Conditional Distribution Adjustment For Truncated Samples

Why Truncation Changes the Distribution

A loss database may include only claims above a reporting threshold, below a policy limit, or inside a specified range. The recorded claims are then not a random sample from the full severity distribution: they are a sample conditional on being in the observation region. A proportion calculated among the recorded claims describes that conditional distribution, not automatically the distribution of all losses.

The adjustment is based on the probability of selection under the underlying model. First calculate the probability of the event among observations that could be recorded. Then, when the model probability of entering the record is known, use it to translate the conditional probability into a population probability. This tutorial focuses on these distribution calculations, not on estimating the model parameters.

Definition: Let $X$ be a loss with cumulative distribution function $F_X$, and let $R$ be the event that a loss is observed. The distribution among observed losses is the conditional distribution $$ P(X\leq x\mid R)=\frac{P(X\leq x,\ R)}{P(R)},\qquad P(R)>0. $$ More generally, for any loss event $A$, $$ P(A\mid R)=\frac{P(A\cap R)}{P(R)}. $$

These formulas apply whether the observation region is determined by a fixed threshold or by fixed lower and upper bounds, provided the selection rule is correctly represented by $R$. The examples below use continuous severity distributions, so the probability of a loss exactly at a boundary is zero. For a distribution with point masses, strict and inclusive boundary conditions must instead be handled explicitly.

Conditional Distributions for Common Truncation Regions

For a continuous loss distribution, the conditional distribution has a convenient form for each common observation region. Write $F_X(x)=P(X\leq x)$ and $S_X(x)=1-F_X(x)$.

Observation region	Conditional probability for a threshold in the region	Probability of the observation region
Left truncation: $X>L$	$P(X\leq x\mid X>L)=[F_X(x)-F_X(L)]/S_X(L)$, for $x>L$	$S_X(L)$
Right truncation: $X<U$	$P(X\leq x\mid X<U)=F_X(x)/F_X(U)$, for $x<U$	$F_X(U)$
Double truncation: $L<X<U$	$P(X\leq x\mid L<X<U)=[F_X(x)-F_X(L)]/[F_X(U)-F_X(L)]$, for $L<x<U$	$F_X(U)-F_X(L)$

For example, under left truncation the probability of a loss between two thresholds $L<a<b$, conditional on being observed, is $P(a<X\leq b\mid X>L)=[F_X(b)-F_X(a)]/S_X(L)$. The numerator is the probability of the event in the original population; the denominator is the probability that a loss passes the observation threshold.

Specify the event and the observation region.
Write the event of interest $A$ and the selection event $R$ separately. This prevents a threshold in the question from being mistaken for the truncation threshold.

Calculate the intersection probability.
Find $P(A\cap R)$ from the underlying model. Only the part of the event inside the observation region can occur in the recorded sample.

Condition, then adjust if needed.
Divide by $P(R)$ to obtain the probability among recorded losses. To obtain a population probability for an event contained in $R$, multiply that conditional probability by $P(R)$.

Why the Adjustment Formula Holds

The key relationship is the definition of conditional probability. If the event $A$ lies entirely inside the observation region $R$, then $A\cap R=A$, so $P(A\mid R)=P(A)/P(R)$. Rearranging gives $P(A)=P(A\mid R)P(R)$. This is the adjustment from the selected-sample probability to the corresponding population probability.

A\subseteq R \quad\Longrightarrow\quad P(A\mid R)=\frac{P(A)}{P(R)} \quad\Longrightarrow\quad P(A)=P(A\mid R)P(R).

If $A$ extends outside $R$, this multiplication recovers only the probability of $A\cap R$. Any probability mass of $A$ outside the observation region must be calculated separately from the underlying model. Also, a conditional proportion by itself does not reveal $P(R)$. Without a model or other information about the probability of selection, the population probability is generally not identified by the selected records alone.

Worked Example: Adjusting a Left-Truncated Exponential Loss

A hypothetical portfolio has losses $X$, measured in thousands of dollars, with an exponential distribution of mean $100$. Losses are recorded only when $X>50$. Calculate (1) the probability that a recorded loss is at most $150$, and (2) the population probability that a loss is at most $150$.

