Radial Kurtosis and Departure From Gaussianity

Why Measure Kurtosis Radially?

For a multivariate observation, unusualness is not determined by any one coordinate alone. A point can be far from the center in a direction where the distribution is naturally variable, yet not be especially unusual overall. The squared Mahalanobis distance accounts for the covariance geometry by measuring distance after standardizing and decorrelating the coordinates. Mardia’s multivariate kurtosis summarizes the second moment of that squared distance, and so focuses on the distribution’s radial behavior.

This summary is useful for comparing an elliptical distribution with the Gaussian benchmark. It describes one aspect of departure from Gaussianity, not a complete test of whether a distribution is Gaussian: different radial distributions can share the same kurtosis.

Definition: Let $X$ be a $p$-dimensional random vector with mean $\mu=E[X]$ and positive-definite covariance matrix $\Sigma=\operatorname{Cov}(X)$. Its squared Mahalanobis radius is $$ Q=(X-\mu)^\mathsf{T}\Sigma^{-1}(X-\mu). $$ If $E[Q^2]$ is finite, Mardia’s population multivariate kurtosis is $$ \beta_{2,p}=E[Q^2]. $$ The Gaussian benchmark is $p(p+2)$; the excess over that benchmark is $\beta_{2,p}-p(p+2)$.

The covariance matrix in this definition is essential. An elliptical model may be described using a scatter matrix that is proportional to its covariance but is not itself the covariance. The radius in Mardia’s definition uses $\Sigma$, so its value does not depend on which proportional scatter matrix was used to describe the model.

Deriving the Radial-Moment Formula

Suppose $X$ has an elliptical radial representation $X=\mu+A R U$, where $A$ is a nonsingular $p$-by-$p$ matrix, $R\geq 0$, and $U$ is independent of $R$ and uniformly distributed on the unit sphere in $\mathbb{R}^p$. Assume $0<E[R^2]<\infty$ and $E[R^4]<\infty$. Write $S=AA^\mathsf{T}$. Rotational symmetry gives $E[U]=0$ and $E[UU^\mathsf{T}]=I_p/p$, and hence $\operatorname{Cov}(X)=E[R^2]S/p$.

Theorem: Under this radial representation and the stated moment conditions, the covariance-based squared Mahalanobis radius and Mardia’s kurtosis satisfy $$ Q=\frac{pR^2}{E[R^2]}, \qquad \beta_{2,p}=p^2\frac{E[R^4]}{(E[R^2])^2}. $$

Proof. Since $\mu=E[X]$, the covariance is $\Sigma=(E[R^2]/p)S$. Its inverse is $\Sigma^{-1}=(p/E[R^2])S^{-1}$. Substitution into the definition of $Q$, using $X-\mu=ARU$ and $S=AA^\mathsf{T}$, gives

\begin{aligned} Q &=R^2 U^\mathsf{T}A^\mathsf{T} \left(\frac{p}{E[R^2]}S^{-1}\right)AU\\ &=\frac{pR^2}{E[R^2]}U^\mathsf{T}U =\frac{pR^2}{E[R^2]}. \end{aligned}

Here $A^\mathsf{T}S^{-1}A=I_p$ and $U^\mathsf{T}U=1$. Squaring the expression for $Q$ and taking expectations gives $\beta_{2,p}=E[Q^2]=p^2 E[R^4]/(E[R^2])^2$. The assumed fourth-moment condition makes this expectation finite. $\square$

This formula makes clear that Mardia’s kurtosis depends on a standardized radial fourth moment. Jensen’s inequality applied to $R^2$ gives $E[R^4]\geq (E[R^2])^2$, and therefore $\beta_{2,p}\geq p^2$. This lower bound is not the Gaussian benchmark: a distribution can have kurtosis between $p^2$ and $p(p+2)$.

