Left-Censored Loss Observations

Why a Loss May Be Known Only to Be Small

A loss database may record the amount of every large claim but suppress amounts below a reporting or measurement threshold. For a suppressed record, the database does not tell us that the loss equaled the threshold, or that it was zero. It tells us only that the loss fell on one side of that threshold. Correctly preserving that limited information matters when describing the data or constructing a likelihood.

This tutorial considers a loss $X$ that is known to be below a stated detection threshold. The loss is not measured exactly, but the record still conveys information about the distribution. That information differs from an exact observation and from a right-censored observation, which says the loss exceeded a stated amount.

Definition: A record is left-censored at threshold $c$ when the observed information is $X\le c$, but the exact value of $X$ is not recorded. Here $c$ is known. The record identifies an event involving $X$, not a numerical loss amount.

For example, if a system records only whether a repair cost is at least \$100, a record stating “below threshold” means $X<\$100$. If the system instead reports that a loss is no more than \$100, the recorded statement is $X\le \$100$. For a continuous loss distribution, the probability of equality at a single point is zero, so these statements have the same probability. For a distribution with a point mass at the threshold, the distinction matters: use the inequality the record actually supports.

What Each Record Tells Us

An exact record gives a particular value of the loss. A right-censored record gives a lower bound: the loss is known to exceed a threshold, but its exact size is unknown. A left-censored record gives an upper bound: the loss is known to be below a threshold, but its exact size is unknown.

Record type	Observed information	Contribution for a continuous loss model
Exact at $x$	$X=x$, recorded as a measured value	$f_X(x)$
Left-censored at $c$	$X\le c$	$F_X(c)$
Right-censored at $c$	$X>c$	$S_X(c)$

Here $F_X(c)=P(X\le c)$ is the distribution function and $S_X(c)=P(X>c)=1-F_X(c)$ is the survival function. For an exact value from a continuous distribution, the likelihood contribution is the density $f_X(x)$; for a censored record, it is the probability of the region known to contain the loss. A density has units inverse to the loss unit, while a probability is dimensionless.

These contributions assume independent sampled losses and a known threshold, with the record mechanism represented by the stated censoring information. If the data set includes only losses that passed a threshold-based inclusion rule, the sample may instead be truncated: the likelihood must then account for the condition that an observation was included. Censoring and truncation are not interchangeable.

Why the Left-Censored Likelihood Contribution Is a Probability

Suppose a model for $X$ has parameter $\theta$, and a record says only that $X\le c$. The likelihood is the probability of seeing the observed information under that model. Since every possible value from the lower end of the loss range through $c$ is consistent with the record, add the probabilities over that region. For a continuous model, adding those infinitesimal probabilities gives the integral of the density:

L(\theta\mid X\le c) =P_\theta(X\le c) =F_X(c;\theta) =\int_{-\infty}^{c} f_X(x;\theta)\,dx.

The record does not select one particular $x$, so substituting the threshold into the density would answer a different question: it would describe the density at $c$, not the probability that the loss is at or below $c$. Likewise, replacing the record by an exact value $c$ would assert information that was never observed.

Worked Likelihood Calculations

Worked Example: One Left-Censored Record and One Exact Loss

A hypothetical portfolio has independent losses following an exponential distribution with rate $\lambda=0.001$ per dollar, so $f_X(x)=0.001e^{-0.001x}$ for $x\ge0$. The records are one loss known to be at most \$200 and one exact loss of \$500. Find their joint likelihood contribution. No interest or mortality assumption applies.

Method. Multiply the left-censoring probability by the exact observation’s density. Independence makes the joint likelihood the product of the two contributions.

For the left-censored record,

P(X\le 200)=F_X(200)=1-e^{-0.001(200)} =1-e^{-0.2} =0.1812692469.

For the exact \$500 loss, the density contribution is

f_X(500)=0.001e^{-0.001(500)} =0.001e^{-0.5} =0.0006065306597 \quad\text{per dollar}.

Therefore the joint likelihood contribution is

(1-e^{-0.2})(0.001e^{-0.5}) =0.00010994536\ldots \approx 0.00010995 \quad\text{per dollar}.

The likelihood has units per dollar because it contains one density contribution and one probability contribution. Check: the left-censoring factor is between 0 and 1, and the exact-loss factor is a density, as required. The common mistake is to use $f_X(200)$ for the first record; that would treat \$200 as an exact loss. It is not the observed information.

Worked Example: Combining Left-Censored, Exact, and Right-Censored Records

A hypothetical equipment insurer models independent loss amounts as exponential with rate $\lambda=0.002$ per dollar. Its records are: one loss known to be at most \$150, one exact loss of \$250, and one loss known to exceed \$400. Write and evaluate their joint likelihood contribution. No interest or mortality assumption applies.

