Biased Cramér-Rao lower bound calculations for inequality-constrained estimators

Area of Science:

• Statistics
• Estimation Theory
• Signal Processing

Background:

• Unbiased Cramér-Rao lower bound (CRB) theory provides variance bounds for unbiased estimators.
• Inequality constraints, like positivity, often lead to biased estimators.
• Calculating CRBs for biased estimators typically requires a bias gradient matrix, which is often unavailable.

Purpose of the Study:

• To develop an alternative method for deriving biased CRB expressions for estimators with inequality constraints.
• To address the limitation of needing a bias gradient matrix for biased CRB calculations.
• To provide a practical approach for scenarios where estimators must satisfy constraints such as positivity.

Main Methods:

• Proposed an alternative approach based on constructing the probability density function of the biased estimate.
• Utilized existing knowledge of estimator properties to define this probability density function.
• Applied the method to calculate biased CRBs for estimators with positivity and support constraints.

Main Results:

• Successfully derived biased CRB expressions without requiring the bias gradient matrix.
• Demonstrated the application of the method for specific measurement models with positivity constraints.
• Evaluated the benefits and limitations of the proposed approach.

Conclusions:

• The presented method offers a viable alternative for calculating biased CRBs under inequality constraints.
• This approach is particularly useful when the bias gradient matrix is unknown or difficult to obtain.
• The findings contribute to more accurate variance bound calculations in constrained estimation problems.