discrimination rates and heightens Pitman efficacy, up to the $3/4$ bound for Greenwood-type statistics (Mirakhmedov, 26 Aug 2025).
• The asymptotic calibration is exact enough to support principled sample size and power calculations.

This suggests that, for nonparametric goodness-of-fit problems where high sensitivity to fine-scale deviations from uniformity is required, overlapping spacings methods with suitably chosen or diverging orders are theoretically preferred.

6. Mathematical Formulas and Summary Table

The principal formulas used in the asymptotic theory are summarized below:

Statistic	Formula under Null and Alternatives	Efficacy / Power
Overlapping $V_{n,m}$	$$\displaystyle \frac{V_{n,m} - n\mathcal{A}_{0,n}}{\sigma_m\sqrt{n}} \to N(0,1)$$	$$\displaystyle e_m^2(h) = \frac{(m+1)\sigma_m^{*2}}{2\sigma_m^2} \mu_m^2(h)$$
Disjoint $V_{n,m}^*$	Same as above, $N = n/m - 1$, $\sigma_m^{*2}$ replaces $\sigma_m^2$	$$\displaystyle e_m^{*2}(h) = \frac{m+1}{2m} \mu_m^2(h)$$

Key parameters:

• $\mathcal{A}_{0,n} = E[h(Z_{0,m})]$
• $\sigma_m^2$ includes variance and covariance terms of $h(Z_{0,m})$
• $\mu_m(h)$: correlation of $\varphi(Z_{0,m})$ with quadratic deviations
• Pitman ARE: $$PE(V_{n,m}(h), V_{n,m}^*(h)) = m \sigma_m^{*2}(h)/\sigma_m^2(h)$$

These precise expressions should be used to calculate sample size, select the order $m$, and choose the tuning function $h$ for optimal uniformity testing.

7. Conclusion

Tests for uniformity based on sum-functions of overlapping spacings, especially those allowing the spacing order to grow with sample size, are theoretically robust and highly efficient. The recent asymptotic theory establishes their normal limiting distribution under mild conditions and clarifies the dependence of their power and efficiency on related disjoint spacings statistics. The framework accommodates flexible choices of $h$ and supports tuning of $m$ for improved discrimination power. The critical link to disjoint spacings statistics, the explicit Pitman efficacy formulas, and the normal limiting distribution position these methods as a theoretically optimal choice for modern nonparametric testing of uniformity in a variety of applied statistical domains (Mirakhmedov, 26 Aug 2025, Mirakhmedov, 15 Apr 2024).