Marginal Distribution Asymmetry

• Marginal distribution asymmetry is the lack of mirror symmetry in univariate distributions, revealing nuanced properties like skewness and tail behaviors.
• Robust measures such as cumulative skew (CS) offer alternatives to classical skewness metrics by mitigating the influence of outliers and reflecting true distribution shape.
• The concept applies broadly—from power system frequency deviations and galaxy spin distributions to advanced statistical modeling in censored data and contingency tables.

Marginal distribution asymmetry refers to the failure of a marginal distribution to be mirror-symmetric, most commonly in the sense of univariate skewness, but the term is also used for asymmetry in empirical marginal laws, frequency probability distributions, and binary observational counts. Across these settings, the common problem is to characterize imbalance in a way that remains informative about shape, tail behavior, or structural bias without collapsing distinct mechanisms into a single symmetric summary (Schlemmer, 2022, Müller et al., 2024, Kerci et al., 2024, Shamir, 2024).

1. Classical statistical meaning and validity criteria

In the univariate setting, a distribution is symmetric if both sides mirror each other, left-skewed if the left tail is longer, and right-skewed if the right tail is longer. The standard third-moment coefficient of skewness, denoted $b_1$, is widely used for this purpose, but it is described as very sensitive to outliers. The motivating example in the robust-skewness literature is that a single unusually large or small value may make a distribution appear strongly skewed even when the bulk of the data is nearly symmetric. Other robust measures, including the Pearson mode skewness and the medcouple, are less affected by outliers, but they rely heavily on medians and quantiles, which can make them less sensitive to extreme tail behavior (Schlemmer, 2022).

A skewness measure is treated as acceptable only if it satisfies standard axioms. The requirements reported for a valid skewness measure are scale and location invariance, sign reversal under reflection, zero value under symmetry, and consistency with van Zwet’s $c$-ordering. In the notation used in the robust-skewness paper, these are expressed through invariance under $a\times F+b$ with $a>0$, reflection changing the sign, $F=0$ under symmetry, and monotonicity under $F<.G$. These requirements matter because they distinguish genuine asymmetry from artifacts of scaling, translation, or coordinate choice; in that sense, they function as structural constraints on any proposed marginal asymmetry statistic (Schlemmer, 2022).

2. Lorenz-based cumulative skew and robustness to outliers

A robust alternative to $b_1$ is the cumulative skew statistic $CS$, introduced through the Lorenz-curve framework. For a sample $X_n=x_1,\ldots,x_n$, let $p_i$ be the cumulative proportion of observations, $c$0 the cumulative proportion of the measured values, and $c$1 the vertical distances between the Lorenz curve and the $c$2-degree line. The proposed statistic is

$c$3

where $c$4 denotes the ranks in the cumulative distribution. The weights are bounded by $c$5, $c$6 is bounded by $c$7, and for finite $c$8 its range becomes $c$9. The geometric interpretation is that asymmetry is extracted from cumulative distances whose signs and magnitudes are weighted differently above and below the center (Schlemmer, 2022).

The reported behavior of $a\times F+b$0 is nearly zero on symmetric distributions, monotone in Tukey’s $a\times F+b$1-distributions, and bounded even for strongly right-skewed cases. In repeated simulation, the mean skewness under a normal distribution was $a\times F+b$2, and under a Cauchy distribution it was $a\times F+b$3. On lognormal data, both $a\times F+b$4 and $a\times F+b$5 increase with the shape parameter $a\times F+b$6, but $a\times F+b$7 remains much more moderate:

Condition	$a\times F+b$8 (average)	$a\times F+b$9 (average)
$a>0$0	0.586	0.104
$a>0$1	1.598	0.241
$a>0$2	3.724	0.448
$a>0$3	7.635	0.741

The main robustness result concerns outliers. With high outliers added to a lognormal sample with $a>0$4, $a>0$5 increases from $a>0$6 to $a>0$7, while $a>0$8 changes from $a>0$9 to $F=0$0. With low outliers added to a lognormal sample with $F=0$1, $F=0$2 becomes negative at $F=0$3, incorrectly suggesting left skewness, whereas $F=0$4 remains positive at $F=0$5. The butter clam example makes the same point operationally: with one extreme observation, the classical skewness becomes $F=0$6, although the sample is approximately symmetric without that outlier. The paper identifies the largest discrepancy between $F=0$7 and $F=0$8 when there is just one strong outlier (Schlemmer, 2022).

3. Structural asymmetry in empirical marginals and ordinal tables

In random $F=0$9-SAT, marginal asymmetry is not summarized by a single skewness coefficient. The object of study is the empirical marginal distribution

$F<.G$0

and for each $F<.G$1 this empirical measure converges weakly in probability to a deterministic probability measure $F<.G$2 on $F<.G$3. The main theorem states that the pure point support of $F<.G$4 is exactly $F<.G$5 for any $F<.G$6; if $F<.G$7, then $F<.G$8 is purely discrete, while if $F<.G$9, then $b_1$0 has a non-trivial continuous part with $b_1$1. The asymmetry here is explicitly contrasted with symmetric models such as random graph coloring or random NAE-SAT, where marginals are forced to be uniform by symmetry. The mechanism is a multi-type Galton–Watson local limit: finite tree neighborhoods generate exact rational atoms, while infinite neighborhoods generate the continuous component (Müller et al., 2024).

