Total Performance Evaluation of Intensity Estimation after Detection

Abstract 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 Cramer-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.