ADF Smooth Test Statistics

• Asymptotically distribution-free smooth test statistics are hypothesis tests whose limiting distribution is independent of nuisance parameters due to an integrated, L2-based structure.
• They utilize integral contrasts like the Cramér–von Mises and Anderson–Darling forms, often paired with transformations (e.g., Khmaladze martingale transforms) to neutralize parameter effects.
• These methods are widely applied across parametric, nonparametric, and multivariate settings, offering robust type-I error control and adaptability to complex data models.

Asymptotically distribution-free smooth test statistics constitute a class of hypothesis testing procedures where the test statistic, under the null hypothesis, converges in distribution to a law that does not depend on unknown model parameters or other nuisance variables. This property enables universal calibration of significance thresholds, independent of the data-generating distribution's specifics, while “smoothness” refers to the integral or quadratic structure of the test statistic as opposed to maximally local (supremum-based) measures. Theoretical advances have produced a variety of such tests for both parametric and nonparametric models, for i.i.d., time series, diffusions, point processes, and multivariate settings.

1. Defining Properties and General Structure

A test statistic is called asymptotically distribution-free (ADF) if, under the null hypothesis, its limiting (often functional) distribution is independent of nuisance parameters. “Smooth” refers to statistics that leverage integrated squared or $L^2$-type contrasts (e.g., Cramér–von Mises, Anderson–Darling, or integral functionals of empirical processes) rather than maxima/minima. Typical ADF smooth test statistics take the form

$$T_n = \int [\hat{F}_n(x) - F_0(x;\theta)]^2 w(x) dx$$

with $\hat{F}_n$ an estimator of the distribution function, $F_0$ the hypothesized model, and $w(x)$ a weight. Variants for densities, diffusions, or processes use empirical densities, local times, spacings, U-statistics, and functionals of other empirical or projected processes.

The fundamental property is that, after proper centering, scaling, and possibly applying a linear or innovation transformation (e.g., Khmaladze’s martingale transformation or its contemporaries), the limiting distribution of $T_n$ is a known law such as a (non-central) chi-squared, normal, Brownian motion functional, or standard Wiener process, with all nuisance parameters eliminated.

2. Construction in Parametric, Diffusion, and Point Process Models

Diffusion Processes

For discretely observed multidimensional ergodic diffusion processes, one constructs statistics by comparing quasi-likelihood approximations of transition densities under the null and alternative. Consider a family of $\phi$-divergence statistics,

$$T_{\phi,n}(\hat{\theta}, \theta_0) = 2n D_{\phi,n}(\hat{\theta}, \theta_0)$$

where

$$D_{\phi, n}(\hat{\theta}, \theta_0) = \frac{1}{n}\sum_{i=1}^n \phi\left(\frac{p_i(\hat{\theta})}{p_i(\theta_0)}\right),$$

and $p_i(\theta)$ are quasi-likelihood approximations based on an Euler–Maruyama or higher-order scheme. For admissible $\phi$ (with $\phi(1)=\phi'(1)=0$, $\phi''(1)=1$), $T_{\phi,n}$ converges to a $\chi^2_{p+q}$ distribution, $p,q$ being model parameters. The corresponding test is ADF since the limit is universal for each $p+q$ and does not depend on unknown nuisance parameters. The choice of $\phi$ affects small sample power, with no uniformly most powerful test; e.g., an AKL divergence, power divergences with various $\lambda$, and the BS divergence demonstrate varying behaviors under simulation (Gregorio et al., 2011).

For ergodic diffusions with parametric trend, tests are built on integrated squared deviations between the empirical local time (density estimator) and the theoretical invariant density computed at the MLE: $A_T = \int [L[S_T](x)]^2 dF(\hat{\theta}, x)$ where $L[\cdot]$ is a linear transformation (often defined via a Fredholm equation for projection) that “standardizes” the limit, making the test statistic converge to a universal Wiener process functional (e.g., $\int_0^1 w(t)^2 dt$), thus attaining the distribution-free property (Kleptsyna et al., 2013).

For point processes, the central object is a compensated empirical counting process, martingale decomposition, and subsequent innovation martingale transformation. After standardization and parameter estimation correction, the transformed process converges to a multivariate standard Wiener process. Functionals such as the Cramér–von Mises or Kolmogorov–Smirnov derived from this process yield exactly ADF tests (Baars et al., 31 Mar 2025).

