Total performance evaluation of intensity estimation after detection

Statistical inference problems where both the hypothesis testing and the parameter estimation are of primary interest arise in various signal processing applications. A special case is a cascaded scheme where first, a signal is detected, and then, its parameters of interest are estimated. In this work, we present a new performance evaluation measure for estimation-after-detection that incorporates the estimation and detection-related errors by considering the mean-squared-selected-error, false-alarm-error, and miss-detection-error. The proposed risk is suitable for parameters representing intensity, since its penalty of estimating a high-intensity value of a non-existing phenomenon (wrongly detected) is higher than the penalty of estimating it by a low value. Similarly, according to this risk, miss-detecting a signal of high intensity is worse than miss-detecting a signal of low intensity. We derive a new Cramér-Rao type bound on the risk that can be used for performance analysis and system design. We present the use of the risk for the Pareto-efficient design of the detection threshold. We demonstrate the results on a simple detection-estimation problem, inspired by the application of rain and humidity estimation, and on a problem of noise source detection and estimation of its intensity (standard deviation). Simulations show that the proposed bound is tight.