Maximum Likelihood Estimation for Loss Distributions
Apply MLE to fit loss distributions with complete and incomplete data for Exam STAM.
The Likelihood Function
For a random sample x_1, ..., x_n from a distribution with PDF f(x|theta), the likelihood function is L(theta) = product of f(x_i|theta) for i = 1 to n. The log-likelihood is l(theta) = sum of ln(f(x_i|theta)). The maximum likelihood estimator (MLE) is the value of theta that maximizes l(theta), found by solving the score equation dl/d(theta) = 0 and verifying that the second derivative is negative.
The MLE has several desirable large-sample properties: consistency (converges to the true parameter value), asymptotic normality (the distribution of the MLE approaches a normal distribution), and asymptotic efficiency (achieves the lowest possible variance among consistent estimators). These properties make MLE the standard estimation method for loss distributions on Exam STAM.
For the exponential with mean theta: l(theta) = -n*ln(theta) - sum(x_i)/theta. Setting dl/d(theta) = -n/theta + sum(x_i)/theta^2 = 0 gives theta_hat = x_bar, the sample mean.
MLE for Common Loss Distributions
For the Pareto(alpha, theta) with theta known: l(alpha) = n*ln(alpha) + n*alpha*ln(theta) - (alpha+1)*sum(ln(x_i + theta)). Setting dl/d(alpha) = 0 gives alpha_hat = n / sum(ln((x_i + theta)/theta)). When both parameters are unknown, the system of score equations must be solved numerically.
For the lognormal(mu, sigma^2): since ln(X_i) ~ Normal(mu, sigma^2), the MLEs are mu_hat = (1/n)*sum(ln(x_i)) and sigma^2_hat = (1/n)*sum((ln(x_i) - mu_hat)^2). Note that the MLE of sigma^2 divides by n, not n-1, so it is slightly biased downward.
For the Weibull(tau, theta): the score equations for tau and theta must be solved simultaneously using numerical methods. On Exam STAM, Weibull MLEs are typically tested with one parameter known, reducing the problem to a single equation in one unknown.
Right-Censored Data
Right censoring occurs when the exact loss value is unknown; we only know it exceeds some censoring point c. In insurance, this arises from policy limits: if a loss exceeds the limit, the insurer records the payment at the limit, but the actual ground-up loss is larger. For a right-censored observation at c, the contribution to the likelihood is S(c|theta) = 1 - F(c|theta) rather than f(x|theta).
The log-likelihood for mixed exact and censored data is: l(theta) = sum over exact observations of ln(f(x_i|theta)) + sum over censored observations of ln(S(c_j|theta)). Each observation contributes either the PDF (if exact) or the survival function (if censored).
Example: Five losses from an exponential distribution. Three are observed exactly: 100, 200, 300. Two are censored at 400 and 500 respectively. l(theta) = 3*(-ln(theta)) - (100+200+300)/theta + (-400/theta) + (-500/theta) = -3*ln(theta) - 1500/theta. Setting dl/d(theta) = 0 gives theta_hat = 1500/3 = 500. The MLE uses the total of all values (including censored values at their censoring points) divided by the number of uncensored observations.
Left-Truncated Data
Left truncation occurs when losses below a threshold d are not observed at all. This arises from deductibles: only losses exceeding d enter the dataset. For a left-truncated observation with truncation point d and observed value x > d, the contribution to the likelihood is f(x|theta) / S(d|theta). The denominator adjusts for the conditional nature of the observation: X was observed only because X > d.
With both truncation at d and censoring at c (deductible d and policy limit c), the likelihood contribution for each observation depends on its type. For an exactly observed value x (d < x < c): f(x|theta) / S(d|theta). For a censored observation at c (only known that X > c): S(c|theta) / S(d|theta).
Example: Exponential data with deductible d = 100. Observed excess amounts: 50, 150, 250. l(theta) = sum of ln(f(x_i|theta)/S(100|theta)) = sum of ln((1/theta)*e^(-x_i/theta) / e^(-100/theta)) = -3*ln(theta) - (50+150+250)/theta = -3*ln(theta) - 450/theta. The MLE is theta_hat = 150. Notice that the effective data are the excess amounts (x_i - d).
Fisher Information and Confidence Intervals
The Fisher information for a single observation is I(theta) = -E[d^2 l/d(theta)^2] = E[(dl/d(theta))^2]. For n independent observations, the total Fisher information is n*I(theta). The asymptotic variance of the MLE is Var(theta_hat) approximately equals 1/(n*I(theta)).
For the exponential: I(theta) = 1/theta^2, so Var(theta_hat) is approximately theta^2/n. For the Pareto with theta known: I(alpha) = 1/alpha^2, so Var(alpha_hat) is approximately alpha^2/n. Approximate confidence intervals use theta_hat +/- z_{alpha/2} * sqrt(1/(n*I(theta_hat))).
On Exam STAM, you may be asked to compute the observed Fisher information (using the actual data rather than the expected value) or construct confidence intervals for parameters of fitted loss distributions. The observed information is -d^2 l/d(theta)^2 evaluated at theta_hat, while the expected information evaluates the expectation of this quantity.
Likelihood Ratio Tests
The likelihood ratio test compares a restricted model (parameter space Theta_0) with a full model (parameter space Theta). The test statistic is T = 2*(l(theta_hat) - l(theta_hat_0)), which follows a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters.
This test is used on Exam STAM to compare distributional assumptions: for example, testing whether the Pareto provides a significantly better fit than the exponential. Since the exponential is a limiting case of the Pareto (as alpha approaches infinity with theta/alpha held constant), the test has one degree of freedom. If T exceeds the chi-squared critical value, the Pareto provides a statistically significant improvement.
Study Tips and Common Pitfalls
The most common mistake is constructing the likelihood incorrectly for censored or truncated data. For each observation, determine: is the exact value known (contribute f(x))? Is it censored (contribute S(c))? Is there truncation (divide by S(d))? Getting the likelihood contributions right is the critical step; the calculus that follows is straightforward.
Another frequent error is forgetting to divide by S(d) for truncated data. Without this adjustment, the MLE is biased because it ignores the unobserved losses below the deductible. Practice at least 6 problems involving different combinations of complete, censored, and truncated data with exponential and Pareto distributions.