In this communication, we describe some interrelations between generalized q-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of entropy and Fisher information. More precisely, a generalized Fisher information ...

Associated with each body K in Euclidean n-space Rn is an ellipsoid 02K called the Legendre ellipsoid of K . It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂...

Suppose X is a random variable with pdf fX(x; θ), θ being an unknown parameter. Let X1, . . . , Xn be a random sample and θ̂ = θ̂(X1, . . . , Xn). We’ve seen that E(θ̂), or rather E(θ̂) − θ, is a measure of how biased θ̂ is. We’ve also seen that V ar(θ̂) provides a measure of efficiency, i.e., the smaller the variance of θ̂, the more likely E(θ̂) will provide an accurate estimate of θ. Given a specific...

Variance and Fisher information are ingredients of the Cramér-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts....

In this paper we review different meanings of the word intrinsic in statistical estimation, focusing our attention on the use of this word in the analysis of the properties of an estimator. We review the intrinsic versions of the bias and the mean square error and results analogous to the Cramér-Rao inequality and Rao-Blackwell theorem. Different results related to the Bernoulli and normal dist...

We investigate a one-parameter family of probability densities (related to the Pareto distribution, which describes many natural phenomena) where the Cramér-Rao inequality provides no information. 1. Cramér-Rao Inequality One of the most important problems in statistics is estimating a population parameter from a finite sample. As there are often many different estimators, it is desirable to be...

summary While all nonsequential unbiased estimators of the normal mean have variances which must obey the Cramer-Rao inequality, it is shown that some sequential unbiased estimators do not.