On lower bounds for non-standard deterministic estimation

Abstract : We consider deterministic parameter estimation and the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean-squared-error (MSE). Actually the general case can be tackled by embedding the initial observation space in an hybrid one where any standard LB can be transformed into a modified one fitted to non-standard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears to include the sub-matrix of hybrid lower bounds which is a LB for the deterministic parameters. Moreover, since in non-standard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal non-standard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a non-standard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the NSMLEs, as well as a comparison with the modified LBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well known Gaussian complex observation models. Index Terms Deterministic parameter estimation, estimation error lower bound, maximum likelihood estimation.
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Nabil Kbayer, Jérôme Galy, Eric Chaumette, François Vincent, Alexandre Renaux, et al.. On lower bounds for non-standard deterministic estimation. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2017, 65 (6), pp.1538-1553. ⟨10.1109/TSP.2016.2645538⟩. ⟨hal-01525498⟩

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