Shape-Constrained and Density Smoothness Tests

Likelihood-based spacings tests for univariate shape-constrained families (e.g., $k$-monotone, log-concave, or completely monotone densities) are constructed as

$$T_n = -\frac{1}{n} \log\left(\prod_{i=1}^n \hat{f}_n(X_i)/\prod_{i=1}^n f^H_{n,\nu}(X_i)\right)$$

where $\hat{f}_n$ is the nonparametric MLE under the constraint, and $f^H_{n,\nu}$ is a histogram (spacings-based) estimator. Under the null, when decomposed, the leading term $M_n$ is a sum over transformed spacings with a limit law independent of the underlying $f_0$. Consequently, centering and scaling provide a statistic converging to a standard normal law, with the distribution-freeness stemming from the uniformity of the transformed spacings under the null. Under alternatives, the remaining terms diverge, ensuring consistency. Bootstrap calibration using the NPMLE ensures robust finite-sample type-I error control (Chan et al., 2022).

Tests for smoothness of a density employ wavelet decompositions. The $L^2$-norm of orthogonal projections onto wavelet subspaces $Q_j f$ decays polynomially with $j$ at a rate dictated by smoothness. For density $f$, an unbiased U-statistic estimator $L_{n,j}$ for $\|Q_j f\|_2^2$ has, under regularity and enrichment (mixture with a smooth component for regularity), a centered and scaled limit law that is normal, with variance independent of nuisance parameters (Ćmiel et al., 2018).

3. Distribution-Free Smooth Tests in Nonparametric and Multivariate Contexts

Two-Sample and Independence Testing

ADF smooth tests for the two-sample problem generalize Neyman’s smooth test by expanding alternatives into an orthonormal basis (e.g., trigonometric or Legendre functions) evaluated on probability integral transforms of the empirical distribution. For univariate data,

$$\hat{\Psi}(d) = \sqrt{\frac{nm}{n+m}}\max_{1\leq k\leq d} |\hat{\psi}_k|,\quad\hat{\psi}_k = \frac{1}{m}\sum_{j=1}^{m} \psi_k(\tilde{Y}_j),$$

where $\tilde{Y}_j = \hat{F}_n(Y_j)$; $\psi_k$ are basis functions over $[0,1]$. For fixed $d$, the null limit is the maximum norm of a $d$-variate normal. Allowing $d\to\infty$ with $n$ increases power for local or high-frequency features but requires correction for multiple testing. The approach is extended to multivariate data via projection pursuit, testing for differences along all directions and aggregating via a supremum, with the limiting null approximated (bootstrap) by simulation of a Gaussian process over a function space (Zhou et al., 2015).

Asymptotically distribution-free two-sample tests for multivariate distributions based on graph structures (e.g., minimum spanning tree, K-nearest neighbors, cross-match, depth-based graphs) standardize statistics derived from the number of edges connecting different groups. Under suitable conditions, the standardized statistic converges to a normal distribution whose variance is determined by graph combinatorics, not the underlying distributions, rendering the test asymptotically distribution-free. However, power against close alternatives (Pitman efficiency) depends critically on the choice of graph and underlying data structure; for example, MST or fixed $K$-NN graphs can have zero efficiency for local alternatives, while growing $K$ can remediate this (Bhattacharya, 2015).

Rank-based multivariate tests constructed via optimal transport generalize the Cramér–von Mises test. The empirical rank map is an optimal assignment of sample points to a uniform grid; in one dimension, this reproduces the univariate Cramér–von Mises test, establishing finite-sample exact distribution-freeness and favorable efficiency properties. Multivariate versions extend these properties under natural sub-families of distributions (Deb et al., 2021).

Kernel-Smoothed and Order Statistic Methods

Smoothed versions of the sign and Wilcoxon tests utilize kernel estimators of the distribution function, e.g.,

$$F_n(x) = \frac{1}{n}\sum_{i=1}^n K\left(\frac{x - X_i}{h_n}\right),$$

with $K$ the integrated kernel and $h_n$ the bandwidth. The smoothed sign statistic,

$\hat{S} = n - \sum_{i=1}^n K(-X_i/h_n),$

has, under smoothness and bandwidth conditions, an asymptotic normal distribution with mean and variance equal to those of the original sign test under $H_0$. Hence the smoothed tests are asymptotically distribution-free, and detailed Edgeworth expansions can be derived with remainder terms free of distribution-specific parameters, provided suitable kernels are chosen (Maesono et al., 2016).

