Diagnosing Density Estimates Under Incomplete Sampling

Why Density Diagnostics Need the Sampling Mechanism

A density estimate can look smooth and still describe the wrong population. If only losses below a reporting threshold are observed, the sample favors smaller losses. If follow-up ends before some losses are settled, those censored observations do not reveal their eventual values. In either case, a conventional kernel density estimate of the recorded numbers can mistake the observation process for the loss process.

The diagnostic task is not simply to choose a smoother bandwidth. It is to ask whether the estimate respects the support of the target distribution, whether its tail is supported by observed information, and whether apparent features persist when incomplete-data adjustments and reasonable smoothing choices are compared. The methods here are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Start With the Target and the Observable Support

Let \(X\) denote the loss of interest, with density \(f_X\), distribution function \(F_X\), and survival function \(S_X\). A kernel density estimate based on complete observations \(X_1,\ldots,X_n\) is

where \(K\) is a kernel with integral one and \(h>0\) is a bandwidth. This formula estimates the density of the sample's generating distribution when the observations are complete and comparably sampled. It does not itself correct for incomplete observation.

For right-truncated losses observed only when \(X\leq T\), the observed values have the conditional density

Thus, a naive estimate from the observed sample targets \(g_T\), not \(f_X\). If \(q=F_X(T)\) is known, then \(f_X(x)=qg_T(x)\) for \(x\leq T\). This does not determine the shape of the density above \(T\); the unobserved tail requires additional information or assumptions. A hard upper edge in the observed sample is therefore not evidence that the underlying loss density ends there.

With varying truncation thresholds, an inverse-probability weighted estimate can reduce the over-representation of observations that were easier to include. If observation \(i\) has known inclusion probability \(\pi_i>0\), set \(w_i=1/\pi_i\). A normalized weighted kernel estimate is

This estimate requires correctly specified inclusion probabilities and positive chances of observation over the target region. If a region has zero inclusion probability, weighting cannot recover its density from these data.

What a Censoring Adjustment Estimates

A right-censored value records that the loss exceeds a known censoring point, not that the loss equals that point. Under independent censoring, the Kaplan-Meier estimator uses the risk set and event counts to estimate the loss distribution. If \(t_j\) are distinct event times, \(d_j\) is the number of events at \(t_j\), and \(Y_j\) is the number at risk immediately before \(t_j\), its survival estimate is

The estimated event probability mass at \(t_j\) is \(\Delta\widehat F_X(t_j)=\widehat S_X(t_j-)d_j/Y_j\). Smoothing these event masses gives

Censoring times contribute to the risk sets but are not event masses. Also, if the largest follow-up time is censored, the estimated event masses may sum to less than one. The remaining estimated survival probability represents unresolved tail mass, not an observed event. Automatically renormalizing the event masses to sum to one would instead describe the event distribution conditional on occurrence by the observed endpoint.

Why the Adjustments Work

The right-truncation adjustment follows directly from conditional probability. For any small interval around \(x\leq T\), conditioning on inclusion divides the probability of that interval by the probability of inclusion:

Dividing by \(\Delta\) and taking the limit as \(\Delta\) approaches zero gives \(g_T(x)=f_X(x)/F_X(T)\). Multiplying by the known \(F_X(T)\) recovers the underlying density below \(T\). No corresponding calculation recovers the shape above \(T\), because those losses are never observed.

For censoring, the Kaplan-Meier product estimates the probability of surviving each event time among those still at risk. The decrease in estimated survival at an event time is exactly the estimated event probability there:

Smoothing these probability increments spreads estimated event mass across nearby loss values. Censoring times do not produce such a decrement, which is why treating them as exact losses changes the target being estimated.

Support, Tail, and Sensitivity Checks

A useful diagnosis compares estimates in several ways rather than treating a single curve as definitive. The following checks help separate features supported by loss information from features driven by incomplete observation.

• Support check. Confirm that the estimated density is being interpreted only where losses could be observed and the adjustment has positive support. A density that abruptly vanishes at a truncation boundary may reflect the observation rule, not the underlying distribution. Ordinary kernels can also place estimated mass outside a known physical support; this is a boundary-smoothing artifact, not evidence of negative or impossible losses.
• Tail check. Ask how much estimated probability lies beyond the largest informative loss. Right truncation supplies no direct observations above its threshold. With censoring, a residual Kaplan-Meier survival probability at the end of follow-up signals unresolved tail mass. A smooth-looking tail is not identified merely because a kernel extends past the last event.
• Sensitivity check. Compare plausible bandwidths and, where appropriate, naive and adjusted estimates on the same scale and region. A narrow spike that disappears under a modest bandwidth change may be sampling noise. A large difference between naive and adjusted curves suggests the inclusion mechanism matters. Neither result alone proves which curve is correct; model assumptions and data support still matter.
• Weight check. Examine the range of inverse inclusion weights. Very large weights mean that a few observations can dominate the adjusted estimate. One summary is the effective sample size, \((\sum_i w_i)^2/\sum_i w_i^2\). A much smaller effective size than the recorded sample size warns that the weighted curve may be unstable.
• Mass check. Check the estimated probability mass as well as pointwise density height. A censoring-adjusted curve built from event increments may integrate to less than one when follow-up ends with positive estimated survival. Treating that deficit as a numerical error and forcing the curve to integrate to one can conceal the unknown tail.

Common Diagnostic Traps

A common error is to treat truncation and censoring as interchangeable. Truncation means a loss is absent from the sample unless it satisfies an inclusion condition; censoring means the record is present but its exact loss is not known. The corresponding corrections are different. Conditional-density or inclusion-probability adjustments address truncation, while event-risk methods address censored outcomes.

Another trap is to overinterpret visual smoothness. Smoothing can make a density curve continuous, but it cannot create information in a region with no chance of observation or resolve a tail that remains censored. Report the target population, the observation mechanism, the adjustment, the bandwidth, and any unresolved tail mass together. When naive and adjusted estimates differ materially, investigate why before selecting one for actuarial use.