Statistical signal processing algorithms often rely upon Gaussianity and time-reversibility, two important notions related to the probability structure of stationary random signals and their symmetry. Parametric models obtained via second-order statistics (SOS) are appropriate when the available data is Gaussian and time-reversible. On the other hand, evidence of nonlinearity, non-Gaussianity, or time-irreversibility favors the use of higher-order statistics (HOS). In order to validate Gaussianity and time-reversibility, and quantify the tradeoffs between SOS and HOS, consistent, time-domain chi-squared statistical tests are developed in this paper. Exact asymptotic distributions are derived to estimate the power of the tests, including a covariance expression for fourth-order sample cumulants. A modification of existing linearity tests in the presence of additive Gaussian noise is discussed briefly. The novel Gaussianity statistic is computationally attractive, leads to a constant-false-alarm-rate test and is well suited for parametric modeling because it employs the minimal HOS lags which uniquely characterize ARMA processes. Simulations include comparisons with an existing frequency-domain approach and an application to real seismic data. Time-reversibility tests are also derived and their performance is analyzed both theoretically and experimentally.
Bibliographical noteFunding Information:
Manuscript received June 8, 1993; revised May 6, 1994. This work was supported in part by NSFMIP-9210230 and ONR Grant N00014-93-1-0485. The associate editor coordinating the review of this paper and approving it for publication was Prof. Jose A. R. Fonollosa. The authors are with the Department of Electrical Engineering, University of Virginia, Charlottesville, VA 22903-2442 USA. IEEE Log Number 9406023.