Small-sample unbiased linear coherence estimators for a complex Gaussian random process

Area of Science:

• Signal Processing
• Statistical Inference
• Complex Gaussian Processes

Background:

• The Pearson correlation coefficient is vital for remote sensing and forecasting but suffers from bias with finite samples.
• Maximum likelihood estimators (MLEs) for jointly Gaussian processes have degraded performance in practical, limited-data scenarios.

Purpose of the Study:

• To analytically derive small-sample unbiased estimators for the correlation coefficient and its squared modulus.
• To address the performance degradation of MLEs due to finite sample sizes in complex Gaussian random processes.

Main Methods:

• Analytical derivation of unbiased estimators for complex Gaussian random processes.
• Characterization of estimators as bijective functions of corresponding MLEs.
• Analysis of estimator performance, including bias and mean square error.

Main Results:

• Developed unique, minimum-variance, small-sample unbiased estimators for correlation coefficient and its squared modulus.
• The squared modulus estimator corrects positive bias, extending its range to negative values.
• An unbiased estimator for the absolute modulus of the complex correlation coefficient does not exist.

Conclusions:

• The derived estimators offer improved statistical properties over traditional MLEs for finite samples.
• Proposed an estimator that enhances both bias and mean square error for the squared sample correlation coefficient.
• Highlighted the impact of bias in ocean acoustic coherence measurements on array design.