On Lower Bounds for Nonstandard 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.

Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model

“…The proposed CRB for mixed parameter vectors (7) has been derived in the context of "standard" deterministic estimation problems for which a closed-form expression of p ( y ; θ) is available. In the context of "non standard" deterministic estimation problems (see [40] and references therein), p ( y ; θ) results from the marginalization of an hybrid p.d.f. depending on both random θ r ∈ R P r and deterministic ( θ) parameters, i.e., p y ; θ = R P r p y , θ r | θ dθ r , which is mathematically intractable and prevents from using the proposed standard CRB (7) .…”

On Cramér-Rao Lower Bounds with Random Equality Constraints

“…In standard deterministic estimation problems [2], the MSE matrix of θ is a Gram matrix (general form of the square of a norm) [17] defined on the vector space of square integrable functions and, therefore, all known standard LBs on the MSE can be formulated as the solution of a norm minimization problem under linear constraints (LCs) [9], [10]. This formulation of LBs does not only provides a straightforward understanding of the hypotheses associated with the different LBs [9], [10], but it also allows to obtain a unique formulation of each LB in terms of a unique set of linear constraints.…”

“…of x given θr, and p (θr; θ) is the prior p.d.f., parameterized by θ. If only an integral form of p (x; θ) (5b) is available, the estimation problem at hand is so-called a "non-standard" estimation problem [2]. In this setting,…”

“…which is the definition of a locally strict-sense unbiased estimator [24]. This is sensible, since, as shown in [2,§IV], Non-Standard CRB are LBs on the "non-standard" MLEs (NSMLEs) defined as ( θr, θ) = arg max…”

“…As introduced in [1, p53], a model of the general deterministic estimation problem has the following four components: 1) a parameter space Θ d ⊂ R P , 2) an observation space X ⊂ R M , 3) a probabilistic mapping from parameter vector space Θ d to observation space X , that is the probability law p (x; θ) that governs the effect of a parameter vector value θ ∈ Θ d on the observation x ∈ X and, 4) an estimation rule, that is the mapping of the observation space X into vector parameter estimates θ θ (x). If a closed-form expression of p (x; θ) is available, the estimation problem at hand is so-called a "standard" deterministic estimation problem [2]. In this setting, minimal performance bounds on the mean square error (MSE) matrix of θ allow for calculation of the best performance that can be achieved, when estimating parameters of a signal corrupted by noise.…”