Tilted Least Squares Robust Estimators.

Area of Science:

• Signal Processing
• Statistical Estimation
• Robust Statistics

Background:

• Ordinary least squares (OLS) estimators struggle with contaminated data, leading to poor performance in practical identification scenarios.
• Robust estimators are crucial for reliable data analysis when outliers and heavy-tailed noise are present.

Purpose of the Study:

• Introduce the tilted least squares (TLS) robust estimator for improved data identification.
• Address the limitations of OLS in the presence of large amplitude disturbances and heavy-tailed noise.

Main Methods:

• Developed the tilted least squares (TLS) robust estimator using a weighted quadratic loss function.
• Constrained weights by Kullback-Leibler (KL) divergence and used an exponential function of negative squared residuals.
• Derived a tuning criterion for automatic KL divergence estimation.

Main Results:

• The TLS estimator effectively mitigates the influence of unexpected disturbances with large amplitudes.
• Demonstrated the equivalence of a TLS variant to the relaxed least trimmed squares (RTLS) estimator.
• Established the almost sure convergence of the RTLS estimator under heavy-tailed noises with infinite variance.

Conclusions:

• The TLS robust estimator offers a superior alternative to OLS for data identification with contaminated datasets.
• The derived tuning criterion facilitates practical implementation of the TLS estimator.
• The theoretical convergence properties of the RTLS estimator are confirmed, enhancing its applicability in challenging noise conditions.