Diagnosing Asymmetry and Heavy Tails in Worked Examples

Two Distinct Features of Nonnormal Data

A bivariate cloud can depart from a Gaussian model in more than one way. One departure is asymmetry: the distribution has a longer or more prominent side in some direction. Another is heavy tails: observations far from the centre occur more often than a Gaussian model predicts. These features can occur separately or together. A useful diagnosis therefore asks two questions: do directional summaries indicate asymmetry, and do distances from the fitted centre have unusually heavy tails?

The relevant fitted models represent different possibilities. A Gaussian model is symmetric with Gaussian radial tails. A multivariate \(t\) model is also centrally symmetric, but has heavier tails. A skew-normal model allows asymmetry without the polynomial tails of a \(t\) distribution. A skew-\(t\) model allows both. No single statistic settles the comparison: directional and radial evidence should be read together with fitted-model diagnostics.

Directional Skewness and What It Can Show

For scalar observations \(y_1,\ldots,y_n\), write \(\bar y=n^{-1}\sum_i y_i\) and \(m_r=n^{-1}\sum_i(y_i-\bar y)^r\). The moment-based sample skewness is \(g_1=m_3/m_2^{3/2}\), provided \(m_2>0\). Positive values indicate a longer or more influential upper side in that projection; negative values indicate the opposite. In a bivariate analysis, one can inspect the two coordinates and scientifically meaningful linear combinations, rather than treating a single coordinate system as definitive.

A basic population fact clarifies the interpretation. A distribution is centrally symmetric about \(\mu\) if \(X-\mu\) and \(-(X-\mu)\) have the same distribution. When the relevant third moments exist, this symmetry forces every projected third central moment to vanish.

Proof. Let \(Y=a^\mathsf{T}(X-\mu)\). Central symmetry implies that \(Y\) and \(-Y\) have the same distribution. The stated integrability makes their third moments finite, so

Thus \(E[Y^3]=0\). The conclusion is a population statement, not a claim that every finite sample from a symmetric distribution has sample skewness exactly zero. \(\square\)

Radial Distances and Model-Specific References

A fitted location \(\hat\mu\) and positive-definite scale matrix \(\hat\Sigma\) turn each observation into a squared Mahalanobis distance \(q_i\). The matrix standardizes distances for both marginal scale and correlation. A large \(q_i\) means an observation is far from the fitted centre relative to the fitted scale, but whether it is unexpectedly large depends on the model.

For a bivariate Gaussian with location \(\mu\) and covariance matrix \(\Sigma\), \(q=(X-\mu)^\mathsf{T}\Sigma^{-1}(X-\mu)\) has a chi-square distribution with two degrees of freedom. For a bivariate \(t\), use its scale matrix \(\Sigma\) in this definition. Under the usual construction with degrees of freedom \(\nu\), \(q\) has distribution \(2F_{2,\nu}\). Here the \(t\) scale matrix is not its covariance matrix: the covariance is \(\nu\Sigma/(\nu-2)\) when \(\nu>2\).

Derivation. Let \(U=Z^\mathsf{T}\Sigma^{-1}Z\). Whitening the normal vector gives \(U\sim\chi^2_2\), and \(U\) is independent of \(W\). Substitution into the distance gives \(q=U/(W/\nu)\). By the definition of the \(F\) distribution, \((U/2)/(W/\nu)\sim F_{2,\nu}\); hence \(q=2F_{2,\nu}\). \(\square\)

Fitted Models: Separate the Evidence Before Comparing

A practical analysis uses directional and radial diagnostics as complementary evidence. A skewness signal with ordinary radial behaviour is consistent with asymmetry without a strong heavy-tail signal. Large radial distances in several directions, together with little directional skewness, suggest heavy tails without clear asymmetry. Evidence in both suggests considering a skew-\(t\) model. These are diagnostic patterns, not proofs of the data-generating mechanism.

For a fitted Gaussian or \(t\) model, the corresponding radial reference distribution gives a direct benchmark. A skew-normal or skew-\(t\) model generally does not have that same simple chi-square or scaled-\(F\) radial law when distances are measured from its location parameter and scale matrix. In that case, compare the observed radial distribution with the fitted model’s own radial distribution, for example by simulating datasets from the fitted model and recomputing the distances. To account for parameter estimation, a useful simulation procedure refits the model to each simulated dataset before calculating its diagnostic.

Model fit can also be compared with AIC, defined as \(\mathrm{AIC}=-2\ell(\hat\theta)+2k\), where \(\ell(\hat\theta)\) is the maximized log-likelihood and \(k\) is the number of fitted parameters. AIC assesses a fit-complexity trade-off; it does not identify which feature caused one model to fit better. The directional and radial diagnostics help explain that distinction.

Common Pitfalls and Practical Interpretation

• Calling one extreme point “skewness.” A large distance indicates extremeness relative to a chosen centre and scale; it does not establish a persistent imbalance across directions. Inspect directional projections as well.
• Using the Gaussian radial reference for every model. A \(t\) model has a heavier radial reference, and skew models may require model-specific simulation. A Gaussian-tail flag can simply reflect a wrong reference distribution.
• Reading sample skewness as a definitive population result. Third moments are sensitive to individual observations, especially in small samples. Assess several relevant directions and interpret the result with the sample size in mind.
• Treating AIC as a diagnosis of the departure. AIC ranks the fitted models in a relative complexity-adjusted comparison. It does not replace the directional and radial checks that help distinguish asymmetry from heavy tails.

A disciplined diagnosis keeps these signals separate until the evidence is combined: first examine directional imbalance, then compare radial behaviour with the fitted model’s reference, and finally assess whether models that allow asymmetry, heavy tails, or both provide a worthwhile improvement. When parameters are estimated from the same observations, simulation with refitting can make diagnostic comparisons more realistic.