Statistics of the MLE and Approximate Upper and Lower Bounds—Part I: Application to TOA Estimation

Abstract

Abstract-In nonlinear deterministic parameter estimation, the maximum likelihood estimator (MLE) is unable to attain the Cramér-Rao lower bound at low and medium signal-to-noise ratios (SNRs) due the threshold and ambiguity phenomena. In order to evaluate the achieved mean-squared error (MSE) at those SNR levels, we propose new MSE approximations (MSEA) and an approximate upper bound by using the method of interval estimation (MIE). The mean and the distribution of the MLE are approximated as well. The MIE consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval. Also, we derive an approximate lower bound (ALB) based on the Taylor series expansion of noise and an ALB family by employing the binary detection principle. The accuracy of the proposed MSEAs and the tightness of the derived approximate bounds are validated by considering the example of time-of-arrival estimation.Index Terms-Nonlinear estimation, threshold and ambiguity phenomena, maximum likelihood estimator, mean-squared error, upper and lowers bounds, time-of-arrival.

Jamming of Wireless Localization Systems

“…As discussed in [35], for sufficiently large signal-to-noise ratios (SNRs) and/or effective bandwidths, the maximum likelihood (ML) location estimator becomes approximately unbiased and efficient, i.e., it achieves a mean-squared error (MSE) that is close to the CRLB. For other scenarios, the CRLB may not be a very tight bound for MSEs of ML estimators [36], [37]. Therefore, when the power allocation strategy for the jammer nodes is optimized according to a CRLB based objective function, the CRLBs corresponding to the optimized value of the specific objective function can be considered to provide performance bounds for the MSEs of the target nodes.…”

“…After obtaining the optimal locations of the jammer nodes, the optimization problem in (36) reduces to a problem which is in the same form as that in (17). Hence, the result in Proposition 2 can be employed to obtain the optimal power allocation strategy in (38) (cf.…”

“…, N J }. In this case, a jammer node or multiple jammer nodes located at z * and transmitting a (total) power of P T yields the solution of (36).…”

Statistics of the MLE and Approximate Upper and Lower Bounds–Part II: Threshold Computation and Optimal Pulse Design for TOA Estimation

“…the unknown parameter is oscillating, five regions can be identified (as can be seen in Fig. 1(b) in Part I [1]): 1) the a priori region, 2) the a priori-ambiguity transition region, 3) the ambiguity region where the envelope CRLB (ECRLB) is achieved [1], 4) the ambiguity-asymptotic transition region, and 5) the asymptotic region.…”

“…In Part I of this work [1], an approximate upper bound and various MSEAs for the MLE are proposed by using the MIE [12], [15], [16], [18]- [26]. Some approximate lower bounds (ALB) are proposed as well by employing the binary detection principle first used by Ziv and Zakai [2].…”