Truncated and Censored Distributions

A fund database only lists funds still operating today. A risk desk records "loss exceeded the $5 million limit" without logging the exact number, because the position was closed the instant the limit fired. Both are missing data, but they're missing for structurally different reasons, and fitting the wrong model to each gives you a distribution that's confidently wrong in a specific, predictable direction.

The analogy: a security camera with two failure modes

Imagine a security camera recording how tall people are as they walk through a doorway. Truncation is a camera that's physically incapable of seeing anyone under five feet tall, those people don't exist in the footage at all, not even as a blur. Censoring is a camera that records everyone, but for anyone over six feet it can only report "off the top of frame" rather than the exact height, the person is still in the data, just with an incomplete measurement. Truncation deletes observations outright; censoring keeps the observation but blurs its value.

Writing it down

Let $X$ be the true random variable with density $f(x)$ and CDF $F(x)$.

Truncation (say, left-truncated at $a$): only values with $X > a$ ever enter the sample, and the density of what you actually observe is rescaled to still integrate to 1:

In words: throw away the probability mass at or below $a$, then blow the remaining shape back up so the area under it is still exactly 1, you're not just cutting off a tail, you're renormalizing what's left.

Censoring (say, right-censored at $b$): every unit is observed, but any true value above $b$ is recorded as exactly $b$ with a flag saying "at least this much." The observed variable is $Y = \min(X, b)$, and the count of censored observations is itself informative, it tells you how much mass sits above $b$, which truncation throws away entirely.

Picture the bell curve above with everything left of some cutoff simply deleted and the remaining area rescaled to 1, that's truncation. Now picture the same curve with everything right of a cutoff collapsed into a single spike exactly at the cutoff, but still counted, that's censoring. Same starting shape, two structurally different pictures.

Worked example 1: fund returns, truncated

A database of hedge fund monthly returns only includes funds still reporting, which effectively truncates out anything below a return of $-50\%$ (funds that blow up that badly close and stop reporting). Suppose true returns are $\mathcal{N}(1\%, 5\%^2)$ and you want the observed mean. The truncation point $a = -50\%$ is $(-0.50 - 0.01)/0.05 = -10.2$ standard deviations below the mean, so far in the tail that $F(a) \approx 0$, and the observed mean barely differs from 1 percent. Now redo it for a stressed fund with true mean $-10\%$ and standard deviation $15\%$: the cutoff is $(-0.50+0.10)/0.15 = -2.67$ standard deviations away, so about $F(-2.67) \approx 0.38\%$ of true outcomes get truncated away, a small fraction, but it's disproportionately the worst outcomes, so the observed average return among surviving funds is measurably higher than the true $-10\%$. This is survivorship bias, mechanically: truncation deletes the worst tail and any average computed afterward is biased upward.

Worked example 2: risk limit breaches, censored

A trading system halts and locks in a loss the moment it hits a $5 million limit, but it does record that a halt occurred. Out of 200 trading days, suppose 8 days hit the limit (recorded as exactly $5 million, flagged as censored) and the other 192 days have their true, smaller losses recorded exactly. A naive analyst who simply averages the 200 recorded numbers, treating the $5 million entries as real observed losses, systematically underestimates how bad the tail truly is, the true losses on those 8 days could have been $5.2 million or $40 million, and censoring hides which. The correct approach (survival-analysis-style likelihood) uses the exact values for the 192 uncensored days and, for the 8 censored days, only the fact that the loss was at least $5 million, contributing the survival probability $1 - F(x)$, evaluated at that $5 million cutoff, to the likelihood instead of a fabricated exact number. That distinction is exactly what a naive "just average the numbers" analysis throws away.

What this means in practice

• Survivorship bias in fund and strategy databases is a truncation problem, the failures are not just underweighted, they are absent, and no reweighting of the visible data alone can fully undo it without external knowledge of what was truncated.
• Risk limits and stop-losses generate censored data, not truncated data, since a flagged event ("limit hit") is still logged, losing that information in a naive average is a modeling choice, not a data limitation.
• Maximum likelihood estimation handles both correctly by writing separate likelihood contributions for observed-exact, truncated, and censored records, mixing them into one plain average is the shortcut that goes wrong.