Computationally Efficient Nonparametric Importance Sampling

The variance reduction established by importance sampling strongly depends on the choice of the importance sampling distribution. A good choice is often hard to achieve especially for high-dimensional integration problems. Nonparametric estimation of the optimal importance sampling distribution (known as nonparametric importance sampling) is a reasonable alternative to parametric approaches.In this article nonparametric variants of both the self-normalized and the unnormalized importance sampling estimator are proposed and investigated. A common critique on nonparametric importance sampling is the increased computational burden compared to parametric methods. We solve this problem to a large degree by utilizing the linear blend frequency polygon estimator instead of a kernel estimator. Mean square error convergence properties are investigated leading to recommendations for the efficient application of nonparametric importance sampling. Particularly, we show that nonparametric importance sampling asymptotically attains optimal importance sampling variance. The efficiency of nonparametric importance sampling algorithms heavily relies on the computational efficiency of the employed nonparametric estimator. The linear blend frequency polygon outperforms kernel estimators in terms of certain criteria such as efficient sampling and evaluation. Furthermore, it is compatible with the inversion method for sample generation. This allows to combine our algorithms with other variance reduction techniques such as stratified sampling. Empirical evidence for the usefulness of the suggested algorithms is obtained by means of three benchmark integration problems. As an application we estimate the distribution of the queue length of a spam filter queueing system based on real data.Comment: 29 pages, 7 figure

Particle Filter-Based On-Line Estimation of Spot Volatility with Nonlinear Market Microstructure Noise Models

Summary. A new technique for the on-line estimation of spot volatility for high-frequency data is developed. The algorithm works directly on the transaction data and updates the volatility estimate immediately after the occurrence of a new transaction. We make a clear distinction between volatility per time unit and volatility per transaction and provide estimators for both. A new nonlinear market microstructure noise model is proposed that reproduces the major stylized facts of high-frequency data. A computationally efficient particle filter is used that allows for the approximation of the unknown efficient prices and, in combination with a recursive EM algorithm, for the estimation of the volatility curves. In addition, the estimators are improved by an on-line bias correction. We neither assume that the transaction times are equidistant nor do we use interpolated prices