Misspecification, Quantile Mobility, and Arc Length

J. Landers

Overview. This note unifies three threads: (i) distributional misspecification bounds for a lognormal truth fit by a moment-matched normal; (ii) quantile mobility difficulty, meaning how hard it is to shift a low-percentile point to a higher percentile under additive (normal) vs multiplicative (lognormal) geometries; (iii) a geometric relation between quantile shifts and the arc length of the probability density function (PDF), which clarifies why mobility differs across shapes.

Intuition. Imagine two maps of the same landscape. One uses straight-line distances; the other measures effort along steep terrain. The normal model is the straight-line map. The lognormal model, with its skew and stretch, is the terrain map. Misspecification tells us how different the maps are; quantile mobility asks how hard it is to move from one landmark to another; arc length explains why some routes feel longer even when the endpoints are fixed.

1. Setup and Notation

We place a lognormal distribution and a moment-matched normal side by side to compare how they allocate mass and how they move percentile locations. Matching moments aligns means and variances so that any remaining differences reflect shape alone.

Let $X\sim\mathrm{Lognormal}(\mu,\sigma^2)$ with density $$ f_{\mathrm{LN}}(x)=\frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x-\mu)^2}{2\sigma^2}\right),\qquad x>0. $$ Consider a moment-matched Gaussian model $Y\sim\mathcal N(m,v)$ with $$ m=\mathbb E[X]=e^{\mu+\sigma^2/2},\qquad v=\mathrm{Var}(X)=(e^{\sigma^2}-1)e^{2\mu+\sigma^2}. $$ Denote CDFs by $F_{\mathrm{LN}},F_{\mathrm N}$, PDFs by $f_{\mathrm{LN}},f_{\mathrm N}$, and quantiles by $Q_{\mathrm{LN}}=F_{\mathrm{LN}}^{-1},\ Q_{\mathrm N}=F_{\mathrm N}^{-1}.$

Why this matters. With moments aligned, any mismatch we observe is not a question of location or scale but of geometry. That geometry controls both how far quantiles sit in absolute units and how much work it takes to move between them.

2. Misspecification via Total Variation Bounds

Total variation measures the maximum mass one must reassign to morph one density into another. It is a conservative but interpretable gauge of discrepancy.

$$ d_{\mathrm{TV}}(f_{\mathrm{LN}},f_{\mathrm N})=\tfrac12\int_0^{\infty}\lvert f_{\mathrm{LN}}(x)-f_{\mathrm N}(x)\rvert\,dx. $$

Classical inequalities tie TV to the Kullback-Leibler divergence $D_{\mathrm{KL}}(f_{\mathrm{LN}}\Vert f_{\mathrm N})$.

(Pinsker upper bound) $$ d_{\mathrm{TV}}(f_{\mathrm{LN}},f_{\mathrm N})\le \sqrt{\tfrac12\,D_{\mathrm{KL}}(f_{\mathrm{LN}}\Vert f_{\mathrm N})}. $$ (Bretagnolle-Huber lower bound) $$ d_{\mathrm{TV}}(f_{\mathrm{LN}},f_{\mathrm N})\ge 1-\exp\big(-D_{\mathrm{KL}}(f_{\mathrm{LN}}\Vert f_{\mathrm N})\big). $$

For a representative choice (for example $\mu=4.2$ and $\sigma=0.5$), a direct evaluation gives $D_{\mathrm{KL}}(f_{\mathrm{LN}}\Vert f_{\mathrm N})\approx 0.52$, hence $0.41\lesssim d_{\mathrm{TV}}\lesssim 0.51$. This is a reminder that even well-matched moments can hide sizable shape differences.

Intuition. If we think of the normal as a straight steel template laid over a curved board, TV is the total sanding required so the board fits the template. When the board is lognormal, the right tail bulges and the sanding bill is not small.

3. Quantile Mobility: Additive vs Multiplicative Geometry

Quantile mobility asks how much effort it takes to move a point from percentile $p_0$ to $p_1$. The answer depends on the geometry of displacements: differences for the normal, ratios for the lognormal.

3.1 Displacements in Original Units

Under a normal model with variance $v$,

$$ \Delta_{\mathrm N}=Q_{\mathrm N}(p_1)-Q_{\mathrm N}(p_0)=\sqrt v\,\Delta z, $$

where $\Delta z=\Phi^{-1}(p_1)-\Phi^{-1}(p_0)$. Under a lognormal model with log-scale $\sigma$,

$$ \kappa=\frac{Q_{\mathrm{LN}}(p_1)}{Q_{\mathrm{LN}}(p_0)}=\exp(\sigma\,\Delta z). $$

Intuition. The same percentile step corresponds to adding a length in the normal world and multiplying by a factor in the lognormal world. Add a little repeatedly and you progress steadily; multiply a little repeatedly and growth compounds. This compounding is what makes high-percentile targets feel far away under lognormal geometry.