The Gaussian Benchmark

If $X\sim N_p(\mu,\Sigma)$ with positive-definite $\Sigma$, then standardizing gives $Z=\Sigma^{-1/2}(X-\mu)\sim N_p(0,I_p)$, and $Q=Z^\mathsf{T}Z$. Thus $Q$ is the sum of squares of $p$ independent standard normal variables. Equivalently, $Q\sim\chi^2_p$. Its second moment is $p(p+2)$.

Worked Example: The Gaussian Benchmark in Three Dimensions

Let $X\sim N_3(\mu,\Sigma)$, where $\Sigma$ is positive definite. Then $Q\sim\chi^2_3$. A chi-squared variable with $p$ degrees of freedom has mean $p$ and variance $2p$, so

E[Q^2]=\operatorname{Var}(Q)+(E[Q])^2 =2p+p^2=p(p+2).

For $p=3$, this is $2(3)+3^2=6+9=15$. Hence $\beta_{2,3}=15$, and the Gaussian excess is $15-15=0$. The result depends on the dimension, not on the particular mean or covariance matrix.

Multivariate Student t Kurtosis

Use the parameterization $X=\mu+\Omega^{1/2}Z/\sqrt{W/\nu}$, where $Z\sim N_p(0,I_p)$, $W\sim\chi^2_\nu$, and $Z$ and $W$ are independent. Here $\Omega$ is the t scale matrix, not its covariance matrix. For $\nu>2$, the covariance is $\Sigma=\nu\Omega/(\nu-2)$. For $\nu>4$, the fourth moment exists, so Mardia’s kurtosis is finite.

To calculate it, first use the scale matrix to form $D^2=(X-\mu)^\mathsf{T}\Omega^{-1}(X-\mu)$. The representation gives $D^2=\nu Z^\mathsf{T}Z/W$. Since $\Sigma^{-1}=((\nu-2)/\nu)\Omega^{-1}$, the covariance-based radius is $Q=((\nu-2)/\nu)D^2$. Independence then lets us take the expectations of the numerator and denominator factors separately.

Result: For a $p$-dimensional Student t distribution with $\nu>4$ degrees of freedom and positive-definite scale matrix $\Omega$, $$ \beta_{2,p}=p(p+2)\frac{\nu-2}{\nu-4}. $$ Its excess over the Gaussian benchmark is $2p(p+2)/(\nu-4)$, which is positive.

Derivation. For $Z^\mathsf{T}Z\sim\chi^2_p$, $E[(Z^\mathsf{T}Z)^2]=p(p+2)$. For $W\sim\chi^2_\nu$, the chi-squared density, or the corresponding gamma integral, gives $E[W^{-2}]=1/((\nu-2)(\nu-4))$ when $\nu>4$. Consequently,

\begin{aligned} \beta_{2,p} &=E[Q^2]\\ &=\left(\frac{\nu-2}{\nu}\right)^2 \nu^2 E[(Z^\mathsf{T}Z)^2]E[W^{-2}]\\ &=\frac{(\nu-2)^2p(p+2)}{(\nu-2)(\nu-4)}\\ &=p(p+2)\frac{\nu-2}{\nu-4}. \end{aligned}

The condition $\nu>4$ is needed for the inverse second moment of $W$, and thus for a finite kurtosis. As $\nu$ increases, the ratio $(\nu-2)/(\nu-4)$ approaches one, so the t kurtosis approaches the Gaussian benchmark.

Worked Example: A Bivariate t Distribution

Suppose $X$ has a bivariate Student t distribution with $\nu=8$ degrees of freedom and any positive-definite scale matrix $\Omega$. Since $8>4$, its fourth moment exists. With $p=2$,

\beta_{2,2}=2(2+2)\frac{8-2}{8-4} =8\cdot\frac{6}{4}=12.

The bivariate Gaussian benchmark is $2(2+2)=8$, so the excess is $12-8=4$. To get this result, the radius must use the covariance $\Sigma=8\Omega/6=4\Omega/3$, not $\Omega$. Using $\Omega$ in the definition of $Q$ would calculate a different, scale-based quantity.