The same theme appears in ordinal multi-way contingency tables, where asymmetry is formulated relative to a complete symmetry model $b_1$2 under which cells related by permutations of coordinates have equal probability. The proposed GLS$b_1$3 model minimizes $b_1$4-divergence from complete symmetry subject to fixed symmetry-class totals, first and second marginal moments, and pairwise cross-moments. The paper further proves that

$b_1$5

Here ME$b_1$6, VE$b_1$7, and CE$b_1$8 denote equality of marginal means, variances, and pairwise correlations, respectively. This decomposition is presented as a way to identify whether the failure of complete symmetry is attributable to unequal marginal means, unequal marginal variances, unequal marginal correlations, or lack of symmetric cell structure. A plausible implication is that marginal distribution asymmetry in categorical settings is often intrinsically multicomponent rather than reducible to a single one-dimensional index (Okahara et al., 2024).

4. Tail asymmetry, copula formulations, and the distinction from dependence asymmetry

For bivariate distributions, one important notion of marginal asymmetry is asymmetry between lower-left and upper-right tail probabilities. The proposed local copula-based measure is

$b_1$9

In copula form,

$CS$0

The interpretation is direct: $CS$1 means the upper-right tail is more likely, $CS$2 means the lower-left tail is more likely, and $CS$3 means equality of the two tail masses at level $CS$4. The measure is local, sign-sensitive, invariant under permutation of coordinates, and reverses sign under copula reflection. The paper also gives boundary formulas in terms of tail dependence coefficients and tail orders, sample analogues based on copula samples or pseudo-observations, Gaussian-process weak convergence for the copula-sample estimator, and bootstrap procedures when margins are unknown. In the S&P500–Nikkei225 application, the estimated asymmetry is mostly negative for small $CS$5, indicating that joint lower-tail events are more likely than joint upper-tail events (Kato et al., 2020).

This tail-asymmetry framework should be distinguished from directional dependence asymmetry. The copula-based measure qad defines asymmetry through $CS$6, with

$CS$7

and interprets the two directions as unequal predictability gains. The authors emphasize that this is a copula property, not a marginal property: qad is designed to isolate dependence structure after removing marginal effects, whereas asymmetry in the marginals themselves is not its target. This distinction is methodologically important because it separates asymmetry of one-dimensional or tail distributions from asymmetry of conditional informativeness in a bivariate association (Junker et al., 2019).

5. Physical and observational manifestations

In power systems, asymmetry of the frequency probability distribution means that observed frequency deviations are not mirror-symmetric around nominal frequency. The paper quantifies this through left and right standard deviations,

$CS$8

and the asymmetry statistic

$CS$9

It also reports the usual skewness

$X_n=x_1,\ldots,x_n$0

The proposed explanation is that symmetric disturbance distributions can be mapped into skewed frequency distributions by nonlinear system physics and nonlinear control, notably network losses, pitch-angle-based frequency control of wind turbines, hard limits, and tight deadbands. Real data from Ireland and Australia, together with IEEE 9-bus simulations, are used to argue that automatic generation control reduces asymmetry, whereas tight $X_n=x_1,\ldots,x_n$1 mHz deadband operation can increase it substantially. For Ireland, the reported $X_n=x_1,\ldots,x_n$2 changes from $X_n=x_1,\ldots,x_n$3 Hz with a $X_n=x_1,\ldots,x_n$4 mHz deadband to $X_n=x_1,\ldots,x_n$5 Hz with a $X_n=x_1,\ldots,x_n$6 mHz deadband; for Australia, the 2023 system has $X_n=x_1,\ldots,x_n$7 Hz and $X_n=x_1,\ldots,x_n$8, even though frequency containment is much better than in 2019. The broader claim is that asymmetry of the frequency distribution is a measurable operational signature of nonlinearity, saturation, and renewable-control design (Kerci et al., 2024).

In capacitive discharges excited by tailored waveforms, asymmetry is not summarized by a single scalar but is quantified physically through DC self-bias, unequal sheath widths, unequal sheath voltages, asymmetric ionization localization, and asymmetric ion energy distribution functions. Under identical base frequency, current density, and gas pressure, the study compares sinusoidal, sawtooth-down, and square waveforms. The square waveform gives the highest central plasma density, about $X_n=x_1,\ldots,x_n$9, whereas the sawtooth waveform gives the strongest asymmetry, with a DC self-bias of about $p_i$0. The sawtooth case also yields the most asymmetric ion energies, with mean ion energy about $p_i$1 at the powered electrode and $p_i$2 at the grounded electrode. The paper’s central point is that maximizing asymmetry and maximizing density are distinct objectives because one-sided ionization localization and unequal sheath dynamics are not the same mechanism as globally efficient electron heating (Sharma et al., 2021).

