Bispectrum Based Tests For Gaussianity And Linearity Of Time Series

The classification of time series as Gaussian versus non-Gaussian or linear versus nonlinear is of great value in modeling and analysis of time series. There are many famous parametric and nonparametric tests of Gaussianity and linearity. A classic nonparametric test of Gaussianity and linearity based on bispectrum was initially pro-posed by Subba Rao and Gabr(1980), and then Hinich(1982) presented more robust tests. But these classic tests still suffer from severe statistical problems.In this paper, we will focus on improving the bispectrum based tests for Gaus-sianity and linearity of time series. First, we will briefly introduce the theory of the classic tests proposed by Hinich. And we will reveal the problems of the tests. After that, we will introduce improved tests using surrogate data proposed by Birkelund and Haiissen(2009). They made a lot of improvements by averaging the bispectrum over hexagonal regions, simplifying the test statistics and utilizing surrogate data to deter-mine the correct false alarm rate. Bootstrap is a favorable method to approximate unknown distribution of statistics through empirical distribution. We propose using bootstrap residuals to approximate the finite-sample null distribution combined with simplified test statistic. Finally, the superiority of the proposed tests is demonstrated through Monte Carlo simulations using synthetic data.