We use an example to motivate the definition of a mixture distribution.
Suppose that the loss arising from an insured randomly selected from a large group of insureds follow an exponential distribution with probability density function (pdf) , , where is a parameter that is a positive constant. The mean claim cost for this randomly selected insured is . So the parameter reflects the risk characteristics of the insured. Since the population of insureds is large, there is uncertainty in the parameter . It is more appropriate to regard as a random variable in order to capture the wide range of risk characteristics across the individuals in the population. As a result, the pdf indicated above is not an unconditional pdf, but, rather, a conditional pdf of . The below pdf is conditional on a realized value of the random variable .
What about the marginal (unconditional) pdf of ? Let’s assume that the pdf of is given by . Then the unconditional pdf of is the weighted average of the conditional pdf.
Several other distributional quantities are also weighted averages, which include the unconditional mean, and the second moment.
As a result, the unconditional variance is . Note that the unconditional variance is not the weighted average of the conditional variance. The weighted average of the conditional variance only produces .
It turns out that the unconditional variance has two components, the expected value of the conditional variances and the variance of the conditional means. In this example, the former is and the latter is . The additional variance in the amount of is a reflection that there is uncertainty in the parameter .
The Definition of Mixture
The unconditional pdf derived in Example 1 is that of a Pareto distribution. Thus the Pareto distribution is a continuous mixture of exponential distributions with Gamma mixing weights.
Mathematically speaking, a mixture arises when a probability density function depends on a parameter that is uncertain and is itself a random variable with density . Then taking the weighted average of with as weight produces the mixture distribution.
A continuous random variable is said to be a mixture if its probability density function is a weighted average of a family of probability density functions . The random variable is said to be the mixing random variable and its pdf is said to be the mixing weight. An equivalent definition of mixture is that the distribution function is a weighted average of a family of distribution functions indexed by a mixing variable. Thus is a mixture if one of the following holds.
Similarly, a discrete random variable is a mixture if its probability function (or distribution function) is a weighted sum of a family of probability functions (or distribution functions). Thus is a mixture if one of the following holds.
See this blog post for practice problems on mixture distributions.
- Klugman S.A., Panjer H. H., Wilmot G. E. Loss Models, From Data to Decisions, Second Edition., Wiley-Interscience, a John Wiley & Sons, Inc., New York, 2004