What are higher-order statistics and when are they useful?

Higher-order statistics (HOS) extend beyond mean and variance to 3rd order (skewness/bispectrum), 4th order (kurtosis/trispectrum), and beyond. Properties: Preserve phase information (unlike power spectrum), Suppress Gaussian noise (HOS of Gaussian is zero), and Detect nonlinear coupling (quadratic phase coupling in bispectrum). Applications: Non-Gaussianity detection, nonlinear system identification, phase retrieval, texture analysis, and ICA source separation. Estimation challenges: Large variance requiring substantial data, computational complexity. Used when Gaussian/linear assumptions fail and nonlinear/non-Gaussian analysis is needed.