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:

$$f_{\text{trunc}}(x) = \frac{f(x)}{1 - F(a)}, \qquad x > a.$$

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.