In observational cosmology, marginal asymmetry is used for the distribution of galaxy spin directions. The reported claim is that multiple surveys, including SDSS, Pan-STARRS, DES, HST, HSC, DECam, DESI Legacy Survey, and JWST deep fields, show more galaxies rotating opposite to the Milky Way than in the same direction, as seen from Earth. One JWST example is the JADES field near the Southern Galactic Pole, with $p_i$3 OMW galaxies and $p_i$4 MW galaxies, corresponding to a binomial probability $p_i$5. The paper further argues that the asymmetry is aligned with the Galactic pole and can change sign between opposite hemispheres, consistent with a dipole-like structure. At the same time, it explicitly cautions that the effect need not be cosmological: the observed asymmetry might arise from internal galaxy structure, observer-relative photometric effects, or the physics of galaxy rotation. The associated methodological dispute is whether a complex alternative test can detect asymmetry at all; the comment paper argues that it cannot, even on synthetic or intentionally biased data (Shamir, 2024).

6. Parametric modeling of asymmetric marginals

One class of asymmetric marginal models arises in censored regression. The tobit-LS framework replaces the classical normal-error tobit model by a tobit model based on log-symmetric distributions, motivated by the observation that censored responses may be right-skewed or left-skewed on the observed scale, heavy-tailed, light-tailed, or bimodal. The construction starts from a symmetric density on the log scale and induces a positive asymmetric response distribution. The family includes log-normal, log-Student-$p_i$6, log-power-exponential, log-Laplace, log-Cauchy, Birnbaum–Saunders, generalized Birnbaum–Saunders, and $p_i$7-type log-symmetric models. Estimation is by maximum likelihood using the BFGS quasi-Newton method, model checking uses the generalized Cox–Snell residual, and inference uses likelihood ratio and gradient statistics. In the Haiti measles vaccine antibody data, with censoring threshold $p_i$8 IU and $p_i$9 censored observations out of $c$00 ($c$01), all log-symmetric models substantially improve on the classical tobit-normal model in AIC and BIC, and the tobit-LN model has the lowest AIC and BIC among the fitted models (Saulo et al., 2018).

A second class of models builds asymmetry directly from symmetric base families by constrained mixtures. The construction splits the domain at a point $c$02, assigns disjoint left and right pieces to separate components, preserves continuity at the split, enforces unit mass, and fixes the left-side weight to $c$03. When $c$04, the density becomes asymmetric. This framework produces generalized asymmetric Laplace and asymmetric normal distributions; for fixed $c$05, both are exponential families with conjugate priors and closed-form maximum likelihood estimates for the original parameters. The paper also states that the asymmetric and symmetric normal distributions were compared in a linear regression example, showing that the asymmetric version performs at least as well as the symmetric one, and in a stock-index hidden Markov model, where the asymmetric version provides higher likelihood and may learn distribution models over states and transition distributions with considerably less entropy. This suggests that marginal asymmetry can be introduced without abandoning tractability or family resemblance to standard symmetric models (Miranda et al., 2015).

7. Related but distinct meanings of “marginal distribution”

The phrase “marginal distribution” is not used uniformly across arXiv literature, and some prominent uses are not about skewness or asymmetry in the statistical-shape sense. In continuous-variable quantum mechanics, a marginal distribution may mean a one-dimensional quadrature marginal of a Wigner function,

$c$06

measured along a phase-space axis. The demarginalization framework constructs fictitious two-dimensional phase-space distributions from a single measured marginal via

$c$07

and uses the unphysicality of the resulting operator as a nonclassicality witness. The paper explicitly notes that it does not frame the test as asymmetry of the marginal distribution in the usual statistical sense; instead, a single marginal may be non-Gaussian, skewed, or otherwise structured, and the relevant conclusion is nonclassicality rather than skewness per se (Park et al., 2017).

In stochastic growth and last passage percolation, a marginal distribution can mean the distribution of a single random variable extracted from a process. The paper on GOE and $c$08 marginal laws treats the one-point marginal distribution of the $c$09 process and the centered-scaled last passage times in point-to-line and point-to-half-line geometries. Here “marginal” denotes the distribution at one fixed endpoint or observation point, not a measure of asymmetry in shape. The relevant distinction is between full-line and half-line endpoint geometries, which lead after scaling to the Tracy–Widom GOE law and the one-point marginal distribution $c$10, respectively. A plausible implication is that discussions of marginal distribution asymmetry must first specify whether “marginal” refers to a one-dimensional projection, a single-coordinate law, a tail comparison, or a skewness property of a univariate distribution (Bisi et al., 2017).