Truncation Versus Censoring in Sampling Designs

Why the Sampling Design Changes the Likelihood

A loss amount can be missing from a data set for two fundamentally different reasons. Under truncation, a loss that fails a selection rule never enters the observed sample. Under censoring, a record remains in the sample, but its loss amount is known only in part. The distinction matters because a likelihood must describe both the loss model and the way observations came to be recorded.

For example, suppose a study includes only claims exceeding a threshold. A claim below that threshold is absent, not recorded as a small claim. By contrast, if every claim is recorded but amounts above a policy cap are reported only as “at least the cap,” those claims are censored. A censored record tells us an event occurred in the sample; a truncated record does not.

Definition: A loss \(X\) is left-truncated at a fixed threshold \(a\) when only losses with \(X>a\) are included, and the observed data contain exact values only for included losses. The appropriate density for an included value is the conditional density \(f_X(x)/S_X(a)\), for \(x>a\), when the sample is conditioned on inclusion.

A loss is right-censored at a threshold \(c\) when the record indicates either an exact loss \(x<c\) or that the loss is at least \(c\). An exact record contributes \(f_X(x)\); a record known only to satisfy \(X\ge c\) contributes \(S_X(c)\), under a continuous model.

These definitions describe common designs, not every possible incomplete-data mechanism. The likelihood must follow the information actually retained. For instance, a reported amount capped at \(c\) is not automatically a right-censored observation: if the data record only the capped payment and do not identify whether the underlying loss exceeded \(c\), the observation mechanism must be described before choosing a contribution. The results below assume fixed thresholds and a continuous severity distribution, so equality at a threshold has probability zero.

Likelihood Contributions Under Each Design

Suppose \(X\) has density \(f_X(x;\theta)\), distribution function \(F_X(x;\theta)\), and survival function \(S_X(x;\theta)=1-F_X(x;\theta)\). The parameter or parameters of the loss model are denoted by \(\theta\). Under left truncation at \(a\), the fact that a value appears in the data tells us that \(X>a\). Thus, if the analysis conditions on having \(n\) included observations, each exact observed value \(x_i>a\) has conditional density

f_{X\mid X>a}(x_i;\theta) =\frac{f_X(x_i;\theta)}{P(X>a)} =\frac{f_X(x_i;\theta)}{S_X(a;\theta)}.

For a sample of \(n\) independent included losses, the conditional likelihood is therefore

L_{\mathrm{trunc}}(\theta) =\prod_{i=1}^{n}\frac{f_X(x_i;\theta)}{S_X(a;\theta)}.

The denominator is essential. It accounts for the fact that values at or below \(a\) could not appear in this sample. If the threshold is fixed but the loss model has a parameter-dependent survival probability at \(a\), that denominator depends on \(\theta\) and cannot be discarded.

Under right censoring, let \(\delta_i=1\) indicate an exact loss and \(\delta_i=0\) indicate that the loss is only known to exceed its censoring time \(t_i\). For exact loss \(x_i\), the record contributes its density. For a censored record, it contributes the probability that the loss exceeds the observed threshold. Under independent, non-informative censoring, censoring-mechanism factors that do not involve \(\theta\) can be omitted from the loss-model likelihood. The resulting contribution is

L_{\mathrm{cens}}(\theta) =\prod_{i=1}^{n} \left[f_X(t_i;\theta)\right]^{\delta_i} \left[S_X(t_i;\theta)\right]^{1-\delta_i}.

For an exact observation, \(t_i\) is its observed loss amount. For a censored observation, \(t_i\) is the censoring threshold. Unlike truncation, censoring retains one likelihood contribution for every sampled record, including records with incomplete amounts.

Derivation: A truncated value is observed conditional on passing the inclusion rule. By the definition of conditional density, \(f_{X\mid X>a}(x)=f_X(x)/P(X>a)\) for \(x>a\), and \(P(X>a)=S_X(a)\). A right-censored record says the unobserved value lies in the set \([c,\infty)\); the probability of that recorded event is \(P(X\ge c)=S_X(c)\) for a continuous loss. Thus truncation uses a conditional density, whereas censoring uses the probability of the observed event.

Worked Likelihood Comparisons

The examples use exponential severity, \(X\sim\operatorname{Exponential}(\lambda)\), with rate \(\lambda>0\) per dollar. Its density and survival function are \(f_X(x)=\lambda e^{-\lambda x}\) for \(x\ge 0\) and \(S_X(x)=e^{-\lambda x}\). No interest or mortality assumptions apply to these severity-sampling examples.

