Departure-from-Normality Functional

• Departure-from-Normality Functional is a quantitative mapping that measures deviations from normality in statistical distributions, linear operators, and topological mappings.
• It employs methods such as moment ratios, empirical characteristic functions, and spectral decompositions to rigorously test non-Gaussianity and operator nonnormality.
• This framework unifies diagnostic tools for diverse applications, providing clear metrics for assessing hypothesis tests, transient dynamics, and functional separation.

A departure-from-normality functional is a quantitative mapping designed to measure or detect the extent to which data, operators, or mappings deviate from normality—a property that may refer to statistical normality of distributions, normality of linear operators, or topological normality of mappings. Such functionals arise in multiple contexts, including statistics, analysis, and matrix theory, each featuring specialized constructions and diagnostic principles informed by the respective notion of normality.

1. Departure-from-Normality in Distributional Testing

Departure-from-normality functionals are central to statistical tests for normality. They operationalize quantitative criteria for the non-Gaussianity of distributions and typically take the form of functionals of empirical distribution functions or summary statistics.

For univariate data, the sample Pearson measure of skewness (spms) introduced by Nakagawa, Hashiguchi, and Niki is a canonical example (Nakagawa et al., 2012). Let $X_1, \dots, X_n$ be a sample, with central moments $m_r = \frac{1}{n} \sum_{i=1}^n (X_i - \bar X)^r$ for $r=2,3,4$. Then, setting

$$\sqrt{b_1} = \frac{m_3}{m_2^{3/2}}, \quad b_2 = \frac{m_4}{m_2^2},$$

the spms statistic is

$$spms = \frac{\sqrt{b_1 (b_2 + 3)}}{2(5 b_2 - 6 b_1 - 9)}.$$

This functional of the data is sensitive to symmetry and tail behavior and forms the basis for a family of highly effective normality tests.

In the multivariate context, Ejsmont, Milošević, and Obradović propose an integral-based functional using the empirical characteristic function: $$T_N = N \int_{\|t\|=1} \left|\varphi_N(t) - e^{-1/2}\right|^2 dS(t),$$ where $\varphi_N$ is the empirical characteristic function. This is fully affine-invariant and acts as an omnibus measure for both normality and independence in multivariate samples (Ejsmont et al., 2021).

Alternative functionals for high-dimensional or functional data exploit spectral decompositions (e.g., spatial principal component scores) and aggregate departures of moments from their Gaussian benchmarks, as in

$\widehat T_p = \sum_{m=1}^p \widehat J_m,$

with each $\widehat J_m$ a spatial Jarque–Bera-type quadratic form (Kuenzer et al., 2021).

2. Operator-Theoretic Departure-from-Normality

In operator theory, departure-from-normality quantifies how far a linear operator or matrix is from being normal (i.e., commuting with its adjoint). The principal functional considered is the numerical abscissa: $$\omega(A) = \max_{\|x\|=1} \Re(x^* A x) = \lambda_{\max}\left(\frac{A + A^*}{2}\right),$$ where $A$ is a complex square matrix. This scalar functional measures how far the numerical range $W(A)$ extends into the right half-plane, which is intimately connected to transient dynamics in linear ODEs and to the rate of singular value decay in Lyapunov equations (Baker et al., 2014).

Traditional bounds on the decay of singular values of Lyapunov solutions involve functionals of the departure from normality, such as the condition number $\|V\|\|V^{-1}\|$ (where $A = V \Lambda V^{-1}$) or the maximal real part $\omega(A)$. The structural principle is that increased nonnormality (as measured by these functionals) up to a certain threshold slows the decay, beyond which faster decay may paradoxically occur.

3. Functional-Analytic Quantification of Departure from Normality in Mappings

In topological and functional-analytic settings, particularly in the theory of mappings between topological spaces, departure-from-normality is quantified via the oscillation of $f$-continuous functions that separate closed sets after lifting along the map $f: X \to Y$. Liseev introduces the following functional (Liseev, 12 Jun 2024): $$J(f) := \sup_{\substack{\mathcal O \subset Y,\, F,T \subset f^{-1}(\mathcal O) \ F \cap T = \varnothing,\, F,T\,\text{closed}}} D_f(\mathcal O;F,T),$$ where

$$D_f(\mathcal O;F,T) = \inf_{\varphi} \sup_{y \in \mathcal O} \inf_{\text{nbhd }\mathcal U \ni y} \operatorname{osc}_\varphi(f^{-1} \mathcal U),$$

the infimum taken over all bounded $\varphi : X \to [0,1]$ that separate $F$ and $T$. Here, $J(f)$ is zero if and only if $f$ is normal, serving as a sharp, quantitative index of departure from topological normality.

4. Asymptotic Properties and Consistency of Statistical Functionals

A crucial aspect of statistical departure-from-normality functionals is their behavior under the null hypothesis of normality and under alternatives.

For spms, the expansion under $X_i \sim N(0,1)$ yields vanishing mean and all odd moments, with variance and excess kurtosis controlled as: $$\operatorname{Var}(spms) = \frac{3}{2} n^{-1} + O(n^{-2}), \quad \beta_2(spms) = 3 + 20 n^{-1} + O(n^{-2}),$$ ensuring approximate normality for large $n$ (Nakagawa et al., 2012). The Johnson $S_U$ transformation further normalizes the null law for moderate sample sizes.

PDE-based and characteristic-function-based functionals (e.g., $T_{n,a}$, $T_N$) admit weak limit distributions: centered Gaussian process integrals under $H_0$, with power against nonnormal alternatives growing with sample size (Henze et al., 2019, Ejsmont et al., 2021). In the case of functional data, quadratic forms in high-dimensional spatial principal component scores converge in law to chi-squared distributions, permitting rigorous inference (Kuenzer et al., 2021).

5. Comparative Performance, Power, and Practicality

The performance of departure-from-normality functionals varies with the alternative hypothesis and data structure.

• For light-tailed, moderately skewed alternatives (e.g., Beta(2,1)), spms provides superior power over classical moment skewness, Shapiro–Wilk, and Lin–Mudholkar tests (Nakagawa et al., 2012).
• For heavy-tailed or multivariate alternatives, characteristic-function-based integrals and PDE-MGF residual norms ($T_{n,a}$, $T_N$) exhibit strong, sometimes best-in-class power and are fully affine-invariant (Ejsmont et al., 2021, Henze et al., 2019).
• Functional approaches combining spectral principal components and Jarque–Bera-type statistics demonstrate high sensitivity to departures in infinite-dimensional functional data, as seen in climate applications (Kuenzer et al., 2021).

Correct application requires attention to sample size, moment estimation accuracy, tail behavior, and, for operator-theoretic functionals, spectral localization.

6. Synthesis and Connections Across Domains

Departure-from-normality functionals unify disparate quantitative frameworks for detecting, testing, or measuring the deviation from normality. In statistics, they provide omnibus diagnostics and rigorous test statistics. In operator theory, they encode fine geometric and spectral information about matrices. In topology and analysis, functionals like $J(f)$ provide quantitative separation criteria for mapping normality.

The explicit construction and analysis of such functionals—ranging from ratios of sample moments, integrals of residuals against analytic PDE constraints, Bessel-based distance kernels, to measure-theoretic oscillations over fibers—play a foundational role in their respective domains, enabling both theoretical insight and practical implementation (Nakagawa et al., 2012, Henze et al., 2019, Ejsmont et al., 2021, Kuenzer et al., 2021, Liseev, 12 Jun 2024, Baker et al., 2014).