The profound impact of negative power law noise on statistical estimation

This paper investigates the profound impact of negative power law (neg-p) noise-that is, noise with a power spectral density L
_p
(f) ¿ | f |
^p
for p < 0 - on the ability of practical implementations of statistical estimation or fitting techniques, such as a least squares fit (LSQF) or a Kalman filter, to generate valid results. It demonstrates that such neg-p noise behaves more like systematic error than conventional noise, because neg-p noise is highly correlated, non-stationary, non-mean ergodic, and has an infinite correlation time ¿
_c
. It is further demonstrated that stationary but correlated noise will also cause invalid estimation behavior when the condition T ¿ ¿
_c
is not met, where T is the data collection interval for estimation. Thus, it is shown that neg-p noise, with its infinite ¿
_c
, can generate anomalous estimation results for all values of T, except in certain circumstances. A covariant theory is developed explaining much of this anomalous estimation behavior. However, simulations of the estimation behavior of neg-p noise demonstrate that the subject cannot be fully understood in terms of covariant theory or mean ergodicity. It is finally conjectured that one must investigate the variance ergodicity properties of neg-p noise through the use of 4th order correlation theory to fully explain such simulated behavior.