ROBUST ESTIMATION FOR NOISY DATA

Abstract

When a data set is corrupted by noise, the model for the data generating process is misspecified and can cause parameter estimation problems. In the case of a Gaussian autoregressive (AR) process corrupted by noise, the data is more accurately modeled as an autoregressive moving average (ARMA) process rather than an AR process. This misspecification leads to bias, and hence, low resolution in AR spectral estimation. However, a new parametric estimator, the realizable information theoretic-estimator (RITE) based on a non-homogeneous Poisson spectral representation, is shown by simulation to be more robust to noise than the asymptotic maximum likelihood estimator (MLE). We therefore conducted an in-depth investigation and analyzed the statistics of RITE and the asymptotic MLE for the misspecified model. For large data records, RITE and the asymptotic MLE are both asymptotically normally distributed. The asymptotic MLE has a slightly lower variance, but RITE exhibits much less bias. Simulation examples of a noise corrupted AR process are provided to support the theoretical properties. This advantage of RITE increases as the SNR decreases. Another topic of interest is Data fusion for estimation. It is a problem that utilizes information from multiple data sets to estimate an unknown parameter or vector. These data sets are usually from multiple sources, for example, multiple sensors. In this case, this problem is called distributed estimation. It uses data from multiple sensors and a fusion center (FC), or central processor, to achieve a more accurate estimation than using a single sensor observation. In this paper, we propose two estimators for data fusion estimation problem. The Fisher information and the observed Fisher information are used to reduce the negative effects of poor estimations and therefore improve the new estimators' performance in terms of mean square error. At the same time, we found that there is a relationship between our new estimators and the second order Taylor expansion of l, which is the log likelihood function of data from all sensors. The solution of the maximum of the second order Taylor expansion of l, turns out to be our new estimator that uses the observed Fisher information. Our simulation results showed that the proposed estimators have obvious advantages in both low and intermediate SNR regions, especially when one or many sensors have much lower SNRs than the others.