In discrete settings, a unitary transformation is applied to Pearson’s chi-squared components to produce an orthogonal system whose partial sum functionals (e.g., discrete empirical process functionals) are asymptotically distribution-free, even when the null is composite or involves unknown parameters. Functionals such as Kolmogorov–Smirnov or Cramér–von Mises can then be constructed directly from these transformed quantities (Khmaladze, 2014).

4. Theoretical Advances and Transformations

Empirical processes theory has supplied the analytical tools for these developments. The Khmaladze martingale transformation and more recent innovation martingale transforms are fundamental to several construction strategies. For instance, for fitted models (with estimated parameters), the empirical process is projected orthogonally to the score functions (and their derivatives), or, equivalently, transformed so that the resulting process' limiting covariance structure is universal (i.e., a Brownian motion or bridge with known covariance). For increasingly complex hypotheses, recent work provides new linear projections (e.g., Khmaladze-2 transforms, K2) and function-parametric empirical processes to ensure that parameter estimation and model selection do not affect asymptotic null distributions (Zhang et al., 4 Aug 2025).

In model selection and high-dimensional settings, projected bootstrapping, which forgoes re-estimation of parameters on every bootstrap replicate (relying on the equivalence induced by projection properties), yields computationally efficient calibration with robust approximation accuracy even at moderate sample sizes.

5. Power, Practical Application, and Comparative Analysis

Power studies, both asymptotic and simulation-based, demonstrate that while all ADF smooth tests achieve the correct size, their relative efficacy depends on the choice of test, model, and sample size. For instance, in discretely observed diffusions, small-sample performance varies across $\phi$-divergence statistics, with no uniformly most powerful test; in mixture and multiple testing models, specially designed distribution-free statistics match the asymptotic power of likelihood-based or oracle procedures (e.g., the CUSUM sign and tail-run tests for sparse mixtures or the Barber–Candès procedure for FDR control in multiple testing) (Arias-Castro et al., 2013, Arias-Castro et al., 2016). For smoothness and shape-constrained density inference, bootstrap techniques rectify slow convergence of secondary terms, ensuring accurate type-I error control and robust power (Chan et al., 2022, Ćmiel et al., 2018). In high-dimensional projection-pursuit and graph-based two-sample tests, the choice of projection, basis, or graph construction directly influences Pitman efficiency and power against local or sparse alternatives (Zhou et al., 2015, Bhattacharya, 2015).

In practical applications (e.g., financial data modeling, point process analysis such as ETAS models for earthquakes or epidemic models), the transformation-based ADF procedures yield robust inference by ensuring that nominal significance levels are meaningful and do not require extensive re-calibration for each scenario.

6. Methodological Extensions and Limitations

ADF smooth test statistics now exist for a broad class of models: from classical i.i.d. problems to dependent time series (e.g., ergodic diffusions, point processes), from univariate to high-dimensional and multivariate scenarios, including settings with unknown parameters or where model selection is performed. The critical regularity requirements are typically that the estimator is asymptotically linear, the indexing function class is regular (e.g., Donsker class), and that the transformation or projection needed for parameter adjustment is computable.

Limitations arise for highly irregular data or models where parameter estimation rates are not standard, where the requisite projection or transformation cannot be easily constructed, or where the model’s structural regularity is insufficient for the theory. For some tests, notably in the high-dimensional regime, careful consideration of the curse of dimensionality and computational complexity is important.

7. Summary Table of Selected ADF Smooth Tests

Model/Class	Test Statistic Form	Asymptotic Null Distribution
One-dimensional diffusion	$\phi$-divergence, $L^2$-quasi-likelihood	$\chi^2$ ($p+q$ d.o.f.)
Two-sample (univariate)	Max of smooth scores over orthonormal basis	Max norm of normal vector
Multivariate distributions	Graph-based edge counts, depth-based ranks	Normal (variance from graph)
Shape-constrained density	Log-likelihood ratio (spacings)	Standard normal
Kernel-smoothed ranks	Smoothed sign/Wilcoxon kernel statistic	Standard normal
Point processes	Martingale-transformed empirical process	Standard Wiener process functional
Optimal transport ranks	Rank-based test on assignment grid	Same as Cramér–von Mises (finite-sample d.f.)

In summary, ADF smooth test statistics provide a robust, theoretically justified, and widely applicable framework for hypothesis testing, with uniformly calibrated significance levels and strong small-sample and asymptotic behaviors. Their methodological advances, especially recent innovations in transformation and projection, continue to extend applicability to high-dimensional, dependent, and composite models.