Radial Diagnostics for Elliptical Model Fit

Why Check the Radial Distribution?

An elliptical model describes more than the direction and shape of its contours. It also specifies how probability is distributed from the center outward. A fitted model can have plausible-looking ellipsoidal contours while assigning too little probability to large radii, or too much probability near the center. Comparing observed Mahalanobis radii with the model’s radial distribution gives a focused diagnostic for this aspect of fit.

The comparison is useful but not a complete assessment of an elliptical model. It checks the distribution of distances from the fitted center relative to the fitted scatter matrix. It does not, by itself, check whether directions are distributed as the model requires. The distinction between radial and directional behavior is important when interpreting any probability plot or goodness-of-fit statistic.

The Model’s Radial Reference Distribution

For an elliptical random vector with a stochastic radial representation \(X=\mu+RAU\), suppose \(A\) is nonsingular, \(U\) is a random unit vector, and \(R\geq 0\) is independent of \(U\). Write \(\Sigma=AA^\mathsf{T}\). In this representation, the squared Mahalanobis distance from the model center is \(R^2\). Thus the model supplies a reference distribution for squared distances, not just a reference shape for the scatter.

Proof. Since \(\Sigma^{-1}=A^{-\mathsf{T}}A^{-1}\),

The equality uses only nonsingularity of \(A\) and the unit length of \(U\). In particular, it does not require a moment condition such as a finite covariance. \(\square\)

For a multivariate normal model, \(Q\) has a chi-squared distribution with \(p\) degrees of freedom, where \(p\) is the vector dimension. For a multivariate \(t\) model with \(\nu\) degrees of freedom, location \(\mu\), and scatter \(\Sigma\), \(Q/p\) has an \(F\) distribution with \(p\) and \(\nu\) degrees of freedom. The scatter parameter in the \(t\) model is not generally its covariance matrix; using the correct scatter parameter is essential to the reference distribution.

Constructing and Reading Probability Plots

A radial Q–Q plot places ordered observed squared distances \(\hat{q}_{(i)}\) on one axis and theoretical quantiles \(H^{-1}(u_i)\) on the other, where \(H\) is the model cumulative distribution function of \(Q\). Either axis may be chosen for the observed values, but the choice should be stated and kept consistent. Under a good match, points should follow an approximately straight, increasing pattern. If the plot uses the same scale on both axes, that pattern should be near the identity line.

A radial probability plot instead compares the empirical cumulative distribution of observed squared distances with \(H\). These two displays encode the same basic comparison in different coordinates. In either display, curvature can indicate a systematic mismatch: for example, the largest observations may rise more sharply than the theoretical quantiles, suggesting a heavier observed radial tail. A shift across much of the plot can reflect a scale mismatch, though fitting the scatter may partly absorb such a discrepancy.

For continuous \(H\), a useful theoretical result explains why the reference quantiles are appropriate when model parameters are known. If \(Q_1,\ldots,Q_n\) are independent with continuous cumulative distribution function \(H\), then \(H(Q_i)\) are independent uniform random variables on \((0,1)\). Their ordered values tend to line up with the uniform quantiles used in a probability plot.

Proof. For \(0<u<1\), strict increase of \(H\) gives

Thus \(H(Q)\) is uniform, and independence is preserved when the same function is applied separately to independent observations. The \(i\)th order statistic \(U_{(i)}\) from \(n\) independent uniforms has density

Integrating \(u\) against this density, or evaluating the corresponding beta integral, gives \(E[U_{(i)}]=i/(n+1)\). Since \(U_{(i)}=H(Q_{(i)})\), the stated result follows. \(\square\)

The expected order-statistic positions \(i/(n+1)\) are a principled choice for a probability plot. Another common choice is \((i-0.5)/n\), which is a practical approximation, especially for small samples. Neither choice makes the plotted points independent: order statistics are dependent, so visual deviations should not be treated as independent tests at each point.

What Changes When Location and Scatter Are Estimated?

The reference distribution above applies to distances from the true model center and scatter. In practice, the same observations are often used to estimate \(\mu\) and \(\Sigma\), then to calculate their fitted distances. The resulting \(\hat{q}_i\) are dependent and are not generally independent draws from the model’s \(Q\) distribution. Estimation pulls the fitted distances toward the estimated center and adapts the scatter to the sample, changing the distribution of the diagnostic.

A simple multivariate normal calculation makes this effect concrete. If the sample covariance is computed with divisor \(n\), and \(\bar{X}\) is the sample mean, define \(S=n^{-1}\sum_{i=1}^n(X_i-\bar{X})(X_i-\bar{X})^\mathsf{T}\). When \(S\) is invertible, the fitted squared distances obey

This exact constraint is incompatible with treating the fitted distances as independent chi-squared observations, whose sum is not fixed at \(np\). If the covariance uses divisor \(n-1\), the corresponding sum of fitted distances is \((n-1)p\). Other estimators and elliptical families impose different dependence and calibration effects.

Goodness-of-Fit Checks and Calibration

A formal check can summarize the discrepancy between the empirical distribution of squared distances and the model radial distribution. For example, the Kolmogorov–Smirnov statistic is the largest absolute difference between the empirical cumulative distribution function and the proposed cumulative distribution function. When parameters are known and the distances are independent draws from a continuous \(H\), the usual one-sample reference distribution applies. When location, scatter, or radial parameters are estimated from the same vectors, that known-parameter calibration generally does not apply.

A practical approach is a parametric bootstrap that repeats the whole fitting and checking procedure. It is an approximation to the null distribution, not a universal exact correction. The simulated data must be generated from the fitted model, and every step that affected the observed statistic must be repeated in each replicate.

Common Pitfall and Scope of the Diagnostic

Wrong reasoning: “The fitted Mahalanobis distances are supposed to follow the model’s radial distribution, so I can apply a known-parameter goodness-of-fit test directly to them.” This ignores that the distances use parameters estimated from those same observations. In many common fits the distances are dependent, and their distribution is altered by estimation.

Correction: distinguish a theoretical radial reference from the sampling distribution of fitted distances. Use the reference distribution to construct an informative plot, but calibrate a formal statistic with a method appropriate to the estimation procedure, such as a parametric bootstrap that refits every replicate. Also check whether the radial model’s assumptions are plausible: a radial plot alone cannot establish that directions are distributed correctly or that the full elliptical model fits.