Mixture and Spliced Distributions for Insurance

Mixture Distributions

A k-point mixture combines k distributions with weights w_1, ..., w_k (summing to 1). The PDF is f(x) = w_1*f_1(x) + w_2*f_2(x) + ... + w_k*f_k(x). Mixtures model heterogeneous populations where different subgroups follow different distributions. The mean of a mixture is the weighted mean of component means. The variance includes both within-component and between-component variation. A common example is mixing two exponentials to model a population with both small frequent losses and large rare losses.

Spliced Distributions

Spliced distributions use different models for different ranges of the loss variable. For example, an exponential below threshold c and a Pareto above c, with weights ensuring the total integrates to 1. The PDF is f(x) = alpha*f_1(x) for x < c and (1-alpha)*f_2(x) for x >= c, where each component is a proper density on its domain. Spliced models allow flexible tail behavior independent of the body. Exam STAM tests your ability to compute moments, probabilities, and limited expected values from mixture and spliced distributions.