Why Mixed Coverage Records Need One Likelihood
A severity data set may contain exact ground-up losses for some claims and only a policy-limit payment for others. It may also include only claims that exceed their deductibles. These features affect what each record says about the underlying loss \(X\): an exact observation contributes a density, a claim recorded because it exceeded a deductible is subject to truncation, and a payment at the policy limit indicates right censoring.
The likelihood must account for all three features at once. In particular, the probability of an exact loss is not generally its unconditional density when the data set contains only losses above a deductible. Likewise, a limit payment is not an exact observation that the loss equals the limit. The methods below assume that the ground-up severity model is common across the records, while each policy’s deductible and limit are known.
For an ordinary policy with deductible \(d\) and maximum payment \(u\), the largest ground-up loss that can be observed exactly is \(c=d+u\). An exact payment below \(u\) can be converted to an exact ground-up loss by adding \(d\). A payment equal to \(u\) is treated as right-censored at \(c\), provided the severity model is continuous and the policy terms do not create a separate atom at that payment.
The conditional likelihood here is for claim-only severity records selected by exceeding the deductible. It does not use the number of policies with no observed claims or model claim frequency. If the data include such exposure or selection information, it must be incorporated using an appropriate model for that information as well.
Likelihood Contributions and Their Justification
Let \(X\) have density \(f_X(x;\theta)\) and survival function \(S_X(x;\theta)=P(X>x)\), where \(\theta\) is an unknown severity parameter. For policy \(j\), let \(d_j\) be its deductible. Write \(c_j\) for its censoring threshold when a policy limit applies. The observation type determines the contribution:
- Exact loss \(x_j\): the claim is known to exceed \(d_j\), and its exact ground-up amount is observed.
- Right-censored loss at \(c_j\): the claim is known to exceed \(d_j\), but the only further information is \(X_j\ge c_j\).
The conditional density for an exact loss follows directly from conditioning on inclusion in the data. For a small interval of width \(dx\) around \(x>d\),
Thus the conditional density is \(f_X(x;\theta)/S_X(d;\theta)\). For a censored record, the observed event is \(X\ge c\), conditional on \(X>d\), so its likelihood contribution is \(P(X\ge c\mid X>d)=S_X(c;\theta)/S_X(d;\theta)\). For continuous severities, using \(>\) or \(\ge\) at a single threshold gives the same probability.
Assuming independent claim severities conditional on their policy terms, multiply the contributions across records. The resulting likelihood is
where \(E\) is the set of exact records and \(C\) is the set of right-censored records. Factors not involving \(\theta\) may be omitted when maximizing, but every parameter-dependent numerator and denominator must be retained.
For every claim, identify its deductible \(d_j\), and identify \(c_j\) if its payment is censored at a limit.
Use an exact density for an exact ground-up loss and a survival probability for a right-censored loss.
Divide each contribution by \(S_X(d_j;\theta)\) because the claim-only record is observed conditional on exceeding its deductible.
Multiply the contributions, take logarithms when convenient, and solve the score equation subject to the model’s parameter constraints.
Worked Examples
Worked Example: Two Exact Claims and One Censored Claim
A hypothetical portfolio records only claims exceeding a deductible of \(2\) thousand dollars. Assume independent ground-up losses from an exponential distribution with mean \(\theta\) thousand dollars, where \(\theta>0\) is unknown. Two losses are observed exactly at \(3\) and \(5\) thousand dollars. A third claim is censored at \(6\) thousand dollars. Estimate \(\theta\) by maximum likelihood.
Method. For an exponential severity, \(f_X(x;\theta)=\theta^{-1}e^{-x/\theta}\) and \(S_X(x;\theta)=e^{-x/\theta}\) for \(x\ge0\). The exact contribution conditional on exceeding \(d\) is \[ \frac{f_X(x;\theta)}{S_X(d;\theta)} =\frac{1}{\theta}e^{-(x-d)/\theta}. \] The censored contribution is \(S_X(c;\theta)/S_X(d;\theta)=e^{-(c-d)/\theta}\).
Calculate the residual amounts above the deductible. The exact records have residuals \(3-2=1\) and \(5-2=3\) thousand dollars. The censored record contributes \(6-2=4\) thousand dollars of survival information.
Form the likelihood. There are two exact records, so there are two factors of \(\theta^{-1}\). The sum of the three residual amounts is \(1+3+4=8\):
The log-likelihood is \(\ell(\theta)=-2\ln\theta-8/\theta\). Its derivative is \(\ell'(\theta)=-2/\theta+8/\theta^2\). Setting the score to zero gives \(-2\theta+8=0\), so \(\widehat{\theta}=4\) thousand dollars.
