Parameter Estimation From Quantized Observations in Multiplicative Noise Environments

Abstract

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramér-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

Asymptotically optimal one-bit quantizer design for weak-signal detection in generalized Gaussian noise and lossy binary communication channel

“…Signal estimation and detection from quantized data continues to attract attention over the past years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In [1], a general result is developed and applied to obtain specific asymptotic expressions for the performance loss under uniform data quantization in several signal detection and estimation problems including minimum mean-squared error (MMSE) estimation, non-random point estimation, and binary signal detection.…”

“…Recently, some variants of the classical signal estimation and detection model from quantized data are studied. One is that the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise [6][7][8]. Another is called the unlabeled sensing where the unknown order of the quantized measurements causes the entanglement of desired parameter and nuisance permutation matrix [9,10].…”

Exploring $Bona~Fide$ Optimal Noise for Bayesian Parameter Estimation

“…Especially, Chen et al considered all possible PDFs of added noise to optimize an arbitrary fixed or variable estimator and proved that the optimal noise, if it exists, is just a finite number of (no more than two) constant vectors by using the properties of convex hull and Caratheodory theorem [28], [29], [32], [38], [39]. Then, this kind of optimal noise PDFs inspired a series of theoretical improvability of estimation under various estimation criteria [26], [33], [37], [40]- [43], [45]- [49]. An interesting question is whether optimal bona fide noise, rather than a constant bias, exists for enhancing the estimator performance or not.…”

Parameter Estimation via Unlabeled Sensing Using Distributed Sensors

“…A variant, where the measurement matrix suffers from errors is also studied, and the corresponding maximum likelihood (ML) estimator can be found efficiently via numerical algorithms [3][4][5][6]. Recently, a parameter estimation problem where the observations are perturbed by an unknown permutation matrix has been studied extensively [7][8][9][10][11].…”