Body, Shoulders, and Tails

Statement¶

For a centered normal density parameterized by variance $v>0$

the body-shoulder-tail diagnostic asks where a small, mean-preserving randomization of $v$ adds density and where it removes density. A local version is the sign of

For the standard normal density, this second derivative changes sign at

These four points split the line into left tail, left shoulder, peak, right shoulder, and right tail regions. Positive variance curvature means a local variance mixture raises density there; negative variance curvature means it lowers density there.

The notation $g_v$ denotes the density at fixed variance $v$; a mixture averages these densities over a positive random variance $V$. The Normix normal-mixtures note supplies the complementary conditional construction. A shared notation page is planned.

Examples¶

• For the standard normal density under variance perturbation, the inner shoulder boundaries are about $\pm0.742$, and the outer tail boundaries are about $\pm2.334$.

• If one perturbs the normal scale $\sigma$ rather than the variance $v$, the corresponding boundaries are different: about $\pm0.662$ and $\pm2.136$ for the standard normal.

• For Student-t densities, a scale-curvature version applies, but the boundary values depend on the degrees of freedom.

• These boundaries are not Pareto thresholds; threshold selection for regular variation is a separate problem.

Diagnostic intuition¶

If a Gaussian variance is randomized while keeping the center fixed, the resulting mixture does not simply “spread out” everywhere. It tends to add mass near the peak and in the far tails, while taking mass from the shoulders. The shoulders are the moderate-deviation region that looks ordinary under a single scale but is depleted when variance uncertainty is introduced.

This is a diagnostic for finite-sample geometry, not a tail-index estimator. It explains why stochastic volatility can create both a sharper center and fatter-looking tails without producing a Pareto tail. Choosing a threshold for tail estimation remains the separate judgment handled by Tail Threshold Selection.

Derivation of the variance-curvature boundaries¶

We derive the standard-normal variance-curvature boundaries by differentiating the density with respect to variance and solving the resulting quadratic in $x^2$. The Taylor expansion explains the local mixture interpretation. The comparison with scale perturbation below uses a different derivative; none of these boundaries is a rule for selecting a Pareto threshold.

For the normal density with variance parameter $v$, write

Set $y=x/\sqrt v$. A direct differentiation gives

and a second differentiation gives

At $v=1$, the sign changes where

Solving the quadratic in $x^2$ gives

This is a local Taylor diagnostic. A sufficient small-perturbation model is

with bounded $Z$ and $\tau$ small enough that $V_\tau>0$. For each fixed $x$, Taylor’s theorem, with bounded third derivative near $v_0$, gives

Thus the curvature sign determines the leading local density change away from its zeros. Small variance of a perturbation alone would not justify this remainder without control of the perturbation family. The expansion is pointwise in $x$, not uniform arbitrarily far into the tails.

Static region and mixture calculations¶

At $v=1$, put $a=\sqrt{3-\sqrt6}\approx0.742$ and $b=\sqrt{3+\sqrt6}\approx2.334$.

For a concrete finite mixture, define

Its mean is zero and its variance is one. The ratio to the unit normal is

At $x=0,1,3$, the ratio is approximately 1.491, 0.617, and 2.754. This exhibits increased center and far-tail density with depleted shoulders. The exact crossings solve the ratio-equals-one equation; they need not be the local curvature boundaries $a,b$. This finite Gaussian mixture has a finite moment generating function at every real argument and is not regularly varying, despite having more far-tail mass than the unit normal.

Density, survival, and scale are different diagnostics¶

For $x>0$, the two-sided normal survival at variance $v$ is $T_v(x)=2\bar\Phi(x/\sqrt v)$, where $\bar\Phi$ is standard normal survival and $\phi$ is its density. Differentiation gives

Its positive-threshold curvature boundary is $x=\sqrt{3v}$, about 1.732 when $v=1$. It differs from both density boundaries.

For scale perturbation, write $h_\sigma(x)=g_{\sigma^2}(x)$ and $z=x/\sigma$. Then

At $\sigma=1$, its positive zeros are $\sqrt{(5-\sqrt{17})/2}\approx0.662$ and $\sqrt{(5+\sqrt{17})/2}\approx2.136$. Mean-preserving scale mixing is a different operation from mean-preserving variance mixing. Student-t scale-curvature comparisons depend on degrees of freedom; a separate worked comparison is planned.

Caveats¶

• The diagnostic uses centered normal components and a small mean-preserving perturbation of their positive variance. Skewed or multimodal distributions need a different interpretation.

• The region labels are not universal definitions of “body” or “tail”. They are tied here to local variance perturbations. Perturbing scale instead of variance changes the numerical boundaries.

• A shoulder/tail boundary is not a threshold for Pareto estimation. It describes density geometry, not regular variation.

• For a large finite mixture, exact density crossings must be calculated from that mixture; the curvature formulas describe a local limit.

References¶

• Taleb, Statistical Consequences of Fat Tails 2020.

Backlinks¶

• Depends on: Normix normal mixtures and the canonical density notation in Notation (planned).

• Used by: Dispersion Ratio Under Fat Tails (planned).

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/methods/body-shoulder-tail.md, revision 9717c9c (2026-09-13 Batch 2 import). Copyright (c) 2023 xshi19. Licensed under MIT. Links, notation, and qualifications were adapted for this site; executable figures and simulations were replaced with static calculations. No upstream execution or formal-proof verification is claimed for this adaptation.