Worked Example: Exact Losses From a Left-Truncated Sample

A hypothetical study includes claims only when the loss exceeds \(700\) dollars. The two included losses are \(900\) dollars and \(1{,}100\) dollars. Treat the included sample size of two as fixed, and estimate \(\lambda\).

Method. Since both claims passed the \(700\)-dollar entry rule, use the conditional density \(f_X(x)/S_X(700)\) for each exact loss. The likelihood is

\begin{aligned} L(\lambda) &=\frac{\lambda e^{-900\lambda}}{e^{-700\lambda}} \frac{\lambda e^{-1100\lambda}}{e^{-700\lambda}}\\ &=\lambda^2 e^{-\lambda[(900-700)+(1100-700)]}\\ &=\lambda^2e^{-600\lambda}. \end{aligned}

The log-likelihood is \(\ell(\lambda)=2\log\lambda-600\lambda\). Its score is \(2/\lambda-600\), so the score equation gives

\widehat{\lambda}=\frac{2}{600} =\frac{1}{300} =0.003333\ldots\ \text{per dollar}.

The second derivative is \(\ell''(\lambda)=-2/\lambda^2<0\), so this stationary point is a maximum. As a check, the fitted total excess above the threshold is \(200+400=600\) dollars, and the exponential rate estimate is the number of observations divided by that total excess. The likelihood at the estimate is \(e^{-2}/90{,}000\approx0.00000150373\), in inverse-dollar-squared units.

Common mistake avoided. Using the untruncated exponential likelihood would give \(2/(900+1{,}100)=0.001\) per dollar. That calculation treats the observed values as an ordinary sample from the full severity distribution and ignores that values at or below \(700\) could not enter the data.

Worked Example: Exact and Right-Censored Losses

A hypothetical claims file contains exact losses of \(400\) dollars and \(1{,}000\) dollars. A third sampled claim is known only to exceed \(700\) dollars. Assume independent, non-informative censoring at \(700\) dollars. Estimate the exponential rate.

Method. The two exact observations contribute densities; the censored record contributes the survival probability at \(700\). Therefore

\begin{aligned} L(\lambda) &=(\lambda e^{-400\lambda}) (\lambda e^{-1000\lambda}) (e^{-700\lambda})\\ &=\lambda^2e^{-2100\lambda}. \end{aligned}

The total exposure in the exponent is \(400+1{,}000+700=2{,}100\) dollars. The log-likelihood is \(2\log\lambda-2{,}100\lambda\), with score \(2/\lambda-2{,}100\). Hence

\widehat{\lambda}=\frac{2}{2100} =\frac{1}{1050} \approx0.0009524\ \text{per dollar}.

The second derivative is \(-2/\lambda^2<0\), confirming a maximum. The censored record contributes \(700\) dollars of exposure, but it does not contribute a factor of \(\lambda\), because no exact loss amount was observed.

Common mistake avoided. Treating the censored claim as an exact \(700\)-dollar loss would produce the incorrect likelihood \(\lambda^3e^{-2100\lambda}\) and estimate \(3/2100=1/700\) per dollar. A record that says \(X\ge700\) contributes \(S_X(700)\), not \(f_X(700)\).

Worked Example: When the Number Screened Is Known

A hypothetical screening study examines exactly three claims. Two exceed a fixed entry threshold of \(100\) dollars and are observed exactly as \(150\) dollars and \(200\) dollars; the third is known to be at or below \(100\) dollars. Compare the likelihood if the analysis conditions on the two included claims with the likelihood if it uses the known count screened and excluded.

Conditioning on inclusion. If only the two selected claims are analyzed and their sample size is treated as fixed, their conditional likelihood is

L_{\mathrm{conditional}}(\lambda) =\frac{f_X(150)f_X(200)}{S_X(100)^2} =\lambda^2e^{-\lambda[(150-100)+(200-100)]} =\lambda^2e^{-150\lambda}.

Using the screening count. If all three screened records are part of the data, the two exact included values contribute their densities and the excluded claim contributes \(F_X(100)\). Thus

L_{\mathrm{screened}}(\lambda) =f_X(150)f_X(200)F_X(100) =\lambda^2e^{-350\lambda}(1-e^{-100\lambda}).

For an arithmetic check at \(\lambda=0.01\) per dollar, the factors are \(f_X(150)=0.01e^{-1.5}\), \(f_X(200)=0.01e^{-2}\), and \(F_X(100)=1-e^{-1}\approx0.6321206\). The screened-data likelihood is therefore

10^{-4}e^{-3}(1-e^{-1}) \approx 0.00000314714.

