Non-Gaussian filter and smoother based on the Pearson distribution system

The Pearson distribution system can represent wide class of distributions with various skewness and kurtosis. We develop a practical approach of using all types of its distribution system including the type-IV distribution which was difficult to implement. We propose an easily implemented algorithm which uses less-memory and performs at a higher speed than other typical methods: using analytic approximation of successive conditional probability density functions for prediction and filtering by the Pearson distribution system in the case of both the system and observation noise being one-dimensional. By using the approximated probability density function and the numerical integration, we obtain mean, variance, skewness and kurtosis of the next distribution. We decide the next approximated distribution from the Pearson distribution system. We adopt these steps for the prediction, filtering and smoothing recursively. Our framework makes it possible to construct time series models with various noise distributions. We apply our non-Gaussian filter to the estimation of non-Gaussian stochastic volatility models of the stock returns. We compare our method with the typical method.