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Illustration of the Cramer-Rao bound: there is no unbiased estimator which is able to estimate the (2-dimensional) parameter with less variance than the Cramer-Rao bound, illustrated as standard deviation ellipse.
The Cramér–Rao bound [9] [10] states that the inverse of the Fisher information is a lower bound on the variance of any unbiased estimator of θ. Van Trees (1968) and Frieden (2004) provide the following method of deriving the Cramér–Rao bound, a result which describes use of the Fisher information.
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a ...
Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ. The Cramér–Rao lower bound is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be. An efficient estimator is also the minimum variance unbiased estimator (MVUE). This is because an efficient ...
In information geometry, the Fisher information metric [1] is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability distributions.
Hence, a lower bound on the entanglement depth is obtained as F Q [ ϱ , J z ] N ≤ k . {\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}\leq k.} A related concept is the quantum metrological gain , which for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher ...
If k exemplars are chosen (without replacement) from a discrete uniform distribution over the set {1, 2, ..., N} with unknown upper bound N, the MVUE for N is +, where m is the sample maximum. This is a scaled and shifted (so unbiased) transform of the sample maximum, which is a sufficient and complete statistic.
The accuracy can be calculated by using the Cramér–Rao bound and taking account of the above factors in its formulation. Additionally, a configuration of the sensors that minimizes a metric obtained from the Cramér–Rao bound can be chosen so as to optimize the actual position estimation of the target in a region of interest. [6]