Interpreting Values Above and Below Gaussian

For an elliptical distribution with finite fourth moments, a value above $p(p+2)$ indicates a larger standardized radial fourth moment than in the Gaussian case. A value below the benchmark indicates a smaller one. This is a statement about the distribution of radius, not about every direction or every marginal distribution separately. In particular, a value equal to the benchmark does not establish that the distribution is Gaussian.

Worked Example: A Non-Gaussian Distribution Below the Benchmark

Let $X$ be uniformly distributed on the unit disk in $\mathbb{R}^2$. Write $r=\lVert X\rVert$. The probability that $r\leq t$ is $t^2$ for $0\leq t\leq1$, so the density of $r$ is $2r$. Therefore

E[r^2]=\int_0^1 2r^3\,dr=\frac12, \qquad E[r^4]=\int_0^1 2r^5\,dr=\frac13.

By rotational symmetry, $\operatorname{Cov}(X)=\tfrac14 I_2$: each diagonal entry is $E[r^2]/2=1/4$, and the off-diagonal entries are zero. Thus $Q=4r^2$, and

\beta_{2,2}=E[(4r^2)^2]=16E[r^4]=\frac{16}{3}.

The bivariate Gaussian benchmark is $8$, and $16/3<8$. The uniform-disk distribution is not Gaussian, despite having finite moments; its lower kurtosis reflects its bounded radius rather than Gaussian radial behavior.

Worked Example: A Gaussian Scale Mixture Above the Benchmark

Let $Z\sim N_2(0,I_2)$, and independently let $S$ equal $1/2$ or $3/2$, each with probability $1/2$. Define $X=\sqrt{S}\,Z$. Its covariance is $E[S]I_2=I_2$, since $E[S]=(1/2+3/2)/2=1$. Its squared Mahalanobis radius is $Q=S Z^\mathsf{T}Z$. Independence and the Gaussian radius moment give

E[S^2]=\frac{(1/2)^2+(3/2)^2}{2}=\frac54, \qquad \beta_{2,2}=E[S^2]E[(Z^\mathsf{T}Z)^2] =\frac54\cdot8=10.

The Gaussian benchmark is $8$, so this mixture has excess $2$. The random scale increases radial variability even though the covariance remains $I_2$.

Common Pitfall and Practical Meaning

Common mistake: “The multivariate t scale matrix is $\Omega$, so I can use $\Omega^{-1}$ in Mardia’s radius.” That confuses a model’s scale parameter with its covariance. For a t distribution with $\nu>2$, the covariance is $\nu\Omega/(\nu-2)$; the Mahalanobis radius in Mardia’s definition must use the inverse of this covariance. The distinction changes the numerical radius, even though using the wrong matrix may still yield a finite expectation.

Mardia’s kurtosis is a compact radial summary: it allows comparison with a dimension-specific Gaussian reference and, for elliptical models, directly reflects the standardized fourth moment of the radial variable. It is not a standalone verdict on distributional form. A complete assessment may require other evidence about the distribution, but choosing such diagnostics is a separate task.

Key takeaway: Mardia’s population kurtosis is $E[Q^2]$, where $Q$ is the covariance-based squared Mahalanobis radius. For elliptical distributions it equals $p^2E[R^4]/(E[R^2])^2$. The Gaussian value is $p(p+2)$, while the multivariate t value exceeds it whenever the fourth moment exists.

Check Your Understanding

Answer each question using the covariance-based radius.

For a Gaussian vector in dimension $p=4$, what is Mardia’s kurtosis and its Gaussian excess?
Why must the covariance matrix, rather than an arbitrary scatter matrix, appear in the definition of $Q$?
For a multivariate t distribution, what condition on $\nu$ makes Mardia’s kurtosis finite?
What does a value below $p(p+2)$ say about radial fourth-moment behavior, and what does it not prove?
In the radial representation, how does the covariance-based radius relate to $R$ and $E[R^2]$?