---
title: 'Epps–Pulley Statistic: Normality Test'
url: https://www.emergentmind.com/topics/epps-pulley-statistic
type: topic
---

# Epps–Pulley Statistic: Normality Test

The Epps–Pulley statistic is a test of normality constructed as a weighted $L^2$-distance between the empirical characteristic function of standardized data and that of the standard normal distribution, with a Gaussian weight parameterized by a bandwidth parameter $\beta>0$. It extends naturally to multivariate (affine-invariant, universally consistent) settings and is closely related, under simple transformations, to the biased Gaussian-kernel maximum mean discrepancy (MMD) between an empirical distribution and the standard normal. The Epps–Pulley statistic is distribution-free under normality and has well-understood asymptotic behavior, efficiency, and practical performance across diverse alternatives to normality [2106.13962, 2109.04897, 1901.03227].

## 1. Definition and Forms of the Epps–Pulley Statistic

For a univariate i.i.d. sample $X_1,...,X_n$ with unknown mean $\mu$ and variance $\sigma^2$, let $\overline X_n = n^{-1}\sum_{j=1}^n X_j$ and $S_n^2 = n^{-1} \sum_{j=1}^n (X_j - \overline X_n)^2$. The scaled residuals are $Y_{n,j} = (X_j - \overline X_n)/S_n$. The empirical characteristic function is $\psi_n(t) = n^{-1} \sum_{j=1}^n e^{i t Y_{n,j}}$, and the Gaussian weight is $\varphi_\beta(t) = (2\pi \beta^2)^{-1/2} \exp(-t^2/(2\beta^2))$.

The univariate Epps–Pulley statistic is
$$
T_{n,\beta} = n \int_{-\infty}^\infty |\psi_n(t) - e^{-t^2/2}|^2 \varphi_\beta(t) \, dt
$$
which can be explicitly computed as
$$
T_{n,\beta} = \frac{1}{n}\sum_{j,k=1}^n \exp\left( -\frac{\beta^2}{2}(Y_{n,j} - Y_{n,k})^2 \right)
- \frac{2}{\sqrt{1+\beta^2}} \sum_{j=1}^n \exp\left( -\frac{\beta^2 Y_{n,j}^2}{2(1+\beta^2)} \right)
+ \frac{n}{\sqrt{1+2\beta^2}}
$$

For the multivariate case, if $X_1, ..., X_n \in \mathbb{R}^d$ have sample mean $\overline X_n$ and covariance matrix $\Sigma_n$, then standardized residuals $Y_{n,j} = \Sigma_n^{-1/2}(X_j - \overline X_n)$ are used and the integration occurs over $\mathbb{R}^d$ with $\varphi_{\beta,d}(t) = (2\pi \beta^2)^{-d/2} \exp( -\|t\|^2/2\beta^2 )$[2106.13962, 2109.04897]. This generalization preserves affine invariance and universal consistency.

## 2. Connection to Maximum Mean Discrepancy (MMD)

The Epps–Pulley statistic coincides, up to notation and centering, with the biased Gaussian-kernel MMD between an empirical distribution $Q_n$ and the standard normal $\mathcal N_d(0,I)$:
$$
\mathrm{MMD}_b^2(Q_n; \gamma) = \left(\frac{\gamma^2}{2+\gamma^2}\right)^{d/2}
- \frac{2}{n} \left(\frac{\gamma^2}{1+\gamma^2}\right)^{d/2} \sum_{i=1}^n \exp\left( -\frac{\|z_i\|^2}{2(1+\gamma^2)} \right)
+ \frac{1}{n^2} \sum_{i,j} \exp\left( -\frac{\|z_i-z_j\|^2}{2\gamma^2} \right)
$$
with the correspondence $\gamma = 1/\beta$ [1901.03227]. The Epps–Pulley (or equivalently, the BHEP) statistic thus inherits the omnibus consistency and interpretability of the MMD, where the smoothing parameter $\beta$ (equivalently, kernel width $\gamma$) modulates sensitivity to different departures from normality.

## 3. Asymptotic Null Distribution and Eigenstructure

Under $H_0\!:\, X_j \sim \mathcal N(\mu, \sigma^2)$, the Epps–Pulley statistic $T_{n,\beta}$ is asymptotically distribution-free. As $n \rightarrow \infty$,
$$
T_{n,\beta} \xrightarrow{d} T_\beta := \int_{\mathbb R} Z(t)^2 \varphi_\beta(t) dt
$$
where $Z(\cdot)$ is a centered Gaussian process with covariance kernel
$$
K(s, t) = \exp\left( -\frac{(s-t)^2}{2} \right) - [1 + st + (st)^2/2] \exp\left( -\frac{s^2 + t^2}{2} \right)
$$
[2106.13962, 2109.04897]. The limiting law admits the series representation
$$
T_\beta \stackrel{d}{=} \sum_{j=1}^\infty \lambda_j(\beta) N_j^2
$$
where $N_j$ are i.i.d. $N(0,1)$ and $\{\lambda_j(\beta)\}$ are the positive eigenvalues of the integral operator on $L^2(\mathbb R, \varphi_\beta)$ defined by $K$. Numerical values for $\beta=1$ include $\lambda_1 \approx 0.0743, \lambda_2 \approx 0.0448, ...$[2109.04897]. This spectral representation underpins tail approximations, p-value computations, and Bahadur slopes.

