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 $Y$. The Normix normal-mixtures note supplies the complementary conditional construction. See the shared notation canon.

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 diagnoses the geometry of a specified population density, not its tail index. 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 local perturbation $H$ and $\tau$ small enough that $Y_\tau>0$. 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 scale-curvature comparisons depend on degrees of freedom, as derived below.

Student scale curvature¶

The scale calculation also has a closed form for a Student density. Write $h_s(x)=C_\nu s^{-1}(1+x^2/(\nu s^2))^{-(\nu+1)/2}$, where $s>0$ is scale, $\nu>1$ is degrees of freedom, and $C_\nu$ normalizes the density. Set $z=x/s$ and $t=z^2/\nu$. Logarithmic differentiation gives

For the second formula, differentiate the first, using $\partial_st=-2t/s$, and add the square of the log derivative. The numerator is a quadratic with two positive roots when $\nu>1$. Solving in $z^2$ gives the four density-curvature boundaries

The leading coefficient and the value at zero are positive, so curvature is positive inside the inner pair and beyond the outer pair, and negative between them. At $\nu=3,s=1$, the positive boundaries are $\sqrt{4-\sqrt{13}}\approx0.628$ and $\sqrt{4+\sqrt{13}}\approx2.758$. The standard deviation is $\sqrt3$; these numbers are in scale units. For $\nu\le2$, the variance is not finite, although the four-boundary formula still applies when $1<\nu\le2$. At $\nu=1$ the quadratic loses its leading term, so that formula’s domain must not be extended through the zero denominator.

Dividing numerator and denominator of the root formula by $\nu$ shows that its two positive roots tend to the Gaussian scale-curvature roots as $\nu\to\infty$. This is an algebraic limit, not a fitted threshold rule. The gamma-precision construction explains the Student tail exponent and moment conditions separately.

Curvature and exact density differences¶

Use normal_peak_shoulder_tail, normal_variance_peak_shoulder_tail, t_peak_shoulder_tail.

from xmath.incerto.stats import (
    normal_peak_shoulder_tail, normal_variance_peak_shoulder_tail,
    t_peak_shoulder_tail,
)

for label, values in (("Normal, scale", normal_peak_shoulder_tail()),
                      ("Normal, variance", normal_variance_peak_shoulder_tail()),
                      ("Student 3, scale", t_peak_shoulder_tail(3))):
    print(f"{label}: [" + ", ".join(f"{v:.4g}" for v in values) + "]")

Output

Normal, scale: [-2.136, -0.6622, 0.6622, 2.136]
Normal, variance: [-2.334, -0.742, 0.742, 2.334]
Student 3, scale: [-2.758, -0.6281, 0.6281, 2.758]

These are local perturbation boundaries at unit scale, not fitted tail thresholds.

Figure regeneration: uv run python demos/incerto/finite_moments.py. These are exact density calculations on 801 fixed points, with no random sampling. The script reports the finite-mixture crossings separately from the local boundaries.

Let $Y$ be a positive mixing variance with $\mathbb EY<\infty$. Jensen’s inequality for survival needs convexity throughout the mixing interval. If $0<Y\le v_{\max}<\infty$ and $x^2\ge3v_{\max}$, the displayed $T_v''(x)$ is nonnegative on that interval, and $\mathbb E T_Y(x)\ge T_{\mathbb EY}(x)$. If instead $Y\ge v_{\min}>0$ and $x^2\le3v_{\min}$, its curvature is nonpositive and the inequality reverses. These sufficient conditions show why a local tail-curvature statement cannot justify global Jensen claims at every threshold under unrestricted mixing.

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 (2025), Statistical Consequences of Fat Tails, third edition, 2025, ch. 4, §§4.3–4.3.1, printed pp. 75–78. The source uses scale curvature; our variance and survival comparisons are derived separately. The calculations above prove the stated normal and Student cases, not a universal crossover theorem for all unimodal densities.

Backlinks¶

• Depends on: Normix normal mixtures and the canonical density notation in Shared notation.

• Used by: Dispersion Ratio Under Fat Tails.