Method. Use the distribution function for the left-censored record, the density for the exact record, and the survival function for the right-censored record.

The left-censoring probability is

F_X(150)=1-e^{-0.002(150)} =1-e^{-0.3} =0.2591817793.

The exact-loss density is

f_X(250)=0.002e^{-0.002(250)} =0.002e^{-0.5} =0.001213061319 \quad\text{per dollar}.

The probability for the right-censored record is

S_X(400)=P(X>400)=e^{-0.002(400)} =e^{-0.8}=0.4493289641.

Thus the joint contribution is

(1-e^{-0.3})(0.002e^{-0.5})e^{-0.8} =0.00014127\ldots \quad\text{per dollar}.

The displayed components give a direct check: the first and third factors are probabilities; the middle factor is a density. Common mistake: using $1-F_X(150)$ for the left-censored record. That is the probability of exceeding \$150, whereas the record says the loss is at most \$150.

Worked Example: What Can Be Said About a Sample Total?

A hypothetical set of three nonnegative losses contains two exact amounts, \$140 and \$230, and one loss recorded only as at most \$100. Find the possible range of the sample total and sample mean. No parametric distribution, interest rate, or mortality assumption is specified or needed.

Method. Retain the censored amount as an unknown loss $x$ satisfying $0\le x\le100$. The two exact amounts sum to

$$ 140+230=370. $$

The sample total is $370+x$. Its minimum occurs at $x=0$, and its maximum occurs at $x=100$:

370\le 370+x\le 470.

There are three losses, so divide both endpoints by 3 to obtain the possible sample-mean range:

\frac{370}{3}\le \bar X\le\frac{470}{3}, \qquad \$123.33\le\bar X\le\$156.67

The bounds are rounded to the nearest cent. Check: the mean bounds are the total bounds divided by the same sample size, 3. Reporting \$100 as the censored loss would give a possible upper endpoint, not an observed sample mean.

Interpretation and Common Traps

A left-censored record contributes information about how much probability a model places below a threshold. It does not reveal where below that threshold the loss lies. In a likelihood calculation, this means using a cumulative probability rather than a density at the threshold. In a simple descriptive calculation with known nonnegative losses, it may support bounds on totals or averages, but not an exact summary.

The threshold and the inequality should be read carefully. “Below \$100” and “at most \$100” are equivalent for a continuous model, but not necessarily for a model with probability concentrated at \$100. Similarly, a record stating that a loss is greater than \$100 is right-censored, not left-censored. The censoring direction comes from the inequality the record establishes, not from the location of the threshold on a data display.

Do not substitute the threshold. $X\le c$ does not mean $X=c$.
Do not substitute zero. A small unreported loss need not be zero.
Do not reverse the tail. $P(X\le c)=F_X(c)$, whereas $P(X>c)=S_X(c)$.
Do not use a density for a censored record. Use a probability for the observed range of possible values.
Check the data collection rule. A sample restricted to losses above or below a cutoff may require a truncation adjustment, not merely a censoring contribution.

Key takeaway: A left-censored loss record at $c$ says $X\le c$. Under a continuous model its likelihood contribution is $F_X(c)$. Preserve the inequality; do not invent an exact loss value.

Check Your Understanding

For each problem, identify the observed information and apply the appropriate probability, density, likelihood, or bound. Probabilities are rounded to four decimal places; likelihoods involving a density are rounded to five significant figures.

A hypothetical loss has an exponential distribution with rate $0.004$ per dollar. A record states that the loss is at most \$50. What probability contribution does the record make? (Answer: $1-e^{-0.2}=0.1813$.)
Under an exponential loss model with rate $0.003$ per dollar, a record says the loss exceeds \$250. What is its contribution? (Answer: $e^{-0.75}=0.4724$.)
A hypothetical loss has an exponential distribution with rate $0.005$ per dollar and is observed exactly as \$120. Give its density contribution. (Answer: $0.005e^{-0.6}=0.002744$ per dollar.)
Independent losses follow an exponential distribution with rate $0.0025$ per dollar. One record is $X\le \$80$, and another is an exact loss of \$200. Give their joint likelihood contribution. (Answer: $(1-e^{-0.2})(0.0025e^{-0.5})\approx0.00027486$ per dollar.)
A sample of four nonnegative losses contains three exact amounts of \$90, \$170, and \$240, and one loss known only to be at most \$60. Give the range of possible sample totals and means. (Answer: total from \$500 to \$560; mean from \$125.00 to \$140.00.)

Record type	Observed information	Contribution for a continuous loss model
Exact at \(x\)	\(X=x\), recorded as a measured value	\(f_X(x)\)
Left-censored at \(c\)	\(X\le c\)	\(F_X(c)\)
Right-censored at \(c\)	\(X>c\)	\(S_X(c)\)

What You'll Learn