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The high-density asymptotic behaviour of a two-level branching system in Rd is studied. In the finite-variance case, a fluctuation limit process is obtained which is characterized as a generalized Ornstein-Uhlenbeck process. In the case of critical branching at the two levels the long-time behaviour of the fluctuation limit process is shown to have critical dimension 2α, where α is the index of the symmetric stable process representing the underlying particle motion. The same critical dimension has been obtained recently for the related (but qualitatively different) two-level superprocess. The fluctuation analysis uses different and simpler tools than the superprocess analysis.
This paper is concerned with the problem of approximating the density of the time at which a Brownian path first crosses a curved boundary in cases where the exact density is not known or is difficult to compute. Two methods are proposed involving the use of images, and the square root boundary provides an example for numerical comparison. Two-sided boundaries are also discussed.
An approximation is derived for tests of one-dimensional hypotheses in a general regular parametric model, possibly with nuisance parameters. The test statistic is most conveniently represented as a modified log-likelihood ratio statistic, just as the R*-statistic from Barndorff-Nielsen (1986). In fact, the statistic is identical to a version of R*, except that a certain approximation is used for the sample space derivatives required for the calculation of R*. With this approximation the relative error for large-deviation tail probabilities still tends uniformly to zero for curved exponential models. The rate may, however, be O(n-1/2) rather than O(n-1) as for R*. For general regular models asymptotic properties are less clear but still good compared to other general methods. The expression for the statistic is quite explicit, involving only likelihood quantities of a complexity comparable to an information matrix. A numerical example confirms the highly accurate tail probabilities. A sketch of the proof is given. This includes large parts which, despite technical differences, may be considered an overview of Barndorff-Nielsen's derivation of the formulae for p* and R*.
The problem of recovery of an unknown regression function f(x), x∈R1, from noisy data is considered. The function f(.) is assumed to belong to a class of functions analytic in a strip of the complex plane around the real axis. The performance of an estimator is measured either by its deviation at a fixed point, or by its maximal error in the L∞-norm over a bounded interval. It is shown that in the case of equidistant observations, with an increasing design density, asymptotically minimax estimators of the unknown regression function can be found within the class of linear estimators. Such best linear estimators are explicitly obtained.
Random censored data consist of i.i.d. pairs of observations (Xi,δi), i=1,...,n. If δi=0, Xi denotes a censored observation, and if δi=1, Xi denotes a survival time, which is the variable of interest. In this paper, we apply the martingale method for counting processes to study asymptotic properties for the kernel estimator of the density function of the survival times. We also derive an asymptotic expression for the mean integrated square error of the kernel density estimator, which can be used to obtain an asymptotically optimal bandwidth.