Method and assumptions. For an exponential loss with mean $100$, the rate is $1/100$, and $S_X(x)=e^{-x/100}$ for $x\geq0$. The threshold is fixed, and the recorded sample is distributed as $X\mid X>50$.

1. Conditional probability among recorded losses. The required event among recorded losses is $50<X\leq150$. Its unconditional probability is $S_X(50)-S_X(150)$. The observation probability is $S_X(50)$. Therefore,

\begin{aligned} P(X\leq150\mid X>50) &=\frac{S_X(50)-S_X(150)}{S_X(50)}\\ &=\frac{e^{-0.5}-e^{-1.5}}{e^{-0.5}}\\ &=1-e^{-1}\\ &=0.6321. \end{aligned}

2. Population probability. The full-population probability includes losses at or below the truncation threshold:

\begin{aligned} P(X\leq150) &=1-S_X(150)\\ &=1-e^{-1.5}\\ &=0.7769. \end{aligned}

As a check, $F_X(50)=1-e^{-0.5}=0.3935$, and the population probability between 50 and 150 is $S_X(50)(0.6321)$, approximately $0.3834$. Their sum is $0.3935+0.3834=0.7769$, as required. The method is conditional probability followed by reconstruction of the population probability. The common mistake it avoids is reporting $0.6321$ as the probability for all losses; it is only the probability among losses exceeding 50.

Worked Example: Right Truncation Under a Uniform Loss Model

Suppose a hypothetical severity model is uniform on $(0,200)$, with loss amounts measured in thousands of dollars. A data system records only losses below $160$. Calculate the conditional probability that a recorded loss is between $40$ and $120$, then find the corresponding population probability.

Method and assumptions. For $X\sim\operatorname{Uniform}(0,200)$, the probability of any interval is its length divided by $200$. The observation region is $R=\{X<160\}$, with probability $160/200=0.8$.

The event $40<X\leq120$ has length $120-40=80$, and it is wholly inside the observation region. Thus its conditional probability among recorded losses is

P(40<X\leq120\mid X<160) =\frac{80/200}{160/200} =\frac{80}{160} =0.5000.

The population probability is the interval length divided by the full support length:

P(40<X\leq120)=\frac{80}{200}=0.4000.

Equivalently, multiply the conditional probability by the probability of observation: $0.5000(0.8000)=0.4000$. This check confirms that the population event probability is smaller than the conditional probability among recorded losses. The method is to condition on the upper truncation limit before comparing the result with the full-support probability. It avoids treating the recorded interval $(0,160)$ as if it were the model's entire support.

Worked Example: A Doubly Truncated Pareto-Type Loss

A hypothetical loss $X$, in thousands of dollars, has survival function $S_X(x)=100/(100+x)$ for $x\geq0$. Only losses satisfying $50<X\leq200$ are recorded. Find the probability among recorded losses that $100<X\leq150$, and the corresponding probability in the full population.

Method and assumptions. The specified survival function is decreasing and gives $F_X(x)=1-S_X(x)$. Both the event and the observation region are intervals under this continuous model.

First calculate the probability of the observation region:

\begin{aligned} P(50<X\leq200) &=S_X(50)-S_X(200)\\ &=\frac{100}{150}-\frac{100}{300}\\ &=\frac{2}{3}-\frac{1}{3}\\ &=\frac{1}{3}. \end{aligned}

The event $100<X\leq150$ has population probability

\begin{aligned} P(100<X\leq150) &=S_X(100)-S_X(150)\\ &=\frac{100}{200}-\frac{100}{250}\\ &=\frac{1}{2}-\frac{2}{5}\\ &=0.1000. \end{aligned}

Because the event is contained in the observation region, the conditional probability is the event probability divided by the observation probability:

P(100<X\leq150\mid 50<X\leq200) =\frac{0.1000}{1/3} =0.3000.

