Parametric Estimation

Parametric estimation assumes losses follow a specific distributional family (e.g., lognormal, Pareto, gamma) and estimates the parameters from data. Methods include maximum likelihood estimation (MLE) and method of moments. Parametric models provide smooth, complete distribution functions useful for extrapolation and pricing. However, if the assumed family is wrong, estimates can be significantly biased. MLE is preferred for Exam STAM because it handles censored and truncated data naturally through likelihood construction.

Maximum Likelihood Estimation

MLE finds the parameter values that maximize the probability of observing the data. For a sample of n independent observations x_1, x_2, ..., x_n from a distribution with parameter theta, the likelihood function is L(theta) = product of f(x_i; theta). Taking the natural log gives the log-likelihood l(theta) = sum of ln(f(x_i; theta)). Setting the derivative equal to zero and solving gives the MLE.

For example, with an exponential distribution with mean theta, f(x) = (1/theta) * exp(-x/theta). The log-likelihood is l(theta) = -n*ln(theta) - (sum of x_i)/theta. Setting dl/dtheta = -n/theta + (sum of x_i)/theta^2 = 0 gives theta_hat = (sum of x_i)/n = x_bar. The MLE of the exponential mean is simply the sample mean.

The power of MLE becomes apparent with complicated data. For right-censored observations (where you know the loss exceeds some value u but not the actual amount), the likelihood contribution is S(u; theta) instead of f(x; theta). For left-truncated observations (where losses below a deductible d are not observed), each contribution is divided by S(d; theta) to account for the conditional nature of the observation. MLE automatically handles these complications through proper likelihood construction.

Method of Moments

The method of moments equates sample moments to population moments and solves for parameters. For a distribution with k parameters, equate the first k sample moments to their theoretical counterparts. For the lognormal distribution with parameters mu and sigma^2, E[X] = exp(mu + sigma^2/2) and E[X^2] = exp(2*mu + 2*sigma^2). Setting these equal to the sample mean and second moment gives two equations in two unknowns.

Method of moments is simpler to compute than MLE but has several disadvantages. It does not easily handle censored or truncated data, it may produce estimates outside the parameter space, and its estimates are generally less efficient (higher variance) than MLE. For Exam STAM, method of moments appears mainly as a comparison point to highlight MLE\'s advantages.

Nonparametric Estimation

Nonparametric methods make no distributional assumptions. The empirical distribution function assigns probability 1/n to each observation. For a sample x_1, x_2, ..., x_n, the empirical CDF is F_n(x) = (number of observations <= x) / n. This is a step function that jumps by 1/n at each data point. The empirical survival function is S_n(x) = 1 - F_n(x).

Kaplan-Meier Estimator

The Kaplan-Meier (product-limit) estimator extends the empirical survival function to handle right-censored data. At each observed loss amount (event time) y_j, the estimated conditional probability of "surviving" past y_j is (r_j - d_j)/r_j, where r_j is the number of observations at risk (still under observation and not yet censored) and d_j is the number of events (losses) at y_j. The survival function estimate is S_hat(t) = product over all y_j <= t of (r_j - d_j)/r_j.

For example, suppose 10 policies are observed. Losses occur at 100, 250, 400, and 700, while censoring (policy limits reached) occurs at 150, 300, and 500. The remaining 3 policies had no loss. At y = 100, r = 10 and d = 1, giving a survival factor of 9/10. At y = 250, after removing the observation censored at 150, r = 8 and d = 1, giving 7/8. The Kaplan-Meier estimate at 250 is (9/10) * (7/8) = 0.7875.

Nelson-Aalen Estimator

The Nelson-Aalen estimator provides an alternative estimate of the cumulative hazard function H(t). The estimated cumulative hazard is H_hat(t) = sum over all y_j <= t of d_j/r_j. The corresponding survival function estimate is S_hat(t) = exp(-H_hat(t)). For small samples, the Nelson-Aalen and Kaplan-Meier estimates differ slightly, with the Nelson-Aalen generally giving slightly higher survival estimates. For large samples, they converge.

Comparing Parametric and Nonparametric Approaches

Parametric models are preferred when you have good reason to believe losses follow a specific distribution and when extrapolation beyond the observed data range is needed (for example, estimating the probability of a loss exceeding the largest observed value). They provide smooth, continuous estimates and are parsimonious, requiring only a few parameters to describe the entire distribution.

Nonparametric methods are preferred when the distributional form is uncertain, when you want to visualize the data without imposing assumptions, or as a diagnostic check on parametric fits. They are flexible and robust but produce jagged step functions that may be noisy for small samples. They cannot estimate probabilities beyond the range of observed data.

Common Pitfalls

A common mistake with the Kaplan-Meier estimator is miscounting the risk set at each event time. Censored observations are removed from the risk set only after their censoring time, not before. If a censoring time equals an event time, conventions vary, but Exam STAM typically specifies the ordering. Another pitfall is attempting to extrapolate a Kaplan-Meier estimate beyond the largest observed value; the estimate is undefined there unless the largest observation is uncensored.

With parametric estimation, students sometimes forget to adjust the likelihood for data modifications. If losses are reported with a deductible of 500, you are observing losses conditional on exceeding 500, and the likelihood must reflect this truncation. Ignoring it biases parameter estimates.

Study Tips

Exam STAM tests your ability to compute both types of estimates and to recognize when each approach is appropriate. Practice constructing Kaplan-Meier tables from scratch with mixed censored and uncensored observations. For parametric estimation, practice writing likelihood functions for data with various combinations of censoring and truncation. Be prepared to compare the two approaches and explain the advantages and disadvantages of each in a given scenario.

Parametric vs. Nonparametric Estimation for Exam STAM