The present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as point$x$in an abstract space;$phases$are points ($x, t$) in the product state×time space; sets of states are denoted by$X$, sets of phases by$S$.

No abstract uploaded!

No abstract uploaded!

In this paper we introduce an appealing nonparametric method for estimating the mean regression function. The proposed method combines the ideas of local linear smoothers and variable bandwidth. Hence, it also inherits the advantages of both approaches. We give expressions for the conditional MSE and MISE of the estimator. Minimization of the MISE leads to an explicit formula for an optimal choice of the variable bandwidth. Moreover, the merits of considering a variable bandwidth are discussed. In addition, we show that the estimator does not have boundary effects, and hence does not require modifications at the boundary. The performance of a corresponding plug-in estimator is investigated. Simulations illustrate the proposed estimation method.

We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam’s inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy- tailed.