3.2 Two Operational Difficulty Models

Fixed-budget success probability. Suppose assistance is an additive boost $A\sim\mathcal N(0,s^2)$ (normal) or a multiplicative boost with $\ln B\sim\mathcal N(0,\tau^2)$ (lognormal). Reaching $p_1$ from $p_0$ requires $A\ge \Delta_{\mathrm N}$ or $B\ge\kappa$, so $$ \Pr_{\mathrm N}(\text{success})=1-\Phi(\Delta_{\mathrm N}/s),\qquad \Pr_{\mathrm{LN}}(\text{success})=1-\Phi\big((\ln\kappa)/\tau\big). $$ Reading. With comparable budgets $(s,\tau)$, success falls off faster in the multiplicative case because the required ratio $\kappa$ grows exponentially in the z-gap.
Transport-style quadratic work. In additive space, take cost $c_{\mathrm N}(x\to y)=(y-x)^2$; in log-space, use $c_{\mathrm{LN}}(x\to y)=(\ln y-\ln x)^2$. Moving between the same percentiles yields costs proportional to $(\sqrt v\,\Delta z)^2$ and $(\sigma\,\Delta z)^2$, respectively. Mapped back to original units, multiplicative geometry penalizes pushes to the far right because ratios outpace differences. Takeaway. Even when quadratic costs are the same in their natural coordinates, the lognormal regime converts them to steeper price tags in raw units.

4. Arc Length-Quantile Relation

The arc length of a PDF over a quantile span records how much the curve bends and rises along the way. It acts like a terrain factor that multiplies the horizontal quantile gap.

$$ L(p_0,p_1)=\int_{Q(p_0)}^{Q(p_1)} \sqrt{1+(f'(x))^2}\,dx = \int_{p_0}^{p_1} \frac{dQ}{dp}\,\sqrt{1+s(Q(p))^{2}}\,dp. $$

Two-sided bound. With $\Delta Q=Q(p_1)-Q(p_0)$ and $$m=\inf_{x\in[Q(p_0),Q(p_1)]}\sqrt{1+s(x)^2},\ M=\sup_{x\in[Q(p_0),Q(p_1)]}\sqrt{1+s(x)^2}, $$ $$ m\,\Delta Q \le L(p_0,p_1) \le M\,\Delta Q. $$

Intuition. If $\Delta Q$ is the distance on a flat map, then $\sqrt{1+s^2}$ is the local steepness. Right-skewed or heavy-tailed shapes exhibit larger steepness on tailward spans, stretching the effective path. This explains why the same percentile shift can feel longer in a lognormal world even when horizontal distances match.

5. Synthesis: Why Mobility Differs Across Shapes

Misspecification and shape. The TV gap between a lognormal and its moment-matched normal reflects systematic shape differences. The very features that increase TV also increase the arc-length multiplier on the right tail.
Quantile shift vs arc length. For the same percentile gap, $\Delta Q$ tends to be larger under right skew, and the arc-length factor inflates it further. Mobility toward higher percentiles is therefore geometrically longer and operationally costlier.
Additive vs multiplicative geometry. Normal mobility buys differences (linear in $\Delta z$); lognormal mobility buys ratios (exponential in $\Delta z$). The arc-length view unifies these pictures by making the geometric premium explicit.

Story in one line. Misspecification tells us the maps are different, quantile mobility tells us which routes are harder, and arc length tells us why the uphill is steeper in one map than the other.

6. Practical Reading Guide

To compare mobility across models for fixed $(p_0,p_1)$: compute $\Delta z$, obtain $\Delta_{\mathrm N}$ and $\kappa$, and evaluate the arc-length bounds on $[Q(p_0),Q(p_1)]$.

If $\kappa$ is large, multiplicative geometry alone creates difficulty.
If $M$ is large over the span, shape amplifies the horizontal gap, signaling a steep route.
If TV is large, no single normal template will capture the terrain, and mobility conclusions drawn from it may be optimistic.

Working principle. Evaluate distance in the coordinates native to the model and then translate costs back to the units that matter. In skewed settings, this translation is where the hidden premium lives.

Appendix: Moment Matching Numerical Illustration Details

With $\mu=4.2$ and $\sigma=0.5$, the moment-matched Gaussian is $\mathcal N(m,v)$ where $m=e^{\mu+\sigma^2/2}\approx 75$ and $v=(e^{\sigma^2}-1)e^{2\mu+\sigma^2}\approx 1125$. A direct computation yields $D_{\mathrm{KL}}(f_{\mathrm{LN}}\Vert f_{\mathrm N})\approx 0.52$, so $0.41\le d_{\mathrm{TV}}\le 0.51$.

Interpretation. Even after matching mean and variance, the curved terrain of the lognormal remains, and it shows up both in the global discrepancy and in the local price of moving quantiles.