Student-t · tail diagnostics

Tail-Fatness Lab — why varentropy outlives kurtosis

Drag the tail index ν. The body of the distribution barely moves; the tails are where the action is. Watch kurtosis hit a wall and vanish while varentropy keeps measuring all the way down to the Cauchy and beyond.

tail index ν = 3.00

0.3 (fattest)1 · Cauchy2430 → Gaussian

Variance

spread · 1st-order

3.00

finite

Differential entropy

mean of −log f

1.42

finite

Kurtosis

4th moment · Gaussian = 3

—

Varentropy

variance of −log f

2.00

finite

The distribution

Standard Student-t (solid) vs standard normal (dashed). Switch the y-axis to log to see the tail.

Every diagnostic, across all tail indices

Log y-axis. The sweep line marks your current ν. Kurtosis dies at ν=4, variance at ν=2 — entropy and varentropy keep going.

Variance Entropy Kurtosis Varentropy

Family: standardized Student-t, ν = degrees of freedom = tail index (ν=1 is Cauchy, ν→∞ is Gaussian). Variance = ν/(ν−2) for ν>2. Kurtosis = 3 + 6/(ν−4) for ν>4. Entropy and varentropy use the digamma/trigamma closed forms — both stay finite where the moment-based measures do not. That wider operating range is the whole point.