CRLB and ML for parametric estimate: New results

Abstract

In the present paper, we calculate the Cramér–Rao Lower Bound (CRLB) for the $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}>0$ case, accounting both for missed detections and false alarms, for an estimation problem in a surveillance system where the measurements are acquired by a radar. The analysis, which is limited to the parameter estimation for a deterministic steady-state target motion, is a further innovation in the computation of the CRLB of maximum likelihood estimators for deterministic steady-state models; in fact, the classical theory applies only in the unrealistic case of $P_{\mathrm{d}}=1$ and $P_{\mathrm{fa}}=0$. The results obtained by means of a Monte Carlo simulation successfully validate our extended enhanced estimation method.

Introduction

The Cramér–Rao Lower Bound (CRLB) plays an important role in the estimation theory because it theoretically predicts the best achievable second-order estimation error performance [1]. Traditionally, the main source of uncertainty in the parameter estimation and the derivation of the corresponding CRLB is the measurement noise. In many engineering problems (e.g. target tracking), due to the imperfections in target detection, there is an additional source of uncertainty in estimation, caused by the unknown measurement origin (target detections or false alarm occurrences). In other words, the assumption of detection probability $P_{\mathrm{d}}=1$ and false alarm probability $P_{\mathrm{fa}}=0$ is unrealistic in these applications; this consideration applies also when the Maximum Likelihood Estimator (MLE) is determined and the CRLB of the same is evaluated.

The effect of uncertain measurements has been first observed in [2], [3], with respect to a deterministic target motion: it is shown that the Fisher Information Matrix (FIM) is scaled by a constant factor less than unity, the so-called Information Reduction Factor (IRF). In mathematical terms, these results are synthesized via the equation (see also Eq. (32) of [2] and Eq. (52) of [3]):

$$\mathrm{FIM}(P_{\mathrm{d}}<1,P_{\mathrm{fa}}>0)\cong \mathrm{FIM}(P_{\mathrm{d}}=1,P_{\mathrm{fa}}=0)·\mathrm{IRF}(P_{\mathrm{d}},P_{\mathrm{fa}})\text{.}$$

The bound resulting from Eq. (1.1) is effective in modeling the measurement origin uncertainty as the product of two distinct factors: the FIM relative to the ideal case ($P_{\mathrm{d}}=1$, $P_{\mathrm{fa}}=0$) and the IRF accounting for missed detections and false alarms.

The CRLB reported in [4] is a generalization allowing for non-linear measurements and non-deterministic dynamics; also in this extended context, the measurement origin uncertainty can be expressed by means of a suitable IRF.

In [5], the theoretical CRLB for $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}=0$ (i.e., in absence of false alarms) has been calculated via the enumeration of all possible sequences of detections and missed detections, given a certain scan number. The computational complexity of this solution grows exponentially with time, but the bound has been verified to be the exact one, by means of numerical simulations, both in linear and non-linear cases. An approximation of the theoretical bound for practical applications, where the number of sensor scans is large, has also been proposed.

The comparison of the two bounds (IRF and enumeration method) has been reported in [6], showing, both theoretically and with numerical examples, that the CRLB computed via the IRF is overly optimistic. The enumeration based CRLB is the true bound for $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}=0$ case, since all possible sequences of detections and missed detections are averaged.

In the present paper, we extend the enumeration method for CRLB computation, to the parameter estimation problem with $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}>0$. A case study, which compares the MLE with this new CRLB, is presented in the context of a ballistic target tracking with a radar. The rationale of the work rests on the features of ML estimators, the most suitable for parametric problems, and on the yardstick for the second-order error performance that the CRLB analysis provides.

The paper is organized as follows:

• Section 2 presents the mathematical formulation of the enumeration method, illustrating all possible sequences of detections and missed detections for true and false targets. The theoretical CRLB for non-linear parameter estimation is calculated with $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}>0$.

• Section 3 presents a case study. It describes the application of the enumeration method, with $P_{\mathrm{d}}<1$ and $P_{\mathrm{fa}}>0$, to the estimation of the impact point of a ballistic target, e.g. an artillery shell.

• Section 4 illustrates the Monte Carlo simulation and compares its results to the CRLB.

• Conclusions are reported in Section 5.