THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING Practical performance of the Maximum Likelihood Estimator for problems characterized by zero information points

In this thesis we study issues related to estimators behavior. We examine the general scalar measurement equation yn = h(θ) + vn, from which we wish to estimate the parameter θ. Specifically, we concentrate on problems where θ is a continuous parameter and the Fisher Information Measure (FIM) equal zero at isolated points θi. The well-known Cramér-Rao Lower Bound (CRLB) on the variance of any unbiased estimator is the inverse of the FIM, hence we define the points where the FIM equal zero as singular points of the CRLB. These conditions are common in many practical applications. One such known case is the problem of Direction Of Arrival (DOA) estimation using a linear array of sensors. In the DOA problem the singularity occurs when the transmitter is located at the end-fire of the array (aligned with the array direction). Our target is to derive the statistical description of the Maximum Likelihood Estimator for the general case of singular points. Maximum Likelihood Estimators (MLE) are very popular since in many cases they are relatively easy to derive, and under regularity conditions (see section 2.2.1) known to be asymptotically unbiased and efficient. These properties led to the common belief that the MLE is useless in the vicinity of singular points of the CRLB. In this thesis we show that the MLE, for this problem, is useful over the entire parameter space. Moreover, we derive the estimator Probability Distribution Function (PDF) from which we derive expressions for its bias and Mean Square Error (MSE), specifically showing that the latter is finite. To solve the apparent contradiction with the bound specified by the CRLB we emphasize a property shared by all (asymptotically) efficient estimators, then show that the MLE does not posses this property. Furthermore, we define the vicinity where the performance of the practical MLE cannot be predicted by the CRLB but is given precisely by our statistical description. In addition, we specify the conditions under which the MLE is locally unbiased at the singular point. We demonstrate our results using three cases: Phase estimation, DOA estimation and a special polynomial case. In each example we define the singular point, derive the estimator PDF and present simulation results. Specifically, we analyze the DOA problem and show that the MSE is reasonable at the array end-fire (this allows a designer to accurately define the array search range). Moreover we provide an upper bound for the MSE at the singular point. For the special polynomial case, we present close-form expression for both the bias and MSE of the estimator. These expressions are used to emphasize the points raised in this thesis.