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

Unbiased Cramér-Rao lower bound (CRB) theory can be used to calculate lower bounds to the variances of unbiased estimates of a set of parameters given only the probability density function of a random vector conditioned on the true parameter values. However, when the estimated parameter values are required to satisfy inequality constraints such as positivity, the resulting estimator is typically biased. To calculate CRBs for biased estimates of the parameter values, an expression for the bias gradient matrix must also be known. Unfortunately, this expression often does not exist. Because expressions for biased CRBs are preferable to sample variance calculations, alternative methods for deriving biased CRB expressions associated with inequality constraints are needed. We present an alternative approach that is based upon creating the probability density function associated with a given biased estimate of these parameters using the available knowledge of the estimator properties. We apply this approach to the calculation of biased CRBs for estimators that use a positivity constraint with and without a support constraint for a specific measurement model and discuss the benefits and limitations of this approach.