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It is natural in many contexts to employ conditioning arguments in order to deduce properties of a multivariate distribution Pn from properties of a lower-order distribution Pn-1. In this paper we show that the pointwise (fractal) dimension of Pn can be updated from the pointwise dimension of Pn-1 via the one-step conditional distributions provided the latter satisfy certain Lipschitz-type properties. Specifically, we prove that pointwise dimension can be computed iteratively according to the conditional additivity rule This approach is then used to analyse the behaviour of pointwise dimension for various stationary stochastic processes; the emphasis is on dynamical systems corrupted by noise. In particular, we show that for functionals of stochastic processes with discrete conditional distributions satisfying the necessary conditions (such as missing-data models of dynamical systems and randomly iterated function systems) pointwise dimension remains bounded over time just as in the strictly deterministic case. On the other hand, we prove that for stochastic dynamical systems with additive diffuse noise, pointwise dimension diverges to infinity over time. An example of a stationary dynamical system where the conditional additivity rule fails is also provided.
A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann-Stieltjes integral equations driven by certain stochastic processes are solved. Boundedness of the p-variation for some 0<p<2 is the only condition on the driving stochastic process. Typical examples of such processes are infinite-variance stable Lévy motion, hyperbolic Lévy motion, normal inverse Gaussian processes, and fractional Brownian motion. The approach used in the paper is based on a chain rule for the composition of a smooth function and a function of bounded p-variation with 0<p<2.
Let be Azéma martingale and its filtration, and let be the local times of the Azéma martingale defined by the following Tanaka formula: Then, for every stopping time T and every p>0, there exist two universal constants cp, Cp >0 depending only on p, such that where .
Using modern theory for semi-parametric models, we provide details for an argument of Robins et al. showing efficiency of the standard logistic regression estimator applied to data from case-control studies. Our elaboration of this argument, and of a related one by Bickel et al., includes a constructive new proof of the result.
The EM algorithm is a much used tool for maximum likelihood estimation in missing or incomplete data problems. However, calculating the conditional expectation required in the E-step of the algorithm may be infeasible, especially when this expectation is a large sum or a high-dimensional integral. Instead the expectation can be estimated by simulation. This is the common idea in the stochastic EM algorithm and the Monte Carlo EM algorithm.
We consider quantile estimation under a two-sample semi-parametric model in which the log ratio of two unknown density functions has a known parametric form. This two-sample semi-parametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new quantile estimator is constructed on the basis of the maximum semi-parametric likelihood estimator of the underlying distribution function. It is shown that the proposed quantile estimator is asymptotically normally distributed with smaller asymptotic variance than that of the standard quantile estimator. Also presented are some results on simulation and on analysis of a real data set.
The paper studies the change-point problem and the cross-covariance function for ARCH models. Bounds for the cross-covariance function are derived and explicit formulae are obtained in special cases. Consistency of a cusum type change-point estimator is proved and its rate of convergence is established. A Hájek-Rényi type inequality is also proved. Results are obtained under weak moment assumptions.
Serial ranks have long been used as the basis for nonparametric tests of independence in time series analysis. We shall study the underlying graph structure of serial ranks. This will lead us to a basic martingale which will allow us to construct a weighted approximation to a serial rank process. To show the applicability of this approximation, we will use it to prove two very general central limit theorems for Wald-Wolfowitz-type serial rank statistics.