Simulated Minimum Cram&amp;#233;r-Von Mises Distance Estimation for Some Actuarial and Financial Models

Andrew Luong,Christopher Blier-Wong

doi:10.4236/ojs.2017.75058

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.

Highlights

IntroductionIn actuarial science or finance we often model losses or log-returns with distri-. A
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
As the method is based on sample distributions instead of density estimates it is easier to implement than simulated Hellinger distance (SMHD) method but it might not be as efficient as SMHD methods for continuous models

Summary

Introduction

In actuarial science or finance we often model losses or log-returns with distri-. A. Like Simulated minimum Hellinger (SMHD) method proposed by Luong and Bilodeau [12], the new method is robust and it is even easier to implement than SMHD method as it makes use of sample distribution functions instead of density estimates It can handle models like the compound Poisson model which displays a probability mass at the origin where SMHD method might not be suitable but comparing to SMHD estimators, the SCVM estimators might not be as efficient as the SMHD estimators for continuous models. For both models, it appears that the SCVM estimators are much more efficient than MM estimators using the overall relative efficiency criterion

The Space l2 and Its Norm

Asymptotic Normality

An Estimate for the Covariance Matrix for SCVM Estimators

MM Estimation for the KWT Model

Conclusion