At the same rate, the conditional likelihood is \(10^{-4}e^{-1.5}\approx0.0000223130\). These likelihoods describe different information: the second uses the fact that a third claim was screened and failed the threshold. The value of the likelihood itself is not a probability of the exact continuous observations; its units here are inverse dollars squared.

Common mistake avoided. Do not add the factor \(F_X(100)\) when the analysis is explicitly conditional on the number of included claims and has no information about how many claims were screened or excluded. Conversely, if the screened and excluded record is known, omitting its contribution discards observed information.

How to Choose the Likelihood

The decisive question is not simply whether some loss amounts are missing. Ask what happened to each potential observation and what information the data retain about it. The following comparison summarizes the distinction.

Sampling or recording design	What appears in the data?	Loss-model contribution
Left truncation at \(a\), conditional on inclusion	Only exact losses exceeding \(a\)	\(f_X(x)/S_X(a)\)
Right censoring at \(c\)	A record known to exceed \(c\)	\(S_X(c)\)
Known screened claim at or below \(a\)	An observed failure to pass the entry rule	\(F_X(a)\)
Exact, uncensored claim	The loss amount \(x\)	\(f_X(x)\)

For a fixed-size sample of truncated observations, the likelihood is conditional on selection. If the original number screened and the number excluded are also observed, those selection outcomes can provide additional information. For example, with \(m\) exact included values \(x_i>a\) and \(r\) known excluded values \(X\le a\), the loss-model likelihood, apart from a parameter-free assignment factor, is

\left[\prod_{i=1}^{m}f_X(x_i;\theta)\right] \left[F_X(a;\theta)\right]^r.

This form uses the actual screened records rather than conditioning away the inclusion rule. The appropriate version depends on the sampling plan and the information available, so the plan should be stated before differentiating a likelihood.

Common Exam Traps

Replacing truncation with censoring. A loss that never entered a threshold-selected sample has no observed record contributing \(F_X(a)\) or \(S_X(a)\) unless the screening outcome or count is known. For an included value in a conditional truncated sample, use \(f_X(x)/S_X(a)\).
Replacing censoring with truncation. A retained record known to exceed \(c\) contributes \(S_X(c)\). It does not disappear from the likelihood, and it is not an exact loss at \(c\).
Dropping a parameter-dependent normalizer. In a truncated model, \(S_X(a;\theta)\) generally depends on \(\theta\). It must remain in the likelihood unless it is constant with respect to the parameters being estimated.
Ignoring the sampling frame. A likelihood conditioned on the number selected and one using the known number screened answer different statistical questions. State which information is observed and condition accordingly.
Using a censoring contribution without its assumptions. The survival contribution for a censored loss presumes the record truly conveys a threshold event, and the loss-model likelihood relies on independent, non-informative censoring as specified in the model.

Key takeaway: Truncation changes which losses can enter the sample, so conditional likelihoods normalize by the probability of selection. Censoring keeps a record but reports only an event such as \(X\ge c\), so its contribution is the probability of that event. If the screening counts are known, use them as part of the observed data.

Check Your Understanding

For each problem, identify the sampling mechanism and write or use the corresponding likelihood contribution.

A hypothetical study retains only losses above \(300\) dollars and records exact included losses of \(500\) and \(900\) dollars. Under an exponential model, what is the conditional maximum likelihood estimate of \(\lambda\)? (Answer: \(2/800=0.0025\) per dollar.)
Two exact exponential losses are \(250\) and \(650\) dollars, and a third sampled loss is right-censored at \(400\) dollars. What is the maximum likelihood estimate of \(\lambda\)? (Answer: \(2/(250+650+400)=1/650\approx0.001538\) per dollar.)
Under left truncation at \(100\) dollars, exact included values are \(160\) and \(240\) dollars. For an exponential rate \(\lambda\), what is their conditional likelihood? (Answer: \(\lambda^2e^{-200\lambda}\).)
A screening frame has six claims, of which two are observed exactly above threshold \(a\), at \(x_1\) and \(x_2\), and four are known to be at or below \(a\). What is the loss-model likelihood up to a parameter-free factor? (Answer: \(f_X(x_1)f_X(x_2)[F_X(a)]^4\).)
In an independent right-censoring study with exponential severity, there are \(d\) exact losses and censored records, with each record contributing its observed exact amount or censoring time \(t_i\). What likelihood form and rate estimate follow? (Answer: \(L(\lambda)=\lambda^d e^{-\lambda\sum_i t_i}\) and \(\widehat{\lambda}=d/\sum_i t_i\).)

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