Simulated Minimum Cramér-Von Mises Distance Estimation for Some Actuarial and Financial Models

Abstract

Minimum Cramér-Von Mises distance estimation is extended to a simulated version. The simulated version consists of replacing the model distribution function with a sample distribution constructed using a simulated sample drawn from it. The method does not require an explicit form of the model density functions and can be applied to fitting many useful infinitely divisible distributions or mixture distributions without closed form density functions often encountered in actuarial science and finance. For these models likelihood estimation is difficult to implement and simulated Minimum Cramér-Von Mises (SMCVM) distance estimation can be used. Asymptotic properties of the SCVM estimators are established. The new method appears to be more robust and efficient than methods of moments (MM) for the models being considered which have more than two parameters. The method can be used as an alternative to simulated Hellinger distance (SMHD) estimation with a special feature: it can handle models with a discontinuity point at the origin with probability mass assigned to it such as in the case of the compound Poisson distribution where SMHD method might not be suitable. As the method is based on sample distributions instead of density estimates it is also easier to implement than SMHD method but it might not be as efficient as SMHD methods for continuous models.