Title: On a projection estimator for Monte Carlo integration
Authors: Huei-Wen Teng - National Chiao Tung University (Taiwan) [presenting]
Abstract: For high-dimensional integration, variance reduction techniques help to improve the efficiency of the standard Monte Carlo simulation, but the reduced variance is difficult to derive theoretically. A novel theoretical perspective is proposed to explain the variance reduction technique by connecting the Monte Carlo estimator with the idea of projection in linear algebra analysis. The mean of a function for a random vector can be seen as a projection of the function on the constant function and the variance is the distance between the above two functions. This framework allows us to propose a new category of estimators for variance reduction with the idea of symmetric group, called the projection estimator. To account for both the amount of variance reduction and computation time, we define the efficiency ratio between two estimators. It is interesting that the commonly known antithetic variates and spherical estimators can be regarded as special cases of projection estimators, and its efficiency ratios can be analysed theoretically. For a projection estimator, the efficiency ratio for a polynomial target function of finite order can be derived exactly. As a result, the proposed framework can be extended to approximate the efficiency ratio of the projection estimator for a general target function. Examples are given both theoretically and numerically.