Check. The estimate is positive and satisfies the score equation: \(-2/4+8/16=0\). The method avoids the mistake of treating the censored claim as an exact loss of \(6\), which would incorrectly add a density factor \(1/\theta\).
Worked Example: Multiple Deductibles and Policy Limits
A hypothetical insurer has five claim-only severity records, all measured in thousands of dollars. Assume independent exponential ground-up losses with a common unknown mean \(\theta\). The records are: an exact loss \(1.8\) above deductible \(1\); a claim censored at \(3\) above deductible \(1\); an exact loss \(4\) above deductible \(2\); an exact loss \(3\) above deductible \(1.5\); and a claim censored at \(5\) above deductible \(2\). Find the maximum likelihood estimate of \(\theta\).
Method. Under the exponential model, each exact record contributes \(\theta^{-1}e^{-(x-d)/\theta}\), and each censored record contributes \(e^{-(c-d)/\theta}\). The deductibles and censoring thresholds differ, so calculate each record’s amount above its own deductible.
Calculate the exact residuals. They are \(1.8-1=0.8\), \(4-2=2\), and \(3-1.5=1.5\) thousand dollars. Their sum is \(0.8+2+1.5=4.3\).
Calculate the censored residuals. They are \(3-1=2\) and \(5-2=3\) thousand dollars. Their sum is \(2+3=5\). The total residual information is therefore \(4.3+5=9.3\) thousand dollars.
There are three exact observations, so the likelihood is
The log-likelihood is \(\ell(\theta)=-3\ln\theta-9.3/\theta\), and its derivative is \(\ell'(\theta)=-3/\theta+9.3/\theta^2\). Solving \(\ell'(\theta)=0\) gives \(-3\theta+9.3=0\), hence \(\widehat{\theta}=9.3/3=3.1\) thousand dollars.
Check. Substitution gives \(-3/3.1+9.3/(3.1)^2=0\), since \(9.3=3(3.1)\). The common mistake is to subtract one deductible from every claim or to count all five records as exact observations. Each threshold is record-specific, and only exact observations supply a factor of \(\theta^{-1}\).
Why the Exponential Result Has This Form
The exponential distribution has the memoryless property: conditional on exceeding \(d\), the excess \(X-d\) has the same exponential distribution with mean \(\theta\). This explains why the deductible-adjusted amounts appear in the preceding likelihoods. For an exact observation, the conditional density is the density of an exponential excess; for a censored observation, the conditional survival probability is the survival probability of that excess.
With \(r\) exact records and \(m\) censored records, define \(z_j=x_j-d_j\) for exact records and \(w_j=c_j-d_j\) for censored records. The likelihood is proportional to
Let \(A=\sum_{j\in E}z_j+\sum_{j\in C}w_j\). Then \(\ell(\theta)=-r\ln\theta-A/\theta\), so \(\ell'(\theta)=-r/\theta+A/\theta^2\). When \(r>0\) and \(A>0\), the score equation gives
The denominator is the number of exact observations, not the total number of records: only exact densities contribute factors of \(\theta^{-1}\). If there are no exact observations, this likelihood increases toward its supremum as \(\theta\) increases and does not produce a finite maximum likelihood estimate.
This convenient formula is specific to the exponential model. For other severity families, the deductible terms generally remain parameter-dependent and cannot be discarded as a fixed adjustment to observed losses.
Worked Example: Pareto Shape Estimation with Mixed Records
A hypothetical claim-only portfolio follows a Pareto Type I model with known minimum \(x_m=1\) thousand dollars and unknown shape \(\alpha>0\). The records are: an exact loss \(x=2\) above deductible \(d=1\); a claim censored at \(c=4\) above \(d=1\); an exact loss \(x=8\) above \(d=2\); and a claim censored at \(c=8\) above \(d=2\). Estimate \(\alpha\).
Method. For \(x\ge x_m\), the Pareto density and survival function are \(f_X(x;\alpha)=\alpha x_m^\alpha x^{-(\alpha+1)}\) and \(S_X(x;\alpha)=(x_m/x)^\alpha\). For an exact observation above deductible \(d\), the conditional density simplifies to \[ \frac{f_X(x;\alpha)}{S_X(d;\alpha)} =\frac{\alpha}{x}\left(\frac{d}{x}\right)^\alpha. \] A censored record contributes \(S_X(c;\alpha)/S_X(d;\alpha)=(d/c)^\alpha\).