## 4. Bahadur Efficiency and Comparative Power

The Bahadur efficiency of the Epps–Pulley statistic against local alternatives $G(\cdot;\theta)$ indexed by $\theta$ is characterized by two quantities:
- $b_{T_\beta}(\theta)$, the asymptotic mean under $G(\cdot; \theta)$,
- $\lambda_1(\beta)$, the leading eigenvalue from the null distribution.

Specifically, for small $\theta$,
$$
b_{T_\beta}(\theta) \approx \Delta(\beta) \theta^2, \qquad c_{T_\beta}^*(\theta) = \frac{b_{T_\beta}(\theta)^2}{\lambda_1(\beta)}, \qquad \ell(\beta) = \frac{\Delta(\beta)^2}{\lambda_1(\beta)}
$$
[2106.13962]. Table 1 in [2106.13962] demonstrates that for $\beta$ in the range $0.5 - 1$, the efficiency of the Epps–Pulley test is competitive with or generally superior to EDF-based tests (e.g., Shapiro–Wilk, Anderson–Darling) for certain alternatives, especially those exhibiting skewness (Ley–Paindaveine alternatives):

| β        | Lehmann | 1st Ley–P. | 2nd Ley–P. | Contam. N(1,1) | Contam. N(.5,1) | Contam. N(0,.5) |
|----------|---------|------------|------------|----------------|-----------------|-----------------|
| 0.25     | .996    | .947       | .824       | .760           | .945            | .084            |
| 0.5      | .895    | .944       | .872       | .649           | .824            | .267            |
| 0.75     | .854    | .998       | .986       | .592           | .766            | .474            |
| 1        | .743    | .937       | .981       | .499           | .654            | .587            |

Values are efficiencies relative to the (infeasible) likelihood-ratio test [2106.13962]. This suggests optimal power for $\beta$ near unity across a range of alternatives.

## 5. Practical Guidelines for Tuning and Implementation

The smoothing parameter $\beta$ modulates the Epps–Pulley test’s sensitivity:
- Small $\beta$ ($0.25$–$0.5$): Emphasizes high-frequency deviations in the characteristic function, suitable for location-type alternatives and mild skew.
- Intermediate $\beta$ ($0.75$–$1$): Effective across both skew and scale-contamination alternatives; $\beta=1$ aligns with classical BHEP settings.
- Large $\beta$ ($2$–$5$): Favours low-frequency discrepancies, enhancing sensitivity to scale or heavy-tailed alternatives at the cost of skew-detection [2106.13962, 1901.03227].

Implementation requires selection or cross-validation of $\beta$, or adoption of empirically motivated heuristics (e.g., Henze–Zirkler’s $\gamma_{d,n}$ for high-dimensional settings). Under the composite null, centering and whitening are performed prior to evaluation [1901.03227].

Monte Carlo can be used to estimate finite-sample thresholds or compute the Bahadur efficiency via plug-in estimates of $b_{T_\beta}(\theta)$ and $\lambda_1(\beta)$. In practice, standardized versions using explicit null mean/variance formulas allow rapid computation and interpretation [1901.03227].

## 6. Comparative and Theoretical Properties

The Epps–Pulley statistic is affine-invariant and, in the multivariate case, universally consistent against alternatives with finite second moments [2106.13962, 1901.03227]. Compared to Shapiro–Wilk and Anderson–Darling, its key distinctions are:
- Direct applicability in arbitrary dimensions,
- Closed-form critical value calculations via eigen-expansions,
- Explicit power modulated by the smoothing parameter,
- One-to-one equivalence (up to bias) with the widely used MMD, situating it within reproducing kernel Hilbert space (RKHS) theoretical frameworks.

Under the simple null $X_i \sim \mathcal N_d(0, I)$, standardized Epps–Pulley values are approximately $N(0,1)$; under the composite null, centering/whitening preserves distribution-freeness up to small sample effects [1901.03227].

A plausible implication is that the flexibility of the smoothing parameter $\beta$ allows the Epps–Pulley statistic to target classes of alternatives of interest, further justifying its use in modern model criticism and generative modeling settings.

## 7. Computational Considerations and Extensions

The computation of the Epps–Pulley statistic is explicit and involves only matrix operations and exponential functions. For limiting distribution evaluation, the eigenvalues $\lambda_j(\beta)$ can be computed numerically by truncating the spectral expansion, with software implementation benefiting from the work of Ebner and Henze [2109.04897]. For p-values, Imhof’s method or cumulant-matching to the Pearson system is effective.

In the context of generative models such as Wasserstein Auto-Encoders, batch-averaged and standardized forms of the Epps–Pulley (MMD) statistic are employed for both training objectives and diagnostics, with the correspondence between kernel width $\gamma$ and smoothing $\beta$ guiding model selection procedures [1901.03227].

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**References:**  
[2106.13962] "Bahadur efficiencies of the Epps--Pulley test for normality"  
[2109.04897] "On the eigenvalues associated with the limit null distribution of the Epps-Pulley test of normality"  
[1901.03227] "Closed-form Expressions for Maximum Mean Discrepancy with Applications to Wasserstein Auto-Encoders"

Source: https://www.emergentmind.com/topics/epps-pulley-statistic