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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.