Evaluate each contribution. The exact loss \(2\) above \(d=1\) contributes \((\alpha/2)(1/2)^\alpha\). The censored record at \(4\) above \(d=1\) contributes \((1/4)^\alpha\). The exact loss \(8\) above \(d=2\) contributes \((\alpha/8)(2/8)^\alpha=(\alpha/8)(1/4)^\alpha\). The censored record at \(8\) above \(d=2\) contributes \((2/8)^\alpha=(1/4)^\alpha\).
Multiply the contributions. There are two exact records, giving the prefactor \(\alpha^2/(2\cdot8)=\alpha^2/16\). The four log-ratio coefficients, in units of \(\ln 2\), are \(1\) for the first exact record, \(2\) for the first censored record, \(2\) for the second exact record, and \(2\) for the second censored record. Their sum is \(1+2+2+2=7\). Therefore,
The log-likelihood, apart from a constant, is \(2\ln\alpha-7\alpha\ln2\). Its derivative is \(2/\alpha-7\ln2\), so the score equation yields \(\widehat{\alpha}=2/(7\ln2)\). Using \(\ln2\approx0.693147\), this is \(2/4.85203\approx0.4122\), to four decimal places.
Check. The second derivative is \(-2/\alpha^2<0\), so this stationary point is a maximum. The critical trap is to miscount the log-ratio coefficients: the exact observations contribute \(1\) and \(2\), while the censored observations contribute \(2\) and \(2\). Their total is \(7\), not \(5\); omitting a censored survival contribution changes the estimate.
Model Checks and Common Errors
Before maximizing a mixed likelihood, confirm that the observation mechanism matches the contribution. An exact observation means the loss amount is known; a censored observation means only that the loss crossed a threshold. Truncation means that claims below a threshold are not included in the data set at all. These are different statements about how the data were generated, and they lead to different likelihoods.
| Record information | Likelihood contribution | Frequent error |
|---|---|---|
| Exact \(x>d\), observed among claims exceeding \(d\) | \(f_X(x;\theta)/S_X(d;\theta)\) | Using the unconditional density without conditioning |
| Only \(X\ge c>d\) is known, among claims exceeding \(d\) | \(S_X(c;\theta)/S_X(d;\theta)\) | Treating \(c\) as an exact loss and using a density |
| Claim is absent because it did not exceed \(d\) | Not a claim-only severity record | Adding an exact or censored severity contribution for an unobserved claim |
A useful exam check is to ask what event the record actually establishes. For an exact record, the likelihood involves probability in a small interval around the observed amount, which produces a density. For a censored record, it involves the probability of exceeding the censoring threshold, which produces a survival probability. For a claim-only sample, both are conditional on exceeding the deductible.
Check Your Understanding
For each problem, identify the observation type and account for the relevant deductible before estimating or evaluating the requested quantity.
- Assume exponential ground-up severity with unknown mean \(\theta\), in thousands of dollars. A claim-only sample has deductible \(2\), exact losses \(4\) and \(7\), and one claim censored at \(6\). Find the maximum likelihood estimate of \(\theta\). (Answer: \(5.5\) thousand dollars)
- A Pareto Type I severity has known \(x_m=1\) and unknown shape \(\alpha\). There is one exact loss \(x=3\) above \(d=1\) and one claim censored at \(c=4\) above \(d=2\). Write the likelihood up to a constant and find the maximum likelihood estimate. (Answer: \(L(\alpha)\propto\alpha e^{-\alpha\ln6}\), \(\widehat{\alpha}=1/\ln6\approx0.5581\))
- Assume exponential severity with unknown mean \(\theta\), in thousands of dollars. The records are an exact loss \(6\) above deductible \(2\), a claim censored at \(5\) above deductible \(1\), and an exact loss \(5\) above deductible \(3\). Find \(\widehat{\theta}\). (Answer: \(5\) thousand dollars)
- A continuous severity model has survival function \(S_X(x;\theta)\). A claim-only record is censored at \(c=10\) after exceeding deductible \(d=4\). State its likelihood contribution and explain why \(S_X(10;\theta)\) alone is not the conditional contribution. (Answer: \(S_X(10;\theta)/S_X(4;\theta)\); the denominator conditions on the claim’s inclusion above the deductible.)
- For exponential severity with unknown mean \(\theta\), a claim-only sample has four exact observations and two right-censored observations. The sum of all exact and censored amounts above their respective deductibles is \(21\) thousand dollars. Find the maximum likelihood estimate and identify the denominator used. (Answer: \(\widehat{\theta}=21/4=5.25\) thousand dollars; the denominator is the four exact observations.)