Abstract
In this paper, we investigate the noise benefits to maximum likelihood type estimators (M-estimator) for the robust estimation of a location parameter. Two distinct noise benefits are shown to be accessible under these conditions. With symmetric heavy-tailed noise distributions, the asymptotic efficiency of the estimation can be enhanced by injecting extra noise into the M-estimators. With an asymmetric contaminated noise model having a convex cumulative distribution function, we demonstrate that addition of noise can reduce the maximum bias of the median estimator. These findings extend the analysis of stochastic resonance effects for noise-enhanced signal and information processing.
Highlights
A N OPTIMAL noise level, obtained by appropriately adding extra noise to a given signal processor or by tuning the existing noise level, can sometimes improve information processing [1]–[6]
We study the enhancement of the asymptotic efficiency and the reduction of the maximum bias by adding noise in robust location M-estimators
For symmetric heavy-tailed noise models, we show that the asymptotic efficiencies of two commonly used M-estimators with the Huber function and the bisquare function are non-monotonic functions of the ratio of the noise scale and the estimator parameter
Summary
A N OPTIMAL noise level, obtained by appropriately adding extra noise to a given signal processor or by tuning the existing noise level, can sometimes improve information processing [1]–[6]. When the optimal noise does not exist in a single M-estimator, it is observed that the asymptotic efficiency can still be enhanced to be very close to the upper bound of unity by an array of M-estimators with added noise selected from a parametric class of noise. By the Cauchy-Schwarz inequality, the upper bound of the asymptotic efficiency of an infinite number of M-estimators is proven to be unity, and the corresponding optimal noise density is the deconvolution of the maximum likelihood estimator and the given M-estimator function. This optimal noise density is frequently unattainable, due to absence of a solution to the deconvolution and the fact that the infinite-size array of M-estimators can only be approached in practice. Theoretical and numerical results show that the maximum bias can be distinctly diminished at an optimal dichotomous noise level
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