The check is $0.3000(1/3)=0.1000$, which returns the population probability. The method is the double-truncation conditional probability formula. It avoids dividing by a one-sided survival probability when records are restricted by both a lower and an upper bound.

Interpreting Sample Proportions and Avoiding Traps

If a sample contains $n$ recorded losses and $k$ fall in an event $A$, then $k/n$ estimates $P(A\mid R)$, assuming the recorded observations represent the conditional distribution. It does not, by itself, estimate $P(A)$. When $A\subseteq R$ and a model gives $P(R)$, the corresponding model-adjusted population probability is $(k/n)P(R)$. This calculation translates a conditional probability; it does not correct for uncertainty in the estimated proportion or model parameters.

The distinction matters when reporting rates, estimating the proportion of all claims in a loss band, or comparing model probabilities with a truncated database. Keep the event and observation region explicit throughout the calculation.

Trap: use the wrong denominator. For left truncation at $L$, divide by $P(X>L)=S_X(L)$, not by $F_X(L)$. For right truncation at $U$, divide by $F_X(U)$.
Trap: treat the selected proportion as population-wide. If $P(A\mid R)=0.30$ and $P(R)=0.40$, with $A\subseteq R$, the population probability is $0.30(0.40)=0.12$, not $0.30$.
Trap: adjust an event that extends outside the recorded region as if it were contained inside it. Multiplication by $P(R)$ recovers $P(A\cap R)$, not the part of $A$ outside $R$.
Trap: change the truncation rule while calculating. A condition such as “recorded only if $X>L$” defines the denominator. It is not the same as asking for the probability of an event whose threshold happens to be $L$.

Key takeaway: A truncated sample describes a conditional distribution. Calculate the event probability within the observation region, divide by the probability of that region for a selected-sample probability, and multiply back by the region probability only when recovering a population probability for an event contained in that region.

Check Your Understanding

For each problem, distinguish the probability among recorded losses from the probability in the full population.

A hypothetical exponential loss has mean $50$ thousand dollars, and only losses exceeding $25$ thousand dollars are recorded. Find $P(25<X\leq75\mid X>25)$ and the population probability $P(25<X\leq75)$. (Answer: $0.6321$ conditional; $0.3834$ population.)
A loss is uniform on $(0,500)$ thousand dollars, but the database records only $X<400$. Find the conditional probability $P(X\leq250\mid X<400)$ and the population probability $P(X\leq250)$. (Answer: $0.6250$ conditional; $0.5000$ population.)
A hypothetical loss has $S_X(x)=[100/(100+x)]^2$, for $x\geq0$. Records satisfy $50<X\leq150$. Calculate $P(75<X\leq125\mid50<X\leq150)$. (Answer: $0.4535$.)
In a sample truncated at $X>30$, 42 of 70 recorded losses satisfy $30<X\leq90$. A fitted model gives $P(X>30)=0.75$. Estimate the population probability $P(30<X\leq90)$ using the observed conditional proportion. (Answer: $0.4500$.)
A severity model is uniform on $(0,1000)$ thousand dollars. Records are restricted to $100<X\leq700$. If the conditional probability $P(300<X\leq500\mid100<X\leq700)$ is estimated as $0.4000$, use the model's probability of the observation region to obtain the adjusted population probability for $300<X\leq500$. (Answer: $0.2400$.)

Observation region	Conditional probability for a threshold in the region	Probability of the observation region
Left truncation: \(X>L\)	\(P(X\leq x\mid X>L)=[F_X(x)-F_X(L)]/S_X(L)\), for \(x>L\)	\(S_X(L)\)
Right truncation: \(X<U\)	\(P(X\leq x\mid X<U)=F_X(x)/F_X(U)\), for \(x<U\)	\(F_X(U)\)
Double truncation: \(L<X<U\)	\(P(X\leq x\mid L<X<U)=[F_X(x)-F_X(L)]/[F_X(U)-F_X(L)]\), for \(L<x<U\)	\(F_X(U)-F_X(